Self-stabilization in certain infinite-dimensional matrix algebras
Gyula Lakos
Analytical tools to $K$-theory; namely, self-stabilization of rapidly decreasing matrices, linearization of cyclic loops, and the contractibility of the pointed stable Toeplitz algebra are discussed in terms of concrete formulas. Adaptation to the $*$-algebra and finite perturbation categories is also considered. The finite linearizability of algebraically finite cyclic loops is demonstrated.
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\title{Self-stabilization in certain infinite-dimensional matrix algebras}
\author{Gyula Lakos}
\address{Department of Geometry, E\"otv\"os University, P\'azm\'any P\'eter s.~1/C,  Budapest, H--1117, Hungary}
\email{lakos@cs.elte.hu}
%\date{June 3, 2005}
\keywords{Linearization of loops, stabilization, Toeplitz algebra, Bott periodicity, locally convex algebras}
\subjclass[2000]{Primary: 19K99, Secondary:  47B35.}
\begin{document}
\begin{abstract}
Analytical tools to $K$-theory; namely, self-stabilization of rapidly decreasing matrices,
linearization of cyclic loops, and the contractibility of
the pointed stable Toeplitz algebra are discussed in terms of concrete formulas.
Adaptation to the $*$-algebra and finite perturbation categories is also considered.
The finite linearizability of algebraically finite cyclic loops is demonstrated.
\end{abstract}
\maketitle
\section{Introduction}
When learning the basics of $K$-theory one likely encounters stabilization
of matrices, linearization of cyclic loops, and the contractibility of
the pointed stable Toeplitz algebra.
Stabilization of matrices is a fundamental feature of $K$-theory;
linearization of cyclic loops is an important method to prove complex Bott periodicity;
the Toeplitz algebra can also be used for the same purpose, but it is also an excellent tool
to construct classifying spaces.

Although considered simple, these basic constructions are often treated in quite awkward
manners in the literature. The purpose of this paper is just to show that how these
topics can be discussed in a unified and simple way.
Our statements are formulated primarily in the setting of locally convex algebras.
That is not for the sake of extreme generality but to show that concrete formulas and
maps can be very successful, without using approximations.

The main  statements of this paper can be summarized as follows:

\begin{state}[Self-stabilization] \label{stat:infstable}
Assume that
$\mathfrak A=\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S)$,
ie.~the locally convex algebra  of rapidly decreasing matrices
over an other locally convex algebra $\mathfrak S$.
Let $r:\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)\rightarrow\mathfrak A$
be an isomorphism which comes from relabeling $\mathcal K^{-\infty}_{\mathbb Z\times\mathbb Z}(\mathfrak S)$ into
$\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S)$. Then we claim:
There is a smooth homotopy
\[E:\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)\times[0,\pi/2]
\rightarrow\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)\]
such that it yields a family of endomorphisms of
$\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)$,
which are isomorphisms for $\theta\in[0,\pi/2)$, and a
closed injective endomorphism for $\theta=\pi/2$; with
\[E(A,0)=A,\qquad\qquad E(A,\pi/2)=r(A)\mathbf e_{0,0}=\left[\begin{array}{cc|ccc}
\ddots&\\&0\\\hline&&r(A)\\&&&0\\&&&&\ddots\end{array}\right].\]
This statement extends to unit groups, showing that $\unit(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A))$
can be pushed down by a homotopy into $\unit(\mathfrak A\mathbf e_{00})$.
\end{state}
(The continuous map $\phi:\mathfrak A\times[0,\pi/2]_\theta\rightarrow\mathfrak B$
is smooth in the variable $\theta$ if the the higher partial derivatives
$\partial_\theta^n\phi:\mathfrak A\times[0,\pi/2]_\theta\rightarrow\mathfrak B$ are still
continuous functions.)

Let $\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$ be the algebra of formal Laurent series with rapidly decreasing
coefficients.
\begin{state}[Linearization of cyclic loops] \label{stat:linloop}
There is a smooth homotopy
\[K:\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)\times[0,\pi/2]
\rightarrow\unit(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)),\]
such that for $a(\mathsf z)=\sum_{n\in\mathbb Z}a_n\mathsf z^n$,
$a(\mathsf z)^{-1}=\sum_{n\in\mathbb Z}b_n\mathsf z^n$,
it yields
\[K(a(\mathsf z),\pi/2)=\left[\begin{array}{ccc|ccc}
\ddots&&&&&\\
&1&&&&\\&&1&&&\\\hline
&&& a(\mathsf z)a(1)^{-1}&&\\&&&&1&\\&&&&&\ddots
\end{array}\right]\]
but
\begin{multline}\notag K(a(\mathsf z),0)=
\left[\begin{array}{ccc|ccc}
\ddots&&&&&\\
&\mathsf z^{-1}&&&&\\&&\mathsf z^{-1}&&&\\\hline
&&&1&&\\&&&&1&\\&&&&&\ddots
\end{array}\right]
\left[\begin{array}{ccc|ccc}
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\
\ddots&a_0&a_{-1}&a_{-2}&a_{-3}&\ddots\\
\ddots&a_1&a_0&a_{-1}&a_{-2}&\ddots\\\hline
\ddots&a_2&a_1&a_0&a_{-1}&\ddots\\
\ddots&a_3&a_2&a_1&a_0&\ddots\\
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots
\end{array}\right] \\
\left[\begin{array}{ccc|ccc}
\ddots&&&&&\\
&\mathsf z&&&&\\&&\mathsf z&&&\\\hline
&&&1&&\\&&&&1&\\&&&&&\ddots
\end{array}\right]
\left[\begin{array}{ccc|ccc}
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\
\ddots&b_0&b_{-1}&b_{-2}&b_{-3}&\ddots\\
\ddots&b_1&b_0&b_{-1}&b_{-2}&\ddots\\\hline
\ddots&b_2&b_1&b_0&b_{-1}&\ddots\\
\ddots&b_3&b_2&b_1&b_0&\ddots\\
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots
\end{array}\right].
\end{multline}

In a more compact notation the large quadruple matrix product can be written as
$\Lambda(\mathsf z,\mathsf Q)^{-1}\mathsf U(a)\Lambda(\mathsf z,\mathsf Q)\mathsf U(a)^{-1}$,
where $\Lambda(\mathsf z,\mathsf Q)$ is the linear loop generated by the Hilbert transform,
and $\mathsf U(a)$ is the operator of multiplication by $a(\mathsf z)$.
\end{state}

The statement about Toeplitz algebras is really simple:
\begin{state}[Toeplitz contractibility] \label{stat:toepcont}
Let $\mathfrak A=\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S)$.
Then, the unit group of the pointed  Toeplitz algebra over  $\mathfrak A$, ie.
 $\unit(\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)^{\mathrm{po}})$, is contractible.
\end{state}

These statements were formulated in the smooth category.
Often, however is useful to work in slightly different categories.
One  case is when $\mathfrak A$ is a $*$-algebra. In those cases
instead of the general unit group $\unit(\mathfrak A)$ of invertible elements
one should work with the group $\unit^*(\mathfrak A)$ of unitary elements.
Another type of restriction occurs in the finite perturbation category, when
the algebra of rapidly decreasing matrices,
$\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S)$ is replaced by
the algebra of matrices with  finitely many  nonzero entries $\mathcal K^{\mathrm f}_{\mathbb Z}(\mathfrak S)$,
and $\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$, the algebra of rapidly decreasing Laurent series is
replaced by the algebra $\mathfrak A[\mathsf z^{-1},\mathsf z]^{\mathrm f}$.
Here one should be careful, because for loops being finite and invertible does
not generally implies that the inverse is finite.



\begin{state}\label{stat:finite}
Statements \ref{stat:infstable}--\ref{stat:toepcont} restrict to the
$*$-algebra and/or finite perturbation categories.
\end{state}

The setting of finite perturbations, may, however, be too restrictive.
Let us call an element $a(\mathsf z)\in\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)$ algebraically finite if
$a=a_s\ldots a_1$ where for each $s$ one of
$a_s$ and $(a_s)^{-1}$ has  finite Laurent series form.
The algebraically finite elements of $\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)$ fall into various finiteness classes $F$
depending on the length of the elements $a_s$ or $(a_s)^{-1}$.
Let ${\mathfrak A}_F$ be the set of decompositions $\{a_j\}_{1\leq j\leq s}$ compatible to $F$.
Then Statement \ref{stat:linloop} can be augmented as follows:

\begin{state}\label{stat:subfinite}
For a finiteness class $F$ there is a smooth homotopy
\[K_F: {\mathfrak A}_F\times[0,\pi/2]\times[0,1]\rightarrow\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty),\]
such that

i.) $K_F(\tilde a,\theta,0)=K(a,\theta)$;

ii.) $K_F(\tilde a,\theta,1)$ differs form $1_{\mathbb Z}$ in finitely many places (depending on $F$);

iii.) $K_F(\tilde a,\pi/2,h)$ is constant;

iv.)
$K_F(\tilde a,0,h)=\Lambda(\mathsf z,\mathsf Q)^{-1}{\mathsf U}_{F}(\tilde a,h)\Lambda(\mathsf z,\mathsf Q)
{\mathsf U}_{F}(\tilde a,h)^{-1}.$
\newline
Here ${\mathsf U}_{F}(\tilde a,h)$ differs from $\mathsf U(a)$ in
a rapidly decreasing matrix.

In particular, $K_F(\tilde a,0,1)$ yields a finite linearization of $a(\mathsf z)a(1)^{-1}$.
\end{state}

These statements are all well-known or, at least, can expected in  various ways:
Statement \ref{stat:infstable}, as stated here in the smooth category (but see \ref{stat:finite}),
follows from Cuntz, \cite{C2}, Section 2.
Statement \ref{stat:linloop} is
a quantitative version of the well-known linearization technique of Atiyah and Bott, \cite{AB};
but much resembling to the formulas of Pressley and Segal, \cite{PS}, Ch.~6, who work with
Hilbert-Schmidt matrices, instead of rapidly decreasing ones. Statement \ref{stat:toepcont}
comes from  the original Toeplitz  argument of Cuntz, \cite{C}, originally stated in the context of
$C^*$-algebras, but subsequently adapted to the smooth case, cf. also \cite{C2}.
Statement \ref{stat:finite} is useful,
because $*$-algebras are prominent in operator algebraic discussions; and the finite
perturbation category is the technically easiest setting to provide large
contractible spaces for the purposes of algebraic topology.
Statement \ref{stat:subfinite} amounts to an explicit computation in the less functorial but more concrete
setting of \cite{AB}.
Our point, however, is that the statements above hold in the  generality indicated
and using only the most elementary and explicit methods.
Moreover, we demonstrate
is that we can prove these statements by  a skillful use of conjugation by matrices
\[\begin{bmatrix}
\pm\sqrt{1-t^2}&\pm t\sqrt{1-t^2}&\pm t^2\sqrt{1-t^2}&\pm t^3\sqrt{1-t^2}&\pm t^4\sqrt{1-t^2}&\cdots\\
-t & 1-t^2 & t(1-t^2) & t^2(1-t^2)  & t^3(1-t^2)&\cdots\\
&-t & 1-t^2 & t(1-t^2) & t^2(1-t^2) &\ddots\\
&&-t & 1-t^2 & t(1-t^2) &\ddots\\
&&&-t & 1-t^2 &\ddots\\
&&&&-t &\ddots\\
&&&&&\ddots\\
\end{bmatrix},\]
which, in turn, are nothing else but just  naive infinite products of rotations.


Sections
\ref{sec:comp}--\ref{sec:CT} form an introduction
into some basics and clarify our conventions.
mentioned. Sections \ref{sec:stabilize}--\ref{sec:practbott} deal with Statements
\ref{stat:infstable}--\ref{stat:subfinite} respectively.


The constructions presented here are more or less improved versions of some
constructions which can be found in my PhD thesis \cite{L}.
I am indebted Prof. Richard B. Melrose, my advisor.
In fact, much of this content was motivated by the geometric idea of Melrose, Rochon \cite{MR}.
I would also like to thank Prof. Joachim Cuntz, who called my attention to some papers.

%\newpage
\section{A general framework for computations }\label{sec:comp}
\paragraph{\textbf{Unit groups}}
\begin{point}
If $\mathfrak A$ is a not necessarily unital algebra then one can formally consider
elements of shape $1+a$, $(a\in\mathfrak A)$, which multiply as $(1+a)(1+b)=1+(a+b+ab)$.
In that way $1+\mathfrak A$ becomes a semigroup. If $\mathfrak A$ was a topological ring,
then there is a natural topology on $1+\mathfrak A$ induced from the topology of $\mathfrak A$.
If $\mathfrak A$ is unital, then it is customary to identify $\mathfrak A$ and $1+\mathfrak A$
by the recipe
$a\in\mathfrak A\,\leftrightarrow\, 1-(1_{\mathfrak A}-a)\in1+\mathfrak A,$
which is compatible with the multiplicative structure. This is also the situation
if there is a natural identity element lurking around $\mathfrak A$, like the
identity matrix in the case of matrix algebras.

The unit group $\unit(\mathfrak A)$ of $\mathfrak A$ is the
unit group  of the semigroup $1+\mathfrak A$. Ie. it is the group of pairs
$(1+a,1+b)\in(1+\mathfrak A)\times(1+\mathfrak A)$
such that $(1+a)(1+b)=(1+b)(1+a)=1$. The group operations are  the multiplication map
$(1+a_1,1+b_1)(1+a_2,1+b_2)=((1+a_1)(1+a_2),(1+b_2)(1+b_1)),$ the inverse
$(1+a,1+b)^{-1}=(1+b,1+a),$ and the identity  $ 1_{\unit(\mathfrak A)}=(1,1).$
If $\mathfrak A$ was a topological ring, then there is a natural topology
on $\unit(\mathfrak A)$ which comes from the product topology of
$(1+\mathfrak A)\times(1+\mathfrak A)$ by restriction.
It is easy to see that if
$(1+a,1+b)\in\unit(\mathfrak A)$ then $1+a$ determines $1+b$ (and the other way around),
so, instead of ``$(1+a,1+b)$'' we can (and generally do) use ``$1+a$'' to denote
the element of $\unit(\mathfrak A)$ is question. But this latter notation does not
properly account for the topology of the unit group.

\label{po:unithom}
The unit group construction is certainly a functor:
If $\phi:\mathfrak A\rightarrow\mathfrak B$
is a (continuous) homomorphism, then it induces a (continuous) map
$\unit\phi:\unit(\mathfrak A)\rightarrow\unit(\mathfrak B)$ defined by
$(1+a,1+b)\mapsto(1+\phi(a),1+\phi(b))$.
When we meet a homomorphism of algebras it will always be understood that
we automatically extend it to unit groups as above.
We will simply write $\phi$ instead of $\unit\phi$.
\end{point}

\paragraph{\textbf{Matrix algebras}}

\begin{point}
In what follows ``a locally convex vector space $\mathfrak A$''  means a
sequentially complete, Hausdorff, locally convex vector space $\mathfrak A$.
The completeness is essential for analytic purposes.
If the topology of $\mathfrak A$ is induced by a set $\Pi_{\mathfrak A}$ of seminorms
then we assume that any positive integral combination of these seminorms also belongs
to the generating seminorm set.
A locally convex algebra $\mathfrak A$ is a locally convex vector space
with continuous linear multiplication. So, for each seminorm  $p\in\Pi_\mathfrak A$
there is an other seminorm $\tilde p\in\Pi_\mathfrak A$ such that for all $X_1,X_2\in\mathfrak A$
the inequality $p(X_1X_2)\leq\tilde p(X_1)\tilde p(X_2)$ holds.

An inductive locally convex vector space $\mathfrak A$ is an indexed family of
locally convex vector spaces $\{\mathfrak A_\lambda\}_{\lambda\in\Lambda}$
such that the following holds:
$\Lambda$ is an upward directed partially ordered set, ie.
for $\lambda,\mu\in\Lambda$ there is an element $\nu\geq\lambda,\mu$.
For all $\mu\geq\lambda$ there exist continuous inclusions
$T_\mu^\lambda:\mathfrak A_\lambda\rightarrow\mathfrak A_\mu$; and for
$\nu\geq\mu\geq\lambda$ one has $T_\nu^\mu\circ T_\mu^\lambda=T_\nu^\lambda$.
Now, $\mathfrak A$ is an inductive locally convex algebra if for each $\lambda,\mu\in\Lambda$ there is
an element $\mathrm{prod}(\lambda,\mu)\in\Lambda$, and for
$\nu\geq\mathrm{prod}(\lambda,\mu)$  bilinear products
$M_{\lambda,\mu}^\nu:\mathfrak A_\lambda\times\mathfrak A_\mu\rightarrow\mathfrak A_{\nu}$
compatible to the inclusions and the usual algebraic prescriptions.
An element of $\mathfrak A$ is an element of $\bigcup_{\lambda\in\Lambda}\mathfrak A_\lambda$ making
identifications along the inclusion maps.
In that way $\mathfrak A$ will be an algebra.
The algebra $\mathfrak A$ has not an ordinary  but an ``inductive'' topology
coming from the filtration $\{\mathfrak A_\lambda\}_{\lambda\in\Lambda}$, such that the
vector space structure respects the filtration but the algebra structure does not.
If  the spaces $\mathfrak A$ and $\mathfrak B$ have inductive topologies with filtrations
$\{\mathfrak A_\lambda\}_{\lambda\in\Lambda}$
and $\{\mathfrak B_\mu\}_{\mu\in M}$ then a map $\phi:\mathfrak A\rightarrow\mathfrak B$
is continuous if for each $\lambda\in\Lambda$ there is an element $\mu\in M$
such that there is a continuous map $\phi_\lambda:\mathfrak A_\lambda\rightarrow
\mathfrak B_\mu$, which is set-theoretically a restriction of $\phi$.
Direct products of spaces with inductive and/or ordinary topologies can easily be defined.
In particular, a deformation $\phi:\mathfrak A\times[0,1]\rightarrow\mathfrak B$ is continuous
if and only if a similar restriction condition holds.
\end{point}
\begin{point}
If $A,B$ are sets then we denote the set of functions from $A$ to $B$ by $\mathcal F(A;B)$.
Suppose that $\Theta_1,\Theta_2$ are sets and $\mathfrak V$ is a vector space.
Then a $\mathfrak V$-valued $\Theta_1\times\Theta_2$ matrix is just a formal sum over
$\Theta_1\times\Theta_2$ with coefficients form $\mathfrak V$ (ie. just a function from
$\Theta_1\times\Theta_2$ to $\mathfrak V$):
\[s=\sum_{(a,b)\in\Theta_1\times\Theta_2}s_{a,b}\mathbf e_{a,b}\in
\mathcal M(\Theta_1,\Theta_2;\mathfrak V).\]
We write $\mathcal
M(\Theta;\mathfrak V)$ instead of $\mathcal M(\Theta,\Theta;\mathfrak V)$ and similarly
for other spaces in the future.
For column and row matrices we use the notation $\mathbf e_a=\mathbf e_{a,*}$ and
$\mathbf e_b^\top=\mathbf e_{*,b}$ respectively and we make the formal identification
$\mathbf e_{a,b}=\mathbf e_a\otimes \mathbf e_b^\top$. For the
column and row spaces we use the notation
$\mathcal S(\Theta;\mathfrak V)=\mathcal M(\Theta,\{*\};\mathfrak V)$ and
$\mathcal S(\Theta;\mathfrak V)^\top=\mathcal M(\{*\},\Theta;\mathfrak V)$, respectively.
We can multiply matrices only if it makes sense positionwise.
We can always take a transpose, however.
We use the notation
${1_\Theta=\sum_{\theta\in\Theta}\mathbf e_{\theta,\theta}}$,
and in general circumstances we consider the identity matrix $1_\Theta$ as the
adjoint unit in a non-unital $\Theta\times\Theta$ matrix algebra.
\end{point}
\begin{point}
On the set $\mathbb N$ of natural numbers there is the natural space
$\mathcal S^{-\infty}(\mathbb N;\mathbb R)^\star$, ie. the
space of multiplicatively invertible polynomially growing functions.
A countable set $\Theta$ is called a set of polynomial growth if it is endowed with
 sets of
functions $\mathcal S^{-\infty}(\Theta;\mathbb R)^\star
\subset \mathcal F(\Theta;\mathbb R)$ such that there is a bijection
$\omega:\Theta\rightarrow\mathbb N$  such that
 $\omega^*\mathcal S^{-\infty}(\mathbb N;\mathbb R)^\star=
\mathcal S^{\infty}(\Theta;\mathbb R)^\star$.
It is notable that $\mathbb N\times\mathbb N$ and $\mathbb N\,\dot\cup\,\mathbb N$ are
sets of polynomial growth naturally;
and that way we can define the direct product $\Theta_1\times\Theta_2$ and direct
sums $\Theta_1\,\dot\cup\,\Theta_2$ of sets of
polynomial growth $\Theta_1$ and $\Theta_2$.
In what follows the sets of polynomial growth we use will be like
$\mathbb N, \mathbb Z,$ or
$\{1,\ldots,n\}\times \mathbb Z$, where the description of the relevant function
spaces is evident, so it will not be detailed.
The main point is that  a set $\Theta$ of polynomial growth is just
like $\mathbb N$ for practical purposes.
\end{point}
\begin{point}
If $\Theta_1,\Theta_2$ are of sets of polynomial growth and $\mathfrak V$ is a locally convex
vector space then we can define some matrix spaces as follows:

a.)
\begin{multline}
\mathcal M^{\infty,\infty}(\Theta_1,\Theta_2;\mathfrak V)=
\biggl\{ s\in \mathcal M(\Theta_1,\Theta_2;\mathfrak V)
\,:\,\forall p\in\Pi_{\mathfrak V} \exists f\in \mathcal S^{\infty}(\Theta_1;\mathbb R)^\star
\\ \exists g\in \mathcal S^{\infty}(\Theta_2;\mathbb R)^\star
\text{ s. t. }|s|_{\frac1f,p,\frac1g}=\sum_{(a,b)\in\Theta_1\times\Theta_2}
\left|\frac1{f(a)}\right|\,p(s_{a,b})\left|\frac1{g(b)}\right|<+\infty\biggr\}.\notag
\end{multline}
With functions $F\in\mathcal F(\Pi_{\mathfrak V};  \mathcal S^{\infty}(\Theta_1;\mathbb R)^\star\times
 \mathcal S^{\infty}(\Theta_2;\mathbb R)^\star)$ and spaces
\begin{multline}
\mathcal M^{\infty,\infty}_F(\Theta_1,\Theta_2;\mathfrak V)=
\biggl\{ s\in \mathcal M(\Theta_1,\Theta_2;\mathfrak V)
\,:\,\forall p\in\Pi_{\mathfrak V}
\text{ s. t. }\\ |s|_{\frac1{F_1(p)},p,\frac1{F_2(p)}}=\sum_{(a,b)\in\Theta_1\times\Theta_2}
\left|\frac1{F_1(p)(a)}\right|\,p(s_{a,b})\left|\frac1{F_2(p)(b)}\right|<+\infty\biggr\}\notag
\end{multline}
the space $\mathcal M^{\infty,\infty}(\Theta_1,\Theta_2;\mathfrak V)$ is filtered and
becomes an inductive locally  convex space.
This space is topologically  the same as the column vector space
$\mathcal M^{\infty,\infty}(\Theta_1\times\Theta_2,\{*\};\mathfrak V)$ which is also the same as
$\mathcal M^{\infty,-\infty}(\Theta_1\times\Theta_2,\{*\};\mathfrak V)$ defined later.

b.)
\begin{multline}
\mathcal M^{\infty,-\infty}(\Theta_1,\Theta_2;\mathfrak V)=
\biggl\{ s\in \mathcal M(\Theta_1,\Theta_2;\mathfrak V)
\,:\,\forall p\in\Pi_{\mathfrak V} \forall g\in \mathcal S^{\infty}(\Theta_2;\mathbb R)^\star
\\ \exists f\in \mathcal S^{\infty}(\Theta_1;\mathbb R)^\star
\text{ s. t. }|s|_{\frac1f,p,g}=\sum_{(a,b)\in\Theta_1\times\Theta_2}
\left|\frac1{f(a)}\right|\,p(s_{a,b})|g(b)|<+\infty\biggr\}.\notag
\end{multline}
With functions $F\in\mathcal F(\Pi_{\mathfrak V}\times
 \mathcal S^{\infty}(\Theta_2;\mathbb R)^\star ;  \mathcal S^{\infty}(\Theta_1;\mathbb R)^\star)$
and spaces
\begin{multline}
\mathcal M^{\infty,-\infty}_F(\Theta_1,\Theta_2;\mathfrak V)=
\biggl\{ s\in \mathcal M(\Theta_1,\Theta_2;\mathfrak V)
\,:\,\forall p\in\Pi_{\mathfrak V} \forall g\in \mathcal S^{\infty}(\Theta_2;\mathbb R)^\star
\text{ s. t. }\\ |s|_{\frac1{F(p,g)},p,g}=\sum_{(a,b)\in\Theta_1\times\Theta_2}
\left|\frac1{F(p,g)(a)}\right|\,p(s_{a,b})|g(b)|<+\infty\biggr\}.\notag
\end{multline}
the space $\mathcal M^{\infty,-\infty}(\Theta_1,\Theta_2;\mathfrak V)$ is filtered and
becomes an inductive locally  convex space.
We can similarly define the space
$\mathcal M^{-\infty,\infty}(\Theta_1,\Theta_2;\mathfrak V)$.

c.) We define
\begin{multline}
\mathcal M^{-\infty,-\infty}(\Theta_1,\Theta_2;\mathfrak V)=
\biggl\{ s\in \mathcal M(\Theta_1,\Theta_2;\mathfrak V)
\,:\,\forall p\in\Pi_{\mathfrak V} \forall f\in \mathcal S^{\infty}(\Theta_1;\mathbb R)^\star
\\ \forall g\in \mathcal S^{\infty}(\Theta_2;\mathbb R)^\star
\text{ s. t. }|s|_{f,p,g}=\sum_{(a,b)\in\Theta_1\times\Theta_2}
|f(a)|\,p(s_{a,b})|g(b)|<+\infty\biggr\}.\notag
\end{multline}
This space is topologically the same as the column  space
$\mathcal M^{-\infty,-\infty}(\Theta_1\times\Theta_2,\{*\};\mathfrak V)$
which is also the same as $\mathcal M^{-\infty,\infty}(\Theta_1\times\Theta_2,\{*\};\mathfrak V)$.

d.) It is natural to define
\[\Psi(\Theta_1,\Theta_2;\mathfrak V)=
\mathcal M^{-\infty,\infty}(\Theta_1,\Theta_2;\mathfrak V)
\cap \mathcal M^{\infty,-\infty}(\Theta_1,\Theta_2;\mathfrak V),\]
the space of matrices of ``pseudodifferential size''.
\end{point}
\begin{point}
If $\mathfrak A\times\mathfrak B\rightarrow \mathfrak C$ is a continuous bilinear pairing
between locally convex spaces then we have induced continuous pairings
\[\mathcal M^{\mathsf X,\infty}(\Theta_1,\Theta_2;\mathfrak A)
\times \mathcal M^{-\infty,\mathsf Y}(\Theta_2,\Theta_3;\mathfrak B)
\rightarrow \mathcal M^{\mathsf X,\mathsf Y}(\Theta_1,\Theta_3;\mathfrak C),\]
\[\mathcal M^{\mathsf X,\mathsf Y}(\Theta_1,\Theta_2;\mathfrak A)
\times \Psi(\Theta_2,\Theta_3;\mathfrak B)
\rightarrow \mathcal M^{\mathsf X,\mathsf Y}(\Theta_1,\Theta_3;\mathfrak C),\]
\[\Psi(\Theta_1,\Theta_2;\mathfrak A)
\times  \Psi(\Theta_2,\Theta_3;\mathfrak B)
\rightarrow \Psi(\Theta_1,\Theta_3;\mathfrak C),\]
etc.
These pairings are associative whenever the original pairings were.
So come the algebra and module structures associated to matrices.
Instead of $\mathcal M^{-\infty,-\infty}(\Theta_1,\Theta_2;\mathfrak A)$ and
$\mathcal M^{-\infty,-\infty}(\Theta ;\mathfrak A)$
we will use the notations $\mathcal K^{-\infty}_{\Theta_1,\Theta_2}(\mathfrak A)$
and $\mathcal K^{-\infty}_{\Theta}(\mathfrak A)$, which are shorter.
Instead of $\mathcal M^{\pm\infty,\mathsf X}(\Theta,\{*\};\mathfrak A)$ it is reasonable to use
$\mathcal S^{\pm\infty}(\Theta;\mathfrak A)$, which is a consistent extension of our notation from earlier.
One can also notice that there are natural isomorphisms like
$\mathcal K^{-\infty}_{\Theta_1\times\Theta_1',\Theta_2\times\Theta_2'}(\mathfrak A)
\simeq\mathcal K^{-\infty}_{\Theta_1,\Theta_2}(\mathcal K^{-\infty}_{\Theta_1',\Theta_2'}(\mathfrak A))$, etc.
\end{point}
\begin{point}
One standard tool one often uses is relabeling of matrices, which is as follows:
Suppose that $\omega:\Omega\rightarrow\Omega'$
is a map between sets of polynomial growth, such that
$\omega^*\mathcal S^{-\infty}(\Omega',\mathbb R)^\star=\mathcal S^{-\infty}(\Omega,\mathbb R)^\star$.
This includes the case when $\omega$
is an isomorphism of sets of polynomial growth,
but also also the natural inclusions $\iota:\Omega\rightarrow\Omega'\dot\cup\Omega''$, where $\Omega''$ is
finite or an other set of polynomial growth.
Let us now consider the matrix
$R_\omega=\sum_{\alpha\in\Omega}\mathbf e_{\alpha,\omega(\alpha)}\in\Psi(\Omega,\Omega';\mathbb R)$.
Then for a matrix $A\in\mathcal M^{\mathsf X,\mathsf Y}(\Theta;\mathfrak A)$ or
$\Psi(\Theta;\mathfrak A)$ we can take the matrix
$r_\omega(A)=R_{\omega}^\top AR_\omega$ which is a matrix of  the same kind as $A$
but $\Omega$ replaced by $\Omega'$.
This relabeling $r_\omega$ is a continuous, smooth operation, which is an isomorphism if $\omega$ is an isomorphism.
We can similarly relabel matrices from
$\mathcal M^{\mathsf X,\mathsf Y}(\Theta_1,\Theta_2;\mathfrak A)$, etc.
\end{point}
\begin{point}
The advantage of the spaces $\mathcal M^{\mathsf X,\mathsf Y}(\Theta_1,\Theta_2;\mathfrak V)$
is that they are fairly large for the purposes of arithmetic calculations. On the other hand,
their topological character (except in the case of
$\mathcal M^{-\infty,-\infty}(\Theta_1,\Theta_2;\mathfrak V))$ is rather obscure.
However we may use these spaces to keep our arithmetical expressions well-defined.
In the what follows  only the algebras $\mathcal K^{-\infty}$
will be used explicitly.  On the other hand, all computations,
except in Lemma \ref{lem:bottinver} and in Section \ref{sec:otheralg}
will be governed by the following
\begin{principle}
Every matrix expression is understood as element of $\Psi(\Omega_1,\Omega_2;\mathfrak A)$,
where $\Omega_i$ are sets of polynomial growth, and $\mathfrak A$ is a locally convex convex algebra;
but we always hope that  our expressions will yield results continuous in stronger topologies.
\end{principle}
Due to the speciality of the algebras which follow we could have used more
concrete topologizations for the computations, but this one fits as any other one.
\end{point}


\section{The environment of cyclic and Toeplitz algebras}\label{sec:CT}
\paragraph{\textbf{The cyclic and Toeplitz algebras}}
\begin{point}
In what follows let $\overline{\mathbb N}=\mathbb Z\setminus\mathbb N$.
Then $\mathbb Z=\overline{\mathbb N}\,\dot\cup\,\mathbb N$.
We make a canonical correspondence between $\mathbb N$ and
$\overline{\mathbb N}$ by relabeling every $n$ to $-1-n$.
Then, in what follows, we may consider every $\mathbb Z\times\mathbb Z$ matrix $U$
as also a $2\times 2$ matrix of $\mathbb N\times\mathbb N$ matrices:
\[U=\left[\begin{array}{c|c}
U|_{\overline{\mathbb N}\times\overline{\mathbb N}} &
U|_{\overline{\mathbb N}\times{\mathbb N}} \\\hline
U|_{{\mathbb N}\times\overline{\mathbb N}} &
U|_{{\mathbb N}\times{\mathbb N}}
\end{array}\right]\simeq
\begin{bmatrix}U^{--}&U^{-+}\\U^{+-}&U^{++}\end{bmatrix},\]
such that on the right side the matrix entries are $\mathbb N\times\mathbb N$ matrices
obtained by the correspondence explained above.
\end{point}
\begin{point}
An element $a=\sum_{i\in\mathbb Z}a_i\mathbf e_i\in\mathcal
S^{-\infty}(\mathbb Z;\mathfrak A)$ can and will, in general,  be identified
with the Laurent series $\sum_{i\in\mathbb Z}a_i\mathsf z^i\in\mathfrak
A[\mathsf z^{-1},\mathsf z]^\infty$ with rapidly decreasing coefficients.
We call this algebra, as the
algebra of cyclic loops, in contrast to the algebra of proper loops
$\mathcal C^{\infty}(\mathrm S^1;\mathfrak A)$.
Then elements $\sum_{i\in\mathbb Z}a_i\mathsf z^i\in\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$ can be
represented by $\mathbb Z\times\mathbb Z$ matrices
\[\mathsf U(a)=\sum_{n,m\in\mathbb Z}a_{n-m}\mathbf e_{n,m}=
\left[\begin{array}{cccc|cccc}
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\
\ddots&a_0&a_{-1}&a_{-2}&a_{-3}&a_{-4}&a_{-5}&\ddots\\
\ddots&a_1&a_0&a_{-1}&a_{-2}&a_{-3}&a_{-4}&\ddots\\
\ddots&a_2&a_1&a_0&a_{-1}&a_{-2}&a_{-3}&\ddots\\\hline
\ddots&a_3&a_2&a_1&a_0&a_{-1}&a_{-2}&\ddots\\
\ddots&a_4&a_3&a_2&a_1&a_0&a_{-1}&\ddots\\
\ddots&a_5&a_4&a_3&a_2&a_1&a_0&\ddots\\
\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots
\end{array}\right].\]


If $\mathcal W^{-\infty}_{\mathbb Z}(\mathfrak V)$ is the image set
of $\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$ through $\mathsf U$ then it is a subset
of $\Psi(\mathbb Z;\mathfrak V)$ algebraically but we put the topology of
$\mathcal S^{-\infty}(\mathbb Z;\mathfrak V)$ to it. If $\mathfrak A$ is an locally
convex algebra then $\mathsf U:\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty\rightarrow\mathcal W^{-\infty}_{\mathbb Z}(\mathfrak A)$
is an isomorphism of algebras, too.
As already explained, we also have the $2\times 2$ decomposition
\[\mathsf U(a)=
\left[\begin{array}{c|c}
\mathsf U(a)|_{\overline{\mathbb N}\times\overline{\mathbb N}} &
\mathsf U(a)|_{\overline{\mathbb N}\times{\mathbb N}} \\\hline
\mathsf U(a)|_{{\mathbb N}\times\overline{\mathbb N}} &
\mathsf U(a)|_{{\mathbb N}\times{\mathbb N}}
\end{array}\right]
=\begin{bmatrix}\mathsf U^{--}(a)&\mathsf U^{-+}(a)\\
\mathsf U^{+-}(a)&\mathsf U^{++}(a)\end{bmatrix}.\]
\end{point}
\begin{point}
As far as the linear structure is concerned we could have just used
the matrices $\mathsf U^{++}(a)$ to represent the elements $a$. The difference is that
in terms of matrix multiplication
\[\mathsf U^{++}(a)\mathsf U^{++}(b)=\mathsf U^{++}(ab)-\mathsf U^{+-}(a)\mathsf U^{-+}(b),\]
so there is an ``anomalous'' term $-\mathsf U^{+-}(a)\mathsf U^{-+}(b)$.
If $a,b\in\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$ then
$\mathsf U^{+-}(a),$ $\mathsf U^{-+}(b)\in\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )$, so
$\mathsf U^{+-}(a)\mathsf U^{-+}(b)\in\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )$.
Let us define $\mathcal W^{-\infty}_{\mathbb N}(\mathfrak A )=
\{\mathsf U^{++}(a)\,:\,a\in \mathfrak A[\mathsf z^{-1},\mathsf z]^\infty \}$.
One can see that
algebraically $\mathcal W^{-\infty}_{\mathbb N}(\mathfrak A )
\cap\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )=0$.
Hence it is reasonable to define the Toeplitz algebra
\[\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )=
\mathcal W^{-\infty}_{\mathbb N}(\mathfrak A )+
\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A ),\]
which is topologically just
$\mathcal W^{-\infty}_{\mathbb N}(\mathfrak A )\oplus
\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )$ but with the product rule
\begin{multline}
(\mathsf U^{++}(a)+p)((\mathsf U^{++}(b)+q))=\\=\mathsf U^{++}(ab)+(
-\mathsf U^{+-}(a)\mathsf U^{-+}(b)+\mathsf U^{++}(a)q+p\mathsf U^{++}(b)+pq),
\notag
\end{multline}
which is induced from the matrix structure. Algebraically,
$\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )$
is just a subset of $\Psi(\mathbb N;\mathfrak A )$, but a locally convex algebra.
\end{point}


\begin{point}
So, there is a short exact sequence of algebras
\[0\rightarrow \mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )
\xrightarrow\iota \mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )
\xrightarrow\sigma \mathcal W^{-\infty}_{\mathbb Z}(\mathfrak A )\rightarrow 0.\]
The map $\iota$ is the inclusion of the ideal of rapidly decreasing matrices
into the Toeplitz algebra,
while $\sigma$ is the symbol map.
In what follows we rather consider the value of the symbol map as an element of
$\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$,
so we have the symbol homomorphism
\[\sigma: \mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )\rightarrow
\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty.\]
We can naturally extend this symbol map for unit groups, as we have seen.

\label{po:bisymbol}
In order to simplify the notation  we will just write $\mathsf W(a)$ instead
of $\mathsf U^{++}(a)$, and $\mathsf Y(a)$ instead of $\mathsf U^{-+}(a)$.
By this notation
\[\mathsf U(a)=\begin{bmatrix}
 \mathsf W(a(\mathsf z^{-1}))& \mathsf Y(a(\mathsf z))\\
 \mathsf Y(a(\mathsf z^{-1})) & \mathsf W(a(\mathsf z))
\end{bmatrix}\in
\begin{bmatrix}
 \mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )& \mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )\\
 \mathcal K^{-\infty}_{\mathbb N}(\mathfrak A ) & \mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )
\end{bmatrix}.\]
It is a small but important observation that the Toeplitz element in the lower right quadrant has
symbol $a(\mathsf z)$, but the Toeplitz element in the upper left quadrant has symbol
$a(\mathsf z^{-1})$.

The Toeplitz algebra can also be understood without matrices.
It is the semigroup algebra of the bicyclic semigroup.
(The bicyclic semigroup is generated by  $\mathsf z_-$, $\mathsf z_+$ and
with the sole relation $\mathsf z_-\mathsf z_+=1$.)
Indeed, it is easy too see that every element of $\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )$
can uniquely be written in the form
${\sum_{n,m\in\mathbb N}a_{n,m}\mathsf W(\mathsf z)^n\mathsf W(\mathsf z^{-1})^m,}$
where $\{a_{n,m}\}_{n,m\in\mathbb N}$ is a rapidly decreasing matrix.
Then, through the correspondences
$\mathsf z_+\leftrightarrow \mathsf W(\mathsf z),\
\mathsf z_-\leftrightarrow \mathsf W(\mathsf z^{-1})$
we obtain a natural isomorphism
$\mathfrak A\lfloor\mathsf z_-,\mathsf z_+\rceil^\infty\simeq \mathcal T^{-\infty}_{\mathbb N}(\mathfrak A ),$
where on the left side we have the semigroup algebra of the bicyclic semigroup with rapidly
decreasing coefficients.
The symbol map sends $\mathsf z_+\mapsto\mathsf z$, $\mathsf z_-\mapsto\mathsf z^{-1}$.
\end{point}

\begin{point}
For technical reasons we  define the algebra
\[\mathcal T^{-\infty}_{\overline{\mathbb N}|\mathbb N}(\mathfrak A )=
\begin{bmatrix}
\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )& \mathcal K^{-\infty}_{\mathbb N}(\mathfrak A )\\
\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A ) & \mathcal T^{-\infty}_{\mathbb N}(\mathfrak A )
\end{bmatrix},\]
which is also naturally a locally convex algebra.

Furthermore, one can see that there are natural isomorphisms like
$\mathcal T^{-\infty}_{\mathbb N}(\mathcal K^{-\infty}_\Omega(\mathfrak A))\simeq
\mathcal K^{-\infty}_\Omega(\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)),$
etc.
In fact, all of our matrix space constructions considered as functors  are naturally ``commutative''.
\end{point}
\begin{point}
Let $\mathfrak A[\mathsf z^{-1},\mathsf z]^{\infty,\mathrm{po}}$ be the set of pointed loops, ie.~where $a(1)=0$.
Then the elements $\tilde a\in\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^{\infty,\mathrm{po}}$)
are those for which $\tilde a(1)=1$.
These pointed spaces are closed subspaces of the unpointed spaces.
We can similarly define the pointed Toeplitz algebra
$\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)^{\mathrm{po}}$,
there the symbols are pointed.
\end{point}
\paragraph{\textbf{The Bott involution map}}
\begin{point}
Beside the natural identification
$\displaystyle{1=\begin{bmatrix}1_{{\mathbb N}}&\\&1_{{\mathbb N} }\end{bmatrix},}$
let us set the involution
\[\mathsf Q=\begin{bmatrix}-1_{{\mathbb N} }&\\&1_{{\mathbb N} }\end{bmatrix}.\]
In what follows we use the abbreviation
$\Lambda(a,b)=\frac12(1+a+b-ab)$. Then
\[\Lambda(\mathsf z,\mathsf Q)=
\begin{bmatrix}\mathsf z1_{{\mathbb N} }&\\&1_{{\mathbb N}}\end{bmatrix}.\]
We also use the delta-function $\delta_{n,m}$ which is 1 if $n=m$ and it is $0$ otherwise.

If $a\in\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)$ then we define the ``Bott'' involution
\[\mathsf B(a)=\mathsf U(a)\mathsf Q\mathsf U(a)^{-1}\in\mathsf Q+
\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)\]
(cf. the symbols). To get an impression about $\mathsf B$ we list some of its properties:
\begin{lemma}\label{lem:bottinver}
$\mathsf B$ can be ``inverted'' by the formula
\[a(\mathsf z_1)a(\mathsf z_2)^{-1}=\frac12
\left(\sum_{n\in\mathbb Z} \mathsf z_1^n\mathbf e_n^\top \right)
\left(\mathsf B(a)-
\mathsf U(\mathsf z)\mathsf B(a)\mathsf U(\mathsf z^{-1})\right)
\left(\sum_{m\in\mathbb Z} \mathsf z_2^m\mathbf e_{-m} \right).\]

The inverse operation is unique with the assumption $a(1)=1$, when substitution by $\mathsf z_2=1$
yields $a(\mathsf z_1)$ explicitly; and substitution by $\mathsf z_1=1$
yields $a(\mathsf z_2)^{-1}$ explicitly.
\begin{proof}
Let $a(\mathsf z)=\sum_{n\in\mathbb N} a_n\mathsf z^n$,
$a(\mathsf z)^{-1}=\sum_{m\in\mathbb N} b_m\mathsf z^m$. Then
\begin{multline}\notag
\tfrac12\left(\mathsf B(a)-\mathsf U(\mathsf z)\mathsf B(a)\mathsf U(\mathsf z^{-1})\right)
=\tfrac12\left(\mathsf U(a)\left(\mathsf Q-\mathsf U(\mathsf z)\mathsf Q\mathsf U(\mathsf z^{-1})\right)
\mathsf U(a)^{-1}\right)=\\
=\mathsf U(a)\mathbf e_{0,0}\mathsf U(a)^{-1}=\sum_{n,m\in\mathbb N}a_n b_m\mathbf e_{n,-m}
\end{multline}
implies the statement. The uniqueness statement is immediate.
\end{proof}
\end{lemma}
\end{point}
\begin{prop}\label{lem:bottiso}
a.) If $c\in\unit(\mathfrak A)$ then $\mathsf B(a)=\mathsf B(ac)$;
ie. $\mathsf B$ is immune for multiplication on the right by a constant.

b.)  $\mathsf B(\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty))$ is
 a closed subspace of $\mathsf Q+\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)$
(in the topology inherited from $\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)$).

c.) The map $\mathsf B$ induces a homeomorphism between the spaces
$\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)^{\mathrm{po}}$ and
$\mathsf B(\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty))$.

In particular,
$\mathsf B(\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty) )$ may be used to represent
$\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)/\unit(\mathfrak A).$
\begin{proof}
(a) is immediate the definition.
(b) follows from the fact that for the pointed preimage $(a(\mathsf z),a(\mathsf z)^{-1})$
we have a continuous formula, whose appropriateness can be tested by continuous,
in fact, arithmetical, identities.
(c) also follows from the continuous inverse formula.
\end{proof}
\end{prop}

\begin{lemma}
Suppose that $a(\mathsf z)=\Lambda(\mathsf z,\tilde Q)$, where $\tilde Q$ is an involution. Then
\[\mathsf B\left(a(\mathsf z))\right)=\left[\begin{array}{cc|ccc}
\ddots&&&&\\
&-1&&&\\\hline
&&\tilde Q&&\\
&&&1&\\
&&&&\ddots
\end{array}\right].\]
\begin{proof}
Direct computation.
\end{proof}
\end{lemma}

\paragraph{\textbf{Some concrete matrices: ``shifting rotations''}}
\begin{point}
For $a\in\mathfrak A$ we define the matrices
$\mathsf E(a)=a\mathbf e_{00}\in\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A)$
and
$\mathsf E_{\mathbb Z}(a)=a\mathbf e_{00}\in\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)$.
Then, as usual, for $\tilde a=1+a\in1+\mathfrak A$ we extend these maps as
$\mathsf E(\tilde a)=1_{\mathbb N}+\mathsf E_{\mathbb N}(a)$
and
$\mathsf E_{\mathbb Z}(\tilde a)=1_{\mathbb Z}+\mathsf E_{\mathbb Z}(a)$.
Visually, for $\tilde a\in1+\mathfrak A$,
\[\mathsf E(\tilde a)
=\begin{bmatrix}
\tilde a\\&1\\&&1\\&&&\ddots
\end{bmatrix},\qquad
\mathsf E_{\mathbb Z}(\tilde a)=
\begin{bmatrix}
1_{\mathbb N}\\&\mathsf E_{\mathbb N}(\tilde a)\end{bmatrix}=
\left[\begin{array}{cc|ccc}
\ddots&&&&\\&1&&&\\\hline&&\tilde a&&\\&&&1&\\&&&&\ddots
\end{array}\right].\]
\end{point}
\begin{point}
Our natural deformation parameter variable, in general, will be $\theta\in[0,\pi/2]$,
or, sometimes, more generally,  $\theta\in\mathrm S^1=\mathbb R/2\pi\mathbb Z$.
To save space we often use $t=\sin \theta$ and $s=\cos \theta$ instead.
It is useful to keep in mind that $s^2=1-t^2$.
\end{point}
\begin{point}
For $\theta\in[0,2\pi]$  we define the matrices
\[\mathsf C(\theta)=\begin{bmatrix}
s&ts&t^2s&t^3s&t^4s&\cdots\\
-t & s^2& ts^2 & t^2s^2  & t^3s^2&\ddots\\
&-t & s^2 & ts^2 & t^2s^2 &\ddots\\
&&-t & s^2 & ts^2 &\ddots\\
&&&-t & s^2 &\ddots\\
&&&&-t &\ddots\\
&&&&&\ddots\\
\end{bmatrix}\in
\mathcal T^{-\infty}_{\mathbb N}(\mathbb R).\]
So,
\[\mathsf C(\theta)=
\begin{cases}
\mathsf E\left(\dfrac{1}{s}\right)\mathsf W\left(
\dfrac{1-t\mathsf z}{1-t{\mathsf z}^{-1}}\right)&\,\,\text{for }-1<t<1\\&\\
 \pm\mathsf W(-\mathsf z)&\,\,\text{for } t=\pm1\qquad\quad.
\end{cases}\]
Some motivation for this definition is as follows.
For $n\in\mathbb N$ let
\[\mathsf A_n(\theta)=\exp\,\theta(\mathbf e_{n,n+1}-\mathbf e_{n+1,n})=
\begin{bmatrix}1&&&&&&\\&\ddots&&&&&\\&&1&&&&\\&&&s&t&&\\&&&-t&s&&\\&&&&&1&\\&&&&&&\ddots
\end{bmatrix}.\]
Then, for a fixed $\theta$,
\[\mathsf C(\theta)=\lim_{n\rightarrow\infty}\mathsf A_0(\theta)
\cdot\ldots\cdot\mathsf A_n(\theta)\]
in some weak sense, say in $\Psi(\mathbb N;\mathbb R)$. In other words,
$\mathsf C(\theta)$ is an infinite product of certain elementary rotations.

Let $\mathsf C(\theta)^\dag$ denote the transpose of $\mathsf C(\theta)$. Then
\begin{lemma}\label{lem:artkey}
a.)\[\mathsf C(\theta)^\dag\mathsf C(\theta)=1_{\mathbb N}.\]

b.)
\[\mathsf C(\theta)\mathsf C(\theta)^\dag=
-\delta_{t,1}\mathbf e_{0,0}-\delta_{t,-1}\mathbf e_{0,0}+1_{\mathbb N}.\]

c.)
\[\mathsf C(\theta)\mathbf e_{n,m}\mathsf C(\theta)^\dag=
\begin{bmatrix}
t^{n+m}s^2&t^{n+m-1}s^3&\cdots& t^{n}s^3
& -t^{n+1}s &\\
t^{n+m-1}s^3&t^{n+m-2}s^4&\cdots& t^{n-1}s^4
& -t^{n}s^2 &\\
\vdots &\vdots &\ddots
& \vdots &\vdots\\
t^{m}s^3&t^{m-1}s^4&\cdots&s^4
& -ts^2 &\\
 -t^{m+1}s&-t^{m}s^2&\cdots& -ts^2&t^2\\
 &&&&&0\\&&&&&&\ddots
\end{bmatrix}.\]

d.) For $n>0$
\[\mathsf C(\theta)\mathsf W(\mathsf z^n)\mathsf C(\theta)^\dag=
-\delta_{t,1}\mathbf e_{0,0}-(-1)^n\delta_{t,-1}\mathbf e_{0,0}+\begin{bmatrix}
t^n&\\
t^{n-1}s&\\
\vdots\\
ts&\\
s&\\
&1\\
&&1\\
&&&1\\
&&&&\ddots
\end{bmatrix}.\]

$\mathsf C(\theta)\mathsf W(\mathsf z^{-n})\mathsf C(\theta)^\dag$ is the transpose of the matrix above.
\begin{proof}
Direct computation.
\end{proof}
\end{lemma}
\end{point}


\section{Stabilizing homotopies}\label{sec:stabilize}
\begin{proposition}\label{prop:rapid} The continuous map
\[T_{\mathcal K}:\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A)\times \mathrm S^1
\rightarrow\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A)\]
given by
\[A,\theta\mapsto T_{\mathcal K}(A,\theta)=\mathsf C(\theta)A\mathsf C(\theta)^\dag\]
is smooth in $\theta$.
It yields a family of endomorphisms of $\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A)$ when $\theta$ is fixed.
These are isomorphisms for $-1<t<1$, and a closed injective endomorphism for $t=\pm1$.

In particular, for $\theta=0$
\[T_{\mathcal K}(A,0)=A=\begin{bmatrix}
a_{11} & a_{12} &\cdots\\
a_{21} & a_{22} &\cdots\\
\vdots&\vdots&\ddots
\end{bmatrix};\]
but for $\theta=\pm\pi/2$ ($t=\pm 1$)
\[T_{\mathcal K}(A,\pm\pi/2)=\mathsf W(\mathsf z)A\mathsf W(\mathsf z^{-1})=
\begin{bmatrix}0\\
&a_{11} & a_{12} &\cdots\\
&a_{21} & a_{22} &\cdots\\
&\vdots&\vdots&\ddots
\end{bmatrix}.\]
\begin{proof}
Well-definedness and smoothness follows from Lemma \ref{lem:artkey}.c.
Lemma \ref{lem:artkey}.a implies that we have a family of endomorphisms.
Furthermore, it also shows that
$A=\mathsf C(\theta)^\dag T_{\mathcal K}(A,\theta)\mathsf C(\theta)$;
from which the statement about the nature of the endomorphisms follows easily.
\end{proof}
\end{proposition}
Hence, taking $\theta\in[0,\pi/2]$ we see that the deformation $T_{\mathcal K}$ does
indeed realize a stabilizing homotopy even if only with one ``extra dimension''.
Nevertheless, after this, stabilization becomes a matter of standard tricks:
\begin{cor}[$\rightarrow$ Statement \ref{stat:infstable}]  \label{cor:infstable}
Let $\Omega_1$ and $\Omega_2$ be set of polynomial growth;
$\Omega=\Omega_1\,\dot\cup\,\Omega_2$, and let $\omega:\Omega\rightarrow\Omega_1\subset\Omega$ be the composition
of an isomorphism $\Omega\simeq \Omega_1$ and the natural inclusion
$\Omega_1\rightarrow\Omega_1\,\dot\cup\,\Omega_2$. Then, we claim:
There is a smooth map
\[\widehat T_{\mathcal K}:\mathcal K^{-\infty}_\Omega(\mathfrak S)\times[0,\pi/2]
\rightarrow\mathcal K^{-\infty}_\Omega(\mathfrak S)\]
such that it yields a family of endomorphisms of $\mathcal K^{-\infty}_\Omega(\mathfrak S)$,
which are isomorphisms for $\theta\in[0,\pi/2)$, and a
closed injective endomorphism for $\theta=\pi/2$, such that
\[\widehat T_{\mathcal K}(A,0)=\id_{\mathcal K^{-\infty}_\Omega(\mathfrak S)},\qquad
\widehat T_{\mathcal K}(A,\pi/2)=r_\omega.\]

The map $\widehat T_{\mathcal K}$ naturally extends (cf. \ref{po:unithom}) to unit groups.
\begin{proof}
Take $\mathfrak A=\mathcal K^{-\infty}_{\mathbb N}(\mathfrak S)$ in the previous statement.
It yields our statement with $\Omega_1=(\mathbb N\setminus\{0\})\times\mathbb N$, $\Omega_2=\{0\}\times\mathbb N$,
$\omega((n,m))=(n+1,m)$.
Now, using an appropriate relabeling $r_\eta$ of $\Omega$ we obtain the general statement.
\end{proof}
\end{cor}

Let us remark that due to the multiplicative structure, the concatenation of group valued homotopies
is particularly simple: If $f,g:Y\times[0,1]\rightarrow G$ yield homotopies $f_0\simeq f_1$, $g_0\simeq g_1$
where $f_1=g_0$ then $h(y,t)=f(y,t)f(y,1)^{-1}g(y,t)$ yields a homotopy between $f_0$ and $g_1$.
The advantage of this construction is that, for example, polynomial/smooth homotopies yield polynomial/smooth
homotopies, and the operation is associative; in contrast to concatenation by reparametrization.
Using this observation and the stabilizing homotopies above can easily prove

\begin{cor}\label{cor:stablehom}
Let $\Omega_1,\Omega_2$ be sets of polynomial growth, and let
$\iota_1:\Omega_1\rightarrow\Omega_1\,\dot\cup\,\Omega_2$
be the natural inclusion.
Assume that
\[H:X\times [0,1]\rightarrow\unit(\mathcal K^{-\infty}_{\Omega_1\,\dot\cup\,\Omega_2}(\mathfrak S))\]
is a smooth homotopy with maps $f_0, f_1:X\rightarrow
\unit(\mathcal K^{-\infty}_{\Omega_1}(\mathfrak S))$
such that
\[H_0=r_{\iota_1}(f_0),\qquad\text{and}\qquad H_1=r_{\iota_1}(f_1).\]
Then, we claim, there is a smooth homotopy
\[f:X\times [0,1]\rightarrow\unit(\mathcal K^{-\infty}_{\Omega_1}(\mathfrak S))\]
extending $f_0$ and $f_1$. This $f$ can be chosen so that there is a
smooth homotopy between $H$ and $r_{\iota_1}(f)$ relative to endpoints.

In other words: ``In stable algebras stable homotopies can be reduced to ordinary homotopies.''
\qed
\end{cor}



\section{Linearization of cyclic loops}
\begin{point}
Let $\mathsf v$ be a cyclic formal variable and let
\[\mathsf V=\sum_{n\in\mathbb N}\mathsf v^{n}\mathbf e_{n,n}=
\begin{bmatrix}1\\&\mathsf v\\&&\mathsf v^2\\&&&\ddots\end{bmatrix},\qquad
\mathsf V^{-1}=\sum_{n\in\mathbb N}\mathsf v^{-n}\mathbf e_{n,n}.\]
Furthermore, let us consider
\[\mathsf G(\theta,\mathsf v)=
\begin{bmatrix}1_{\mathbb N}&\\&\mathsf V^{-1}\mathsf C(\theta)\mathsf V\end{bmatrix}
%=1_{\mathbb Z^-}+\mathbf e_{0,0}\mathsf U\left(\frac{s}{1-t\mathsf v\mathsf z^{-1}} \right)
%+1_{\mathbb Z^+}\mathsf U\left( \frac{1-t\mathsf z\mathsf v^{-1}}{1-\mathsf v\mathsf z^{-1}}\right)
\qquad\text{and}\qquad %\]\[
\mathsf G^\dag(\theta,\mathsf v)=
\begin{bmatrix}1_{\mathbb N}&\\&\mathsf V^{-1}\mathsf C(\theta)^\dag\mathsf V\end{bmatrix}
%=1_{\mathbb Z^-}+\mathbf e_{0,0}\mathsf U\left(\frac{s}{1-t\mathsf z\mathsf v^{-1}} \right)
%+1_{\mathbb Z^+}\mathsf U\left( \frac{1-t\mathsf v\mathsf z^{-1}}{1-\mathsf z\mathsf v^{-1}}\right)
.\]
Let us now define for $a\in\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$
\[ U(a,\theta,\mathsf v)=
\delta_{t,1}a(\mathsf v)\mathbf e_{00}+\delta_{t,-1}a(-\mathsf v)\mathbf e_{00}
+\mathsf G(\theta,\mathsf v)\mathsf U(a)\mathsf G^\dag(\theta,\mathsf v).\]
\begin{point}\label{po:udesc}
a.) For $n>0$ it yields $U(\mathsf z^n, \theta,\mathsf v)=$
\[=\left[\begin{array}{cccccccc|ccc}
\ddots&&&&&&&&&&\\
&1&&&&&&&&&\\
\hline
&&s&ts\mathsf v&t^2s\mathsf v^2&t^3s\mathsf v^3&\cdots&t^{n-1}s\mathsf v^{n-1}&t^n\mathsf v^n&&\\
&&-t\mathsf v^{-1}&s^2&ts^2\mathsf v&t^2s^2\mathsf v^2&\cdots&t^{n-2}s^2
           \mathsf v^{n-2}&t^{n-1}s\mathsf v^{n-1}&&\\
&&&\ddots&\ddots&\ddots&\ddots&\vdots&\vdots&&\\
&&&&\ddots&\ddots&\ddots&t^2s^2\mathsf v^2&t^3s\mathsf v^3&&\\
&&&&&\ddots&\ddots&ts^2\mathsf v&t^2s\mathsf v^2&&\\
&&&&&&\ddots&s^2&ts\mathsf v&&\\
&&&&&&&-t\mathsf v^{-1}&s&&\\
&&&&&&&&&1&\\
&&&&&&&&&&\ddots
\end{array}\right].\]

b.) $U(1, \theta,\mathsf v)=1_{\mathbb Z}$.

c.) For $n>0$ it yields $U(\mathsf z^{-n}, \theta,\mathsf v)=$
\[=\left[\begin{array}{cc|ccccccccc}
\ddots&&&&&&&&&&\\
&1&&&&&&&&&\\
&&s&-t\mathsf v&&&&&&&\\
&&ts\mathsf v^{-1}&s^2&\ddots&&&&&&\\
&&t^2s\mathsf v^{-2}&ts^2\mathsf v^{-1}&\ddots&\ddots&&&&&\\
&&t^3s\mathsf v^{-3}&t^2s^2\mathsf v^{-2}&\ddots&\ddots&\ddots&&&&\\
&&\vdots&\vdots&\ddots&\ddots&\ddots&\ddots&&&\\
&&t^{n-1}s\mathsf v^{1-n}&t^{n-2}s^2\mathsf v^{2-n}&\cdots&
          t^2s^2\mathsf v^{-2}&ts^2\mathsf v^{-1}&s^2&-t\mathsf v&&\\\hline
&&t^n\mathsf v^{-n}&t^{n-1}s\mathsf v^{1-n}&\cdots&t^3s\mathsf v^{-3}&t^2s\mathsf v^{-2}&ts\mathsf v^{-1}&s&&\\
&&&&&&&&&1&\\
&&&&&&&&&&\ddots
\end{array}\right].\]
\end{point}
\begin{lemma}\label{lem:linearkey}
The continuous map
\[U:\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty\times\mathrm S^1\rightarrow
\mathcal W^{-\infty}_{\mathbb Z}(\mathfrak A)+
\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)[\mathsf v^{-1},\mathsf v]^\infty\]
defined by
\[ a(\mathsf z),\theta\mapsto U(a,\theta,\mathsf v)\]
is smooth in $\theta$. It yields a family of homomorphisms with fixed $\theta$.
The symbols remain constant.
For $\theta=0$
\[U(a,0,\mathsf v)=
\begin{bmatrix}
\mathsf W(a(\mathsf z^{-1})) & \mathsf Y(a(\mathsf z))\\
\mathsf Y(a(\mathsf z^{-1})) & \mathsf W(a(\mathsf z))
\end{bmatrix}
=\mathsf U(a(\mathsf z))\]
while for $\theta=\pi/2$
\[U(a,\pi/2,\mathsf v)=\left[\begin{array}{c|cc}
\mathsf W(a(\mathsf z^{-1}))&&-\mathsf v\mathsf Y(a(\mathsf z))
\\\hline
&a(\mathsf v)&\\
-\mathsf v^{-1}\mathsf Y(a(\mathsf z^{-1}))&&\mathsf W(a(\mathsf z))
\end{array}\right].\]
\begin{proof}
This is immediate from \ref{po:udesc} by taking linear combinations.
\end{proof}
\end{lemma}
\begin{proposition}[$\rightarrow$ Statement \ref{stat:linloop}]\label{prop:linear}
The continuous map
\[K:\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)\times\mathrm S^1\rightarrow
\unit(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A)[\mathsf v^{-1},\mathsf v]^{\infty,\pointed})\]
defined by
\[ a,\theta\mapsto K(a,\theta,\mathsf v)=
\Lambda(\mathsf v,\mathsf Q)^{-1}U(a,\theta,\mathsf v)\Lambda(\mathsf v,\mathsf Q)
U(a,\theta,1)^{-1}\]
is smooth in the variable $\theta$.
Here
\[K(a,0,\mathsf v)= \begin{bmatrix}\mathsf v^{-1}1_{\mathbb N}&\\&1_{\mathbb N}\end{bmatrix}
\mathsf U(a)
\begin{bmatrix}\mathsf v1_{\mathbb N}&\\&1_{\mathbb N}\end{bmatrix}
\mathsf U(a)^{-1}=\Lambda(\mathsf v,\mathsf Q)^{-1}
\Lambda(\mathsf v,\mathsf B(a))\]
but
\[K(a,\pi/2,\mathsf v)=
\begin{bmatrix}1_{\mathbb N}&\\&\mathsf E(a(\mathsf v)a(1)^{-1})\end{bmatrix}=
\mathsf E_{\mathbb Z}(a(\mathsf v)a(1)^{-1}).\]
\begin{proof} It immediately follows from the previous lemma.\end{proof}
\end{proposition}
\end{point}
\begin{remark}
It is safe to call this phenomenon as linearization of loops because multiplied by
$\Lambda(\mathsf v,\mathsf Q)$ it yields a homotopy between
\[\begin{bmatrix}\mathsf v1_{\mathbb N}&\\&\mathsf E(a(\mathsf v)a(1)^{-1})\end{bmatrix}\qquad\text{and}\qquad
\Lambda\left(\mathsf v,\mathsf U(a)\mathsf Q\mathsf U(a)^{-1}\right) =
\Lambda(\mathsf v,\mathsf B(a))\]
in
\[\begin{bmatrix}\mathsf v1_{\mathbb N}&\\&1_{\mathbb N}\end{bmatrix}+
\mathcal K^{-\infty}_{\mathbb Z}(
\mathfrak A)[\mathsf v^{-1},\mathsf v]^\infty  .\]
\end{remark}
\begin{remark}
Although we do not plan to discuss $K$-theory here as such, let us remind ourselves to the
geometrical significance of this statement:
For a locally convex algebra $\mathfrak A$ we can define
\[K_0(\mathfrak A)=\pi_0^{\mathrm{smooth}}(\invol(\mathsf Q+\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A))),\]
the smooth path components of the involutions, which are perturbations of $\mathsf Q$.
One can similarly define
\[K_1(\mathfrak A)=\pi_0^{\mathrm{smooth}}(\unit(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak A))).\]
Now, $\mathsf B$, though this linearization argument induces an isomorphism
\[\mathsf B_*:K_1(\mathfrak A[\mathsf z^{-1},\mathsf z]^{\infty,\mathrm{po}})\rightarrow K_0(\mathfrak A).\]
This is the ``hard part'' of Bott periodicity.
\end{remark}
\begin{lemma}
Suppose that $a(\mathsf z)=\Lambda(\mathsf z,\tilde Q)$, where $\tilde Q$ is an involution.
Then $K(a,\theta)$ is constant in $\theta$:
\[K(a,\theta,\mathsf v)=\mathsf E_{\mathbb Z}(a(\mathsf v)).\]
\begin{proof}
Direct computation.
\end{proof}
\end{lemma}

The lemma above is of significance, because in $K$-theory
one prefers to linearize ``small'' perturbations of loops like $\Lambda(\mathsf z,\tilde Q)$.
The lemma shows that the linearizing homotopy is not leaving that class of perturbations.
\begin{remark}
Although $U(a,\theta,\mathsf v)$ is a rather symmetrical object,
its definition involves an apparent asymmetry.
One may wonder if this asymmetry can be avoided.

Let us consider the ``conjugation involution'' $\widehat{\mathsf Q}=\sum_{n\in\mathbb N}\mathbf e_{-n,n}$.
Then $\widehat{\mathsf Q}\mathsf U(a(\mathsf z))\widehat{\mathsf Q}=\mathsf U(a(\mathsf z^{-1}))$, or
more generally
$\widehat{\mathsf Q}U(a(\mathsf z),\theta,\mathsf v) \widehat{\mathsf Q}=
U(a(\mathsf z^{-1}),\theta,\mathsf v^{-1})$. So,
\[U(a(\mathsf z),\theta,\mathsf v) =
\widehat{\mathsf Q} \mathsf G(\theta,\mathsf v^{-1})\widehat{\mathsf Q}
\mathsf U(a)\widehat{\mathsf Q}\mathsf G(\theta,\mathsf v^{-1})^\dag\widehat{\mathsf Q},\]
ie. the conjugating matrix $\mathsf G(\theta,\mathsf v)$ can be replaced by
$\widehat{\mathsf G}(\theta,\mathsf v)
=\widehat{\mathsf Q} \mathsf G(\theta,\mathsf v^{-1})\widehat{\mathsf Q}$.
In particular, $\widehat{\mathsf G}(\theta,\mathsf v)^{-1}
\mathsf G(\theta,\mathsf v)$ must commute by $\mathsf U(a)$.
In fact, as the symbol suggests
\[\mathsf U\left(\frac{1-t\mathsf z\mathsf v^{-1}}{1-t\mathsf v\mathsf z^{-1}}\right)
=\widehat{\mathsf G}(\theta,\mathsf v)^{-1}\mathsf G(\theta,\mathsf v).\]
%It is rather interesting to see that how such a multiplication action
%decomposes into a ``holomorphic action'' $\mathsf G(\theta,\mathsf v)$
%and an ``antiholomorphic action''
%$\widehat{\mathsf G}(\theta,\mathsf v)^{-1}$.

Now, $\widehat{\mathsf G}(\theta,\mathsf v)$ has the same kind of asymmetry as
$\mathsf G(\theta,\mathsf v)$.
In order to get a more symmetric conjugating matrix we may take the geometric mean
of these matrices.
In general, if $A$, $B$ are invertible elements and $BA^{-1}$ allows an appropriate
square root then we may take the geometric mean
\[\sqrt{A\cdot B}=A\sqrt{A^{-1}B}=B\sqrt{B^{-1}A}=\sqrt{BA^{-1}}A=\sqrt{AB^{-1}}B.\]
In the present case with $A=\mathsf G(\theta,\mathsf v)$, $B=\widehat{\mathsf G}(\theta,\mathsf v)$,
$|t|<1$ it yields the geometric mean
\[ \widetilde{\mathsf G}(\theta,\mathsf v)=\widehat{\mathsf G}(\theta,\mathsf v)
\mathsf U\left(\sqrt{\frac{1-t\mathsf z\mathsf v^{-1}}{1-t\mathsf v\mathsf z^{-1}}}\right)=
\mathsf G(\theta,\mathsf v)
\mathsf U\left(\sqrt{\frac{1-t\mathsf v\mathsf z^{-1}}{1-t\mathsf z\mathsf v^{-1}}}\right)\]
\[=1_{\mathbb Z^-}\mathsf U\left(\sqrt{\frac{1-t\mathsf v\mathsf z^{-1}}{1-t\mathsf z\mathsf v^{-1}}}\right)+
\mathbf e_{0,0}\mathsf U\left(\frac{s}{\sqrt{(1-t\mathsf z\mathsf v^{-1})(1-t\mathsf v\mathsf z^{-1})}}\right)+
1_{\mathbb Z^+}\mathsf U\left(\sqrt{\frac{1-t\mathsf z\mathsf v^{-1}}{1-t\mathsf v\mathsf z^{-1}}}\right).\]
The extra symmetry of $\widetilde{\mathsf G}(\theta,\mathsf v)$ is expressed by the
identity
$\widehat{\mathsf Q}\widetilde{\mathsf G}(\theta,\mathsf v)\widehat{\mathsf Q}=
\widetilde{\mathsf G}(\theta,\mathsf v^{-1})$.
In particular, $\widetilde{\mathsf G}(\theta,1)$ commutes with $\widehat{\mathsf Q}.$
Unfortunately, the elements of the matrix are not particularly simple, because
they can be expressed as certain complete elliptic integrals, cf.
\[ \frac{1}{\sqrt{(1-t\mathsf z)(1-t\mathsf z^{-1})}}=
\ldots+\frac2\pi \frac{\bo K(t)-\bo E(t)}t\mathsf z^{-1}
+\frac2\pi\bo K(t)\mathsf z^0+\frac2\pi \frac{\bo K(t)-\bo E(t)}t\mathsf z+\ldots\]
\[\sqrt{\frac{1-t\mathsf z}{1-t\mathsf z^{-1}}}=
\ldots+\frac2\pi\frac{(1-t^2)\bo K(t)+(2t^2-1)\bo E(t)}{3t}\mathsf z^{-1}+\qquad\qquad\qquad\qquad\qquad\qquad\]
\[\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
+\frac2\pi\bo E(t)\mathsf z^0+\frac2\pi\frac{(1-t^2)\bo K(t)-\bo E(t) }{t} \mathsf z+\ldots \]
\end{remark}



\section{The contractibility of the pointed stable Toeplitz unit group} \label{sec:TP}




\paragraph{\textbf{A reformulation of linearizability}}~\\

When we extend the stabilization procedure of Proposition \ref{prop:rapid} to Toeplitz algebras,
something interesting happens: The symbol suddenly appears in the result.

Let  $A\in\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)$ with symbol $\sigma$.
Then we define
\[T(A,\theta,\mathsf v)=\delta_{t,1}\sigma(\mathsf
v)\mathbf e_{0,0}+\delta_{t,-1}\sigma(-\mathsf v)\mathbf e_{0,0}+
\mathsf V^{-1}\mathsf C(\theta)\mathsf VA\mathsf V^{-1}\mathsf
C(\theta)^\dag\mathsf V.\notag\]
\begin{proposition}\label{prop:toeploop} The continuous map
\[T:\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)\times \mathrm S^1\rightarrow
\mathcal W^{-\infty}_{\mathbb N}(\mathfrak A)+
\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A)[\mathsf v^{-1},\mathsf v]^\infty\subset
\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)[\mathsf v^{-1},\mathsf v]^\infty\]
defined by
\[A,\theta\mapsto T(A,\theta,\mathsf v)\]
is smooth in the variable $\theta$. It yields a family of homomorphisms of
$\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)$ to
$\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A)[\mathsf v^{-1},\mathsf v]^\infty$.
The map leaves the symbol invariant.

For $\theta=0$
\[T(A,0,\mathsf v)=A=\begin{bmatrix}
a_{11} & a_{12} &\cdots\\
a_{21} & a_{22} &\cdots\\
\vdots&\vdots&\ddots
\end{bmatrix};\]
but for $\theta=\pm\pi/2$ ($t=\pm 1$)
\[T(A,\pm\pi/2,\mathsf v)=
\sigma(\pm\mathsf v)\mathbf e_{0,0}+\mathsf W(\mathsf z)A\mathsf W(\mathsf z^{-1})=
\begin{bmatrix}
\sigma(\pm\mathsf v)\\
&a_{11} & a_{12} &\cdots\\
&a_{21} & a_{22} &\cdots\\
&\vdots&\vdots&\ddots
\end{bmatrix}.\]
\begin{proof}
It follows from the direct inspection of the matrices in question.
\end{proof}
\end{proposition}



\begin{proposition}\label{prop:toepsymbol}
The continuous map
\[Z:\unit(\mathcal T^{-\infty}_{\mathbb N}(\mathfrak A))\times\mathrm S^1\rightarrow
\unit(\mathcal K^{-\infty}_{\mathbb N}(\mathfrak A)[\mathsf v^{-1},\mathsf v]^{\infty,\pointed})\]
defined by
\[A,\theta\mapsto Z(A,\theta,\mathsf v)=T(A,\theta,\mathsf v)T(A,\theta,1)^{-1}.\]
is smooth in  $\theta$. Here
\[Z(A,0,\mathsf v)=1,\]
but for $\theta=\pm\pi/2$ ($t=\pm 1$)
\[Z(A,\pm\pi/2,\mathsf v)=\mathsf E(\sigma(\pm \mathsf v)\sigma(1)^{-1}).\]

Consequently, the symbols $\sigma(\mathsf z)$ of invertible Toeplitz algebra elements
are stably homotopic to constant loops $\sigma(1)$.
If $\mathfrak A= \mathcal K^{-\infty}_{\mathbb N}(\mathfrak S)$ then
(according to Corollary \ref{cor:infstable}) stable homotopy implies the existence of ordinary homotopies.
\begin{proof}
This immediately follows from the previous proposition.
\end{proof}
\end{proposition}

\begin{point}
Suppose that $Q$ is an involution, ie. $Q^2=1$, and $k\in\mathfrak A$.
We will use the shorthand notation
$k^+_Q=\frac12(k+QkQ)$, $k^-_Q=\frac12(k-QkQ)$,
$k^{++}_Q=\frac{1+Q}2k\frac{1+Q}2$,
$k^{+-}_Q=\frac{1+Q}2k\frac{1-Q}2$,
$k^{-+}_Q=\frac{1-Q}2k\frac{1+Q}2$,
$k^{--}_Q=\frac{1-Q}2k\frac{1-Q}2$.

Let us define
\[L(Q,k)= \mathsf W(\Lambda(\mathsf z,Q))k\mathsf W(\Lambda(\mathsf z^{-1},Q))=
\begin{bmatrix}
k^{++}_Q & k^{+-}_Q \\
k^{-+}_Q & k^{+}_Q & k^{+-}_Q \\
&k^{-+}_Q & k^{+}_Q &\ddots\\
&&\ddots&\ddots
\end{bmatrix}.\]
This is a homomorphism in $k$ and we can extend it to $\tilde k\in1+\mathfrak A$ as before.
\end{point}

\begin{point}\label{po:constru}
Let us assume that $Q=\mathsf Q$ and $k\in \mathcal T^{-\infty}_{\overline{\mathbb N}|\mathbb N}(\mathfrak S)$.
Set
\[\widetilde{L}(k)=\left[\begin{array}{ccc|cccc}
\ddots&\ddots&&&&&\\
\ddots&k^{+}_Q &k^{-+}_Q&&&&\\
&k^{+-}_Q&k^{+}_Q&k^{-+}_Q\mathbf e_{00}&k^{-+}_Q\mathbf e_{10}&k^{-+}_Q\mathbf e_{20}&\cdots\\\hline
&&\mathbf e_{00}k^{+-}_Q&\mathbf e_{00}k^{++}_Q\mathbf e_{00}&
\mathbf e_{00}k^{++}_Q\mathbf e_{10}&\mathbf e_{00}k^{++}_Q\mathbf e_{20}&\cdots\\
&&\mathbf e_{01}k^{+-}_Q&\mathbf e_{01}k^{++}_Q\mathbf e_{00}&
\mathbf e_{01}k^{++}_Q\mathbf e_{10}&\mathbf e_{01}k^{++}_Q\mathbf e_{20}&\cdots\\
&&\mathbf e_{02}k^{+-}_Q&\mathbf e_{02}k^{++}_Q\mathbf e_{00}&
\mathbf e_{02}k^{++}_Q\mathbf e_{10}&\mathbf e_{02}k^{++}_Q\mathbf e_{20}&\cdots\\
&&\vdots&\vdots&\vdots&\vdots&\ddots
\end{array}\right]\]
Again, this is a homomorphism in $k$ and we can extend it to $\tilde k\in1+
\mathcal T^{-\infty}_{\overline{\mathbb N}|\mathbb N}(\mathfrak S)$.
(What happened here compared to $L(k,Q)$ is the following: We inflated the first row and column
to infinitely many rows and columns, and reordered the matrix.)

Assume now that $k\in\unit(\mathcal T^{-\infty}_{\overline{\mathbb N}|\mathbb N}(\mathfrak S))$, and
the symbol of its lower right quadrant is $a(\mathsf z)$.
We find that
\begin{multline}
\widetilde{L}(k)\Lambda(k^{-1},\mathsf Q_{\mathcal T^{-\infty}_{\overline{\mathbb N}|\mathbb N}(\mathfrak S)})
\mathsf U( k\Lambda(\mathsf z^{-1},\mathsf Q)k^{-1}\Lambda(\mathsf z,\mathsf Q))=
\notag\\
\widetilde{L}(k)
\left[\begin{array}{ccc|cc}
\ddots&&&&\\
&k^{-1}&&&\\
&&k^{-1}&&\\\hline
&&&1&\\
&&&&\ddots
\end{array}\right]
\left[\begin{array}{ccc|cc}
\ddots&\ddots&&&\\
\ddots&k(k^{-1})^{+}&k(k^{-1})^{-+}&&\\
&k(k^{-1})^{+-}&k(k^{-1})^{+}&k(k^{-1})^{-+}&\\\hline
&&k(k^{-1})^{+-}&k(k^{-1})^{+}&\ddots\\
&&&\ddots&\ddots
\end{array}\right]
\\
=\left[\begin{array}{ccc|cccc}
\ddots&&&&&&\\
&1&&&&&\\
&&e_{00}&e_{01}&e_{02}&e_{03}&\cdots\\\hline
&&e_{10}&e_{11}&e_{12}&e_{13}&\cdots\\
&&e_{20}&e_{21}&e_{22}&e_{13}&\cdots\\
&&\vdots&\vdots&\vdots&\vdots&\ddots
\end{array}\right]
\in\unit(\mathcal T^{-\infty}_{\overline{\mathbb N}|\mathbb N}(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S))).
\end{multline}

The lower right quadrant will have symbol
\[\mathsf E_{\mathbb Z}(a(\mathsf z))
k\Lambda(\mathsf z^{-1},\mathsf Q)k^{-1}\Lambda(\mathsf z,\mathsf Q).\]
Then, it is reasonable to define
\[N(k)=\begin{bmatrix}
e_{00}&e_{01}&e_{02}&\cdots\\
e_{10}&e_{11}&e_{12}&\cdots\\
e_{20}&e_{21}&e_{22}&\cdots\\
\vdots&\vdots&\vdots&\ddots
\end{bmatrix},\]
which is of symbol $ \mathsf E_{\mathbb Z}(a(\mathsf z))
k\Lambda(\mathsf z^{-1},\mathsf Q)k^{-1}\Lambda(\mathsf z,\mathsf Q)$.
Let us set $G(a)=N(\mathsf U(a))$.
\end{point}
\begin{proposition}\label{prop:symsection} The continuous map
\[G:\unit(\mathfrak S[[z]])\rightarrow \unit(\mathcal T^{-\infty}_{\mathbb N}
(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S)))\]
is such that  the symbol of $G(a)$ is
\[\mathsf E_{\mathbb Z}(a(\mathsf z))
\mathsf U(a)\Lambda(\mathsf z^{-1},\mathsf Q)\mathsf U(a)^{-1}\Lambda(\mathsf z,\mathsf Q).\]
\begin{proof}
That immediately follows from the construction.
\end{proof}
\end{proposition}

Now, according to Proposition \ref{prop:toepsymbol} and  Corollary \ref{cor:stablehom}
the mere existence of the map above implies that the symbol $\mathsf E_{\mathbb Z}(a(\mathsf z))
\mathsf U(a)\Lambda(\mathsf z^{-1},\mathsf Q)\mathsf U(a)^{-1}\Lambda(\mathsf z,\mathsf Q)$
is homotopic to $1$.
That yields a different manifestation of linearizability.~\\



\paragraph{\textbf{Contractibility}}
\begin{proposition}[$\rightarrow$ Statement \ref{stat:toepcont}]
The unit group
$\unit(\,\mathcal T^{-\infty}_{\mathbb N}(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S))^{\mathrm{po}})$
is smoothly contractible.
\begin{proof}
We prove the statement up to stabilization. Then stabilization can be removed according to
Corollary \ref{cor:stablehom}.

a.) First of all, consider any element
$A\in \unit(\,\mathcal T^{-\infty}_{\mathbb N}(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S))^{\mathrm{po}})$.
According to Proposition \ref{prop:toepsymbol} its symbol $a$ is (stably) homotopic to the
constant loop $1$.
Applying Proposition \ref{prop:symsection} to this homotopy we see that it is enough to prove
that Toeplitz units with symbol
$\Lambda(\mathsf z^{-1},\mathsf Q)\mathsf U(a)\Lambda(\mathsf z,\mathsf Q)\mathsf U(a)^{-1}$
can be contracted.

b.) Consider $A$ further as above. Let
\[Q=\begin{bmatrix}1_{\mathbb Z}&\\&\mathsf Q \end{bmatrix}=
\left[\begin{array}{cc||cc}-1_{\mathbb N}&&&\\
&-1_{\mathbb N}&&\\\hline&&&\vspace{-11pt}\\\hline&&-1_{\mathbb N}&\\&&&1_{\mathbb N}\end{array}\right];\]
here the double lines show how we decompose this block matrix of $\mathbb Z\times\mathbb Z$ matrices
to a block matrix of $\mathbb N\times\mathbb N$ matrices.
Furthermore, let
\begin{multline}\notag
S(\theta)=\begin{bmatrix}s&&&-t\\&1&&\\&&1&\\t&&&s\end{bmatrix}
\begin{bmatrix}1_{\mathbb N}&&&&\\&1_{\mathbb N}&&\\&&\mathsf W(a(\mathsf z^{-1})) & \mathsf Y(a(\mathsf z))\\
&&\mathsf Y(a(\mathsf z^{-1})) & \mathsf W(a(\mathsf z))\end{bmatrix}\\
\begin{bmatrix}1_{\mathbb N}&&&&\\&\mathsf W(a^{-1}(\mathsf z))&
\mathsf Y(a^{-1}(\mathsf z^{-1}))&\\&\mathsf Y(a^{-1}(\mathsf z))&\mathsf W(a^{-1}(\mathsf z^{-1})) & \\
&& &1_{\mathbb N} \end{bmatrix}
\begin{bmatrix}1_{\mathbb N}&&&\\&A&&\\&&1_{\mathbb N}&\\&&&A^{-1}\end{bmatrix}
\begin{bmatrix}s&&&t\\&1&&\\&&1&\\-t&&&s\end{bmatrix}\\
\in\unit(\mathcal K^{-\infty}_{\{1,2\}\times\mathbb Z}(\mathfrak S))
 \end{multline}
and take $L(-Q,S(\theta))S(\theta)^{-1}\in
\unit(\mathcal T^{-\infty}_{\mathbb N}(\mathcal K^{-\infty}_{\{1,2\}\times\mathbb Z}(\mathfrak S)))$.
That yields a homotopy between $L(-Q,S(0))S(0)^{-1}$ and $L(-Q,S(\pi/2))S(\pi/2)^{-1}$.
These elements have symbols
\[\Lambda(\mathsf z,-Q)S(0)\Lambda(\mathsf z^{-1},-Q)S(0)^{-1}=\begin{bmatrix}
1_{\mathbb Z}&\\&\Lambda(\mathsf z^{-1},\mathsf Q)\mathsf U(a)\Lambda(\mathsf z,\mathsf Q)\mathsf U(a)^{-1}
\end{bmatrix}\] and
\[\Lambda(\mathsf z,-Q)S(\pi/2)\Lambda(\mathsf z^{-1},-Q)S(\pi/2)^{-1}=\begin{bmatrix}
1_{\mathbb Z}&\\&1_{\mathbb Z}
\end{bmatrix},\]
respectively. So,  Toeplitz units with symbol
$\Lambda(\mathsf z^{-1},\mathsf Q)\mathsf U(a)\Lambda(\mathsf z,\mathsf Q)\mathsf U(a)^{-1}$
can be deformed to Toeplitz units with trivial symbols.
According to part (a) now it is enough to show that elements with
trivial symbol can be contracted.

c.) Now, if the symbol of a Toeplitz unit $A$ is  $1$. According to standard
stabilization arguments we can assume that $A=\mathsf E(k)$
where $k=\begin{bmatrix}k_0&\\&1_{\mathbb N}\end{bmatrix}\in\unit(\mathcal K^{-\infty}_{\mathbb Z}(\mathfrak S))$.
Let
\[k(\theta)=
\begin{bmatrix}s&t\\-t&s\end{bmatrix}
\begin{bmatrix}k_0&\\&1_{\mathbb N}\end{bmatrix}
\begin{bmatrix}s&-t\\t&s\end{bmatrix}.\]
Then $k(\theta)L(\mathsf Q,k(\theta))^{-1}$ yields a homotopy
between $k(0)L(\mathsf Q,k(0))^{-1}=\mathsf E(k)=A$ and $k(\pi/2)L(\mathsf Q,k(\pi/2))^{-1}=1$.
\end{proof}
\end{proposition}
\begin{remark}
If the  locally convex algebra $\mathfrak A$ is what like in \cite{L} is
called strong, ie. for all seminorm $p$ there is a seminorm $\tilde p$
such that $p(X_1\ldots X_n)\leq \tilde p(X_1)\ldots \tilde p( X_n)$ holds for all $n$ then
the proof can be simplified greatly: In that case the associated algebras
are also strong, and the smooth homotopy lifting property holds for the symbol map.
So, using Proposition \ref{prop:toepsymbol} the proof of the
contractibility statement immediately reduces to point (c), hence making
points (a) and (b), and the construction of \ref{po:constru} unnecessary.
\end{remark}
\begin{remark}
Stabilization was an important assumption in the previous statement. For example,
$\unit(\,\mathcal T^{-\infty}_{\mathbb N}(\mathbb C)^{\mathrm{po}})$ is not contractible.
In fact, $\unit(\,\mathcal K^{-\infty}_{\mathbb N}(\mathbb C))$
cannot be contracted in $\unit(\,\mathcal T^{-\infty}_{\mathbb N}(\mathbb C))$.
Indeed, one can show that in $\unit(\,\mathcal T^{-\infty}_{\mathbb N}(\mathbb C))$
the regularized determinant
\[\mathrm{reg}\det A=\det( A\,\sigma(1)^{-1}\exp\mathsf W(-\log (\sigma(\mathsf z)/\sigma(1)))\]
is well-defined and
\[\mathrm{reg}\det \mathsf E(a)=a.\]
(To show well-definedness we use that the index of $\mathsf \sigma$ is $0$ and so
logarithm can be taken. Another point is that in nice algebras one can exponentiate.)
From this it follows that $\mathsf E$ restricted to the unit circle is not null-homotopic.
\end{remark}


\section{Other types of algebras and modifications}\label{sec:otheralg}
\begin{point}
Quite often the algebras one works with are $*$-algebras, which means that
there is a continuous contravariant involution ${}^*$.
Then for matrix algebras, instead of simple transposition ${}^{\top}$ we can
take the (formal) adjoint ${}^{\dag}$, which is transposition combined by ${}^*$
in the usual manner. Elements $A\in\unit(\mathfrak A)$ with the property $A^\dag=A^{-1}$
are called unitary. Let us denote their set by $\unit^*(\mathfrak A)$.
In accordance to the matrix case
the adjoint operation extends to $\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$ by $\mathsf z^\dag=\mathsf z^{-1}$
and to $\mathfrak A\lfloor\mathsf z_-,\mathsf z_+\rceil^\infty$ by $(\mathsf z_+)^\dag=\mathsf z_-$.
Then the natural question arises: Can we execute the stabilizations, linearizations, and contractions
discussed inside $\unit^*(\cdot)$, without stepping out into $\unit(\cdot)$?

The answer is affirmative: all of our constructions are formulated in such a way that
they respect unitarity.
\end{point}

\begin{point}
Another question is the following: Can we repeat our constructions in a category slightly
different from the rapidly decreasing matrices?

First is the case of ``algebraic topology''. In that case we allow only finite matrices
instead of rapidly decreasing ones, and only finite Laurent series instead of rapidly
decreasing ones. So ${}^{-\infty}$ should be replaced by ${}^{\mathrm f}$ everywhere.
Then, however, our algebras will be only inductive locally convex algebras, filtered by
matrix size, and/or Laurent series length.
Otherwise our constructions remain valid. True, $\mathsf C(\theta)$ is not in
$\mathcal T^{\mathrm f}_{\mathbb Z}(\mathbb R)$,
but conjugating by it leaves that algebra invariant, as  Lemma \ref{lem:artkey} shows.
On the level of individual elements the notion of homotopy becomes polynomial homotopy, but
it is also continuous in the sense of inductive locally convex algebras.
\end{point}
\begin{point}
Also often considered is the case of special matrix classes over $\mathbb R$,
when we take the closure of finite matrices under a special norm invariant to
conjugation by unitary matrices.
Examples include the spaces of compact operators (under the usual norm) and the
Hilbert-Schmidt operators (with the Hilbert-Schmidt norm).
Then, because they were
essentially achieved via conjugation by unitary matrices, our stabilization
constructions extend, giving, say, continuous homotopies.

A more intriguing question is what to do with the cyclic loops,  what is the best way
to topologize them.
In general, the key is to topologize the corresponding Toeplitz algebras; but also see
Pressley and Segal, \cite{PS}, Ch. 6.
On the other hand, in algebras where we can approximate (which include Banach algebras, and much more)
such questions are not so important, because we can always smooth out loops by taking, say,
approximations by Fej\'er's method.

For example, if $\overline{\phantom{X}}^{\mathtt N}$ denotes norm closure then it is immediate
that the space $\overline{\unit(\,\mathcal T^{-\infty}_{\mathbb N}(\mathcal K^{-\infty}_{\mathbb Z}(
\mathfrak S))^{\mathrm{po}})}^{\mathtt N}$ is contractible. On the other hand, it also not hard to
see that this space
$\simeq\unit\left(\overline{\mathcal T^{-\infty}_{\mathbb N}(\mathcal K^{-\infty}_{\mathbb Z}(
\mathfrak S))^{\mathrm{po}}}^{\mathtt N}\right)
\simeq \unit\left(\overline{\mathcal T^{\mathrm f}_{\mathbb N}(\mathcal K^{\mathrm f}_{\mathbb Z}(
\mathfrak S))^{\mathrm{po}}}^{\mathtt N}\right)$, etc., getting a connection to the
original setting of Cuntz \cite{C}.
\end{point}
\begin{point}
We mention that the pair of
$\mathsf C(\theta)$ and $\mathsf C(\theta)^\dag$ can be modified for $-1<t<1$ as
\begin{multline}
\tilde{\mathsf C}(\theta)=\mathsf W\left(
\dfrac{1-t\mathsf z}{1-t{\mathsf z}^{-1}}\right)=
\mathsf W(1-t\mathsf z^{-1})^{-1}\mathsf W(1-t\mathsf z)=\\
=\begin{bmatrix}
{1-t^2}&t({1-t^2})&t^2({1-t^2})&t^3({1-t^2})&t^4({1-t^2})&\cdots\\
-t & 1-t^2 & t(1-t^2) & t^2(1-t^2)  & t^3(1-t^2)&\cdots\\
&-t & 1-t^2 & t(1-t^2) & t^2(1-t^2) &\ddots\\
&&-t & 1-t^2 & t(1-t^2) &\ddots\\
&&&-t & 1-t^2 &\ddots\\
&&&&-t &\ddots\\
&&&&&\ddots\\
\end{bmatrix}
\notag
\end{multline}
and
\begin{multline}
\tilde{\mathsf C}'(\theta)=\mathsf W\left(
\dfrac{1-t{\mathsf z}^{-1}}{1-t{\mathsf z}}\right)\mathsf E\left(\frac{1}{1-t^2}\right)=
\mathsf W(1-t\mathsf z)^{-1}\mathsf W(1-t\mathsf z^{-1})=\\=
\begin{bmatrix}
1&-t\\
t&1-t^2&-t\\
t^2&t(1-t^2)&1-t^2&-t\\
t^3&t^2(1-t^2)&t(1-t^2)&1-t^2&-t\\
t^4&t^3(1-t^2)&t^2(1-t^2)&t(1-t^2)&1-t^2&-t\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\ddots\notag
\end{bmatrix}.
\end{multline}
Similarly, the conjugating matrices $\begin{bmatrix}s&t\\-t&s\end{bmatrix}$
can be replaced by $\begin{bmatrix}1-t^2&t\\-t&1\end{bmatrix}$.

That way we loose unitarity but all the expressions involve only
polynomial functions  of $t$.
In fact, using the matrices  above one can formulate some of our statements formally
for general rings.
\end{point}


\section{Algebraically finite cyclic loops}\label{sec:practbott}
\begin{point} Although otherwise quite satisfying, a  practical disadvantage
of $\mathsf B(a)$ is that it is, in general, an infinite perturbation of $\mathsf Q$.
The exception is when $a\in\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^{\mathrm f})$, but this
is a rather restrictive condition from geometrical viewpoint. (On the other hand,
$\unit(\mathcal T^{\mathrm f}_{\mathbb N}(\mathcal K^{\mathrm f}_{\mathbb Z}(\mathfrak A))^{\mathrm{po}})$
is a technically simpler construction for a large contractible space than in the smooth case.)
Here we argue that we can do well in the case when $a$ can be represented by finite
loops but it is not in $\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^{\mathrm f})$.
\end{point}
\begin{point} A loop $a(\mathsf z)\in \unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)$ is an $L(m,n)$-finite loop if
$a(\mathsf z)=\sum_{m\leq j\leq n}a_j\mathsf z^k$.
A loop $a(\mathsf z)$ is an $R(m,n)$-finite loop, if it inverse $a(\mathsf z)^{-1}$ is an
$L(-n,-m)$-finite loop.
For a finite sequence $F=\{(m_j,n_j)\}_{1\leq j\leq s}$ let
\[\mathfrak A_F=\{(a_s,\ldots,a_1)\,:\,a_j\in\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty)
 \text{ is $L(m_j,n_j)$ or $R(m_j,n_j)$-finite}\}.\]
We say that
$a\in\unit(\mathfrak A[\mathsf z^{-1},\mathsf z]^\infty$ is algebraically finite of type $F$ if
$a=a_s\ldots a_1$, for an element $(a_s,\ldots,a_1)\in\mathfrak A_F$.
\end{point}

\begin{point}
For $m\leq 0\leq n$
we say that a matrix $A$ is an $L(m,n)$-perturbation of $A_0$ if
\[A=A_0+\sum_{m\leq i\leq n,\,j\in\mathbb Z}a_{i,j}\mathbf e_{i,j},\]
for $a_{i,j}$ chosen suitably. Similarly we can define $R(m,n)$-perturbations by interchanging the
role of $i$ and $j$ in the expression above. An $(m,n)$-perturbation  is a matrix
which is both an  $L(m,n)$-perturbation and an $R(m,n)$-perturbation.

In what follows we will always be concerned of perturbations of
$\Lambda(\mathsf s,\mathsf Q)=\begin{bmatrix}\mathsf s1_{\mathbb N}&\\&1_{\mathbb N}\end{bmatrix}$,
where $\mathsf s$ is equal to $1$, $-1$, or an other formal variable $\mathsf v$.
We can notice that both $L(m,n)$-perturbations and $R(m,n)$-perturbations of
$\Lambda(\mathsf s,\mathsf Q)$ can be reduced to
$(m,n)$-perturbations by taking direct cut-offs of unwanted matrix elements:
\[\begin{bmatrix}\mathsf s&&\\L^-&M&L^+\\&&1 \end{bmatrix}\xrightarrow{\Red_{(m,n)}}
\begin{bmatrix}\mathsf s&&\\&M&\\&&1 \end{bmatrix}\xleftarrow{\Red_{(m,n)}}
\begin{bmatrix}\mathsf s&R^-&\\&M&\\&R^+&1 \end{bmatrix}.\]
\end{point}
\begin{lemma}
Suppose that $A$ is an $L(m,n)$ or $R(m,n)$-perturbation of $\Lambda(\mathsf s,\mathsf Q)$ and
let $\tilde A$ be the reduction of $A$ to an $(m,n)$-perturbation of
$\Lambda(\mathsf s,\mathsf Q)$. Let us use the notation
\[C_{A,\tilde A}(t)=A+t(\tilde A-A).\]
In particular, $C_{A,\tilde A}(0)=A$, $C_{A,\tilde A}(1)=\tilde A$. We claim:


a.) If $A$ was  invertible, then $C_{A,\tilde A}(t)$ ($t\in\mathbb R$) is an invertible:
\[C_{A,\tilde A}(t)^{-1}=A^{-1}C_{A,\tilde A}(-t)A^{-1}.\]

Moreover, it is true that $A^{-1}$ is an $L(m,n)$ or $R(m,n)$-perturbation, respectively,
whose $(m,n)$-reduction is $\tilde A^{-1}$; and
$C_{A,\tilde A}(t)^{-1}=C_{A^{-1},\tilde A^{-1}}(t)$.
The identity \[C_{A,\tilde A}(t)=C_{A,\tilde A}(t/2)A^{-1}C_{A,\tilde A}(t/2)\] also holds.

b.) If $A$ was  an involution then $C_{A,\tilde A}(t)$ ($t\in\mathbb R$) is an involution. In fact,
\[C_{A,\tilde A}(t)=C_{A,\tilde A}(t/2)A C_{A,\tilde A}(t/2)=C_{A,\tilde A}(t/2)A C_{A,\tilde A}(t/2)^{-1}.\]
\begin{proof}
It is enough the examine the $L$ case because the $R$ case is completely analogous.
It is not hard to see that the only possible shape for  $A^{-1}$ is
\[\begin{bmatrix}\mathsf s^{-1}&&\\-\mathsf s^{-1}M^{-1}L^-&M^{-1}&-M^{-1}L^+\\&&1 \end{bmatrix}.\]
Then all calculations follow from the fact that $\tilde A A^{-1}=1+C$, where
the elements $1+tC$ form a one-parameter group.
\end{proof}
\end{lemma}
\begin{lemma}\label{lem:bu}
If $a(\mathsf z)=\sum_{m\leq j\leq n}a_j\mathsf z^k$ then $U(a,\theta,\mathsf v)$ is an
$(m,n)$-perturbation of $\mathsf U(a)$.
\begin{proof}
This is immediate from \ref{po:udesc}.
\end{proof}
\end{lemma}
\begin{lemma}\label{lem:looppert}
Suppose that $A$ is an $(m',n')$-perturbation of $\Lambda(\mathsf s,\mathsf Q)$, $m'\leq0\leq n'$.
Then, we claim,

If $a$ is an $L(m,n)$- or $R(m,n)$-finite loop then
\[U(a,\theta,\mathsf v)A U(a,\theta,\mathsf w)^{-1}\] is an
$L(m+m',n+n')$- or $R(m+m',n+n')$-perturbation of $\Lambda(\mathsf s,\mathsf Q)$, respectively.

\begin{proof}
a.) The $L$ case:
Let $a(\mathsf z)=\sum_{m\leq j\leq n}a_j\mathsf z^k$.
Then $U(a,\theta,\mathsf v)$ is a $(m,n)$-perturbation of $\mathsf U(a)$.
Consider $\mathbf e_k^\top$, where $k>n+n'$ or $k<m+m'$. Being $k>n$  or $k<m$ we find that
$\mathbf e_k^\top U(a,\theta,\mathsf v)=\mathbf e_k^\top \mathsf U(a)
=\sum_{m\leq j\leq n}\mathbf e_{k-j}^\top a_j$.
Now $k-j>n'$ or $k-j<m'$ respectively, and $A$ is an $(m',n')$-perturbation of
$\Lambda(\mathsf s,\mathsf Q)$, so
$\mathbf e_{k-j}^\top A=h\mathbf e_{k-j}^\top$ where $h=1$ if $k>n+n'$ and $h=\mathsf s$ if
$k<m+m'$.
This implies
\[\mathbf e_k^\top U(a,\theta,\mathsf v)A=\biggl(\sum_{m\leq j\leq n}a_j\mathbf e_{k-j}^\top\biggr)A
=h\sum_{m\leq j\leq n} a_j\mathbf e_{k-j}^\top=h\mathbf e_k^\top U(a,\theta,\mathsf w),\]
from which $\mathbf e_k^\top U(a,\theta,\mathsf v)AU(a,\theta,\mathsf w)^{-1}=h\mathbf e_k^\top=
\mathbf e_k^\top\Lambda(\mathsf s,\mathsf Q)$.
This latter equality which holds for appropriate $k$ is exactly the statement
of having an $L(m+m',n+n')$-perturbation.
The $R$ case is similar.
\end{proof}
\end{lemma}
\begin{point}
Now, let $F=\{(m_j,n_j)\}_{1\leq j\leq s}$ be a finiteness type and $\tilde a=(a_s,\ldots, a_1)\in\mathfrak A_F$.
Let $a=a_s\ldots a_1$.
Set $M_k=m_1+\ldots+m_k$, $N_k=n_1+\ldots+n_k$.

a.) Let us define $H_0(\tilde a,\theta,\mathsf v)=\Lambda(\mathsf v,\mathsf Q)$,
which is an $(0,0)$-perturbation of $\Lambda(\mathsf v,\mathsf Q)$. Then set
\[\hat H_k(\tilde a,\theta,\mathsf v)=
U(a_k,\theta,\mathsf v)H_{k-1}(\tilde a,\theta,\mathsf v)U(a_k,\theta,1)^{-1}\]
and
\[H_{k}(\tilde a,\theta,\mathsf v)= \Red_{(M_k, N_k)}(\hat H_k(\tilde a,\theta,\mathsf v)). \]
According to Lemma \ref{lem:looppert} these matrices $\hat H_k$, $H_k$  are well-defined;
$\hat H_k$ is always an $L(M_k, N_k)$ or $R(M_k, N_k)$-perturbation
of $\Lambda(\mathsf v,\mathsf Q)$, while  $H_k$ is  an $(M_k, N_k)$-perturbation
of $\Lambda(\mathsf v,\mathsf Q)$.

b.) Let us define
\begin{multline}
U_F(\tilde a,\theta,h,\mathsf v)=
C_{\hat H_s(\tilde a,\theta,\mathsf v), H_s(\tilde a,\theta,\mathsf v)}(h)
C_{\hat H_s(\tilde a,\theta,-1), H_s(\tilde a,\theta,-1)}(h/2)^{-1}\notag\\\hfill
\hat H_s(\tilde a,\theta,-1)\hat H_s(\tilde a,\theta,\mathsf v)^{-1} U(a_s,\theta,\mathsf v)\ldots
\\\ldots
C_{\hat H_1(\tilde a,\theta,\mathsf v), H_1(\tilde a,\theta,\mathsf v)}(h)
C_{\hat H_1(\tilde a,\theta,-1), H_1(\tilde a,\theta,-1)}(h/2)^{-1}\\\hfill
\hat H_1(\tilde a,\theta,-1)\hat H_1(\tilde a,\theta,\mathsf v)^{-1} U(a_1,\theta,\mathsf v).
\end{multline}
\end{point}
\begin{point}
In order to get the next proposition in a more compact form we make some additional definitions:

a.) We  define series of involutions by $Q_0(\tilde a)=\mathsf Q$,
\[\hat Q_k(\tilde a)=\mathsf U(a_k)Q_{k-1}(\tilde a)\mathsf U(a_k)^{-1}
\qquad\text{and}\qquad Q_{k}(\tilde a)= \Red_{(M_k, N_k)}(\hat Q_k(\tilde a)). \]
One may immediately notice that $Q_{k}(\tilde a)=H_k(\tilde a,0,-1)$.
We set
\[\mathsf B_F(\tilde a)=Q_s(\tilde a).\]

Consider the involutions
$\tilde Q_k(\tilde a)=\mathsf U(a_s\ldots a_{k+1})Q_k(\tilde a) \mathsf U(a_s\ldots a_{k+1})^{-1}$.
Notice that $\tilde Q_0(\tilde a)=\mathsf B(a)$, $\tilde Q_s(\tilde a)=\mathsf B_F(\tilde a)$.

b.) We define
\[\mathsf U_F(\tilde a,h)=C_{\hat Q_s(\tilde a),Q_s(\tilde a)}(h/2)\hat Q_s(\tilde a)\mathsf U(a_s)
\ldots C_{\hat Q_1(\tilde a), Q_1(\tilde a)}(h/2)\hat Q_1(\tilde a) \mathsf U(a_1).\]
We see that
\[ \mathsf U_F(\tilde a,h)=C_{\tilde Q_{s-1}(\tilde a),\tilde Q_s(\tilde a)}(h/2)\tilde Q_{s-1}(\tilde a)
\ldots C_{\tilde Q_0(\tilde a), \tilde Q_1(\tilde a)}(h/2)\tilde Q_{0}(\tilde a)\mathsf U(a).\]
In particular, $h=0$ yields $\mathsf U_F(\tilde a,0)=\mathsf U(a)$,   but  $h=1$  yields
\[\mathsf U_F(\tilde a,1)=\biggl(\frac{1+\tilde Q_s(\tilde a)\tilde Q_{s-1}(\tilde a)}2\biggr)
\ldots\biggl(\frac{1+\tilde Q_1(\tilde a)\tilde Q_{0}(\tilde a)}2\biggr)\mathsf U(a).\]
\end{point}
\begin{lemma}\label{lem:caplemma} It yields

o.) $H_k(\tilde a,\theta,1)=1$,

i.) $U_F(\tilde a,\theta,0,\mathsf v)=U(a,\theta,\mathsf v)$,

ii.) $U_F(\tilde a,\theta,1,\mathsf v)=H_s(\tilde a,\theta,\mathsf v)
U_F(\tilde a,\theta,1,1)\Lambda(\mathsf v,\mathsf Q)^{-1}$,

iii.) $U_F(\tilde a,\pi/2,h,\mathsf v)=U(a,\pi/2,\mathsf v)$,

iv.) $U_F(\tilde a,0,h,\mathsf v)=\mathsf U_F(\tilde a,h)$.
\begin{proof}
o.) By induction $H_k(\tilde a,\theta,1)=\hat H_k(\tilde a,\theta,1)=1$ is clear.

i.) We see that $C_{\hat H_k,H_k}(0/2)\hat H_k^{-1}=1$, and
$C_{\hat H_k,H_k}(0)\hat H_k^{-1}=1$, so
$U_F(\tilde a, \theta,0,\mathsf v)$ naturally gets simplified to
$U(a_s,\theta,\mathsf v)\ldots U(a_1,\theta,\mathsf v)=U(a,\theta,\mathsf v)$.

 ii.) If $h=1$ then we have the identity
\[C_{\hat H_k(\tilde a,\theta,\mathsf v), H_k(\tilde a,\theta,\mathsf v)}(1)
C_{\hat H_k(\tilde a,\theta,-1), H_k(\tilde a,\theta,-1)}(1/2)^{-1}
\hat H_k(\tilde a,\theta,-1)\hat H_k(\tilde a,\theta,\mathsf v)^{-1} U(a_k,\theta,\mathsf v)\]
\[=H_k(\tilde a,\theta,\mathsf v)
C_{\hat H_k(\tilde a,\theta,-1), H_k(\tilde a,\theta,-1)}(1/2)^{-1}
\hat H_k(\tilde a,\theta,-1)U(a_k,\theta,1) H_{k-1}(\tilde a,\theta,\mathsf v)^{-1}. \]
Multiplying these terms for all $k$, and taking (o) into account, we obtain (ii).

iii.) If $\theta=\pi/2$ then all our matrices split into two independent blocks.
One is the $\{0\}\times \{0\}$ block, the other is the
$(\mathbb Z\setminus \{0\})\times (\mathbb Z\setminus \{0\})$ block.
The $\{0\}\times \{0\}$ block is entirely unaffected by reduction.
The other block can be reindexed into $\mathbb Z\times \mathbb Z$ by pushing down positive indices by $1$.
Let us denote this reindexed block conjugated by $\mathsf Q$ by $\#$. Then
\[U(a_k,\pi/2,\mathsf v)^\#=\Lambda(\mathsf v,\mathsf Q)\mathsf U(a_k)\Lambda(\mathsf v,\mathsf Q)^{-1}\]
(cf. Lemma \ref{lem:linearkey}), and, of course $\Lambda(\mathsf v,\mathsf Q)^\#=\Lambda(\mathsf v,\mathsf Q)$.
Then,
\begin{multline}
\hat H_1(\tilde a,\pi/2)^\#=U(a_k,\pi/2,\mathsf v)^\#\Lambda(\mathsf v,\mathsf Q)^\#
(U(a_k,\pi/2,1)^{-1})^\#=\notag\\
=\left(\Lambda(\mathsf v,\mathsf Q)\mathsf U(a_1)
\Lambda(\mathsf v,\mathsf Q)^{-1}\right)\Lambda(\mathsf v,\mathsf Q)
\mathsf U(a_1)^{-1}=\Lambda(\mathsf v,\mathsf Q),
\end{multline}
and so $H_1(\tilde a,\pi/2)=\Lambda(\mathsf v,\mathsf Q)$. In that way we see that all the $\hat H_k(\tilde a,\pi/2)^\#$
and $H_k(\tilde a,\pi/2)^\#$ are all equal to $\Lambda(\mathsf v,\mathsf Q)$.
So we can see that no reduction happens ever at all.
But then $C_{\hat H_k,H_k}(0/2)\hat H_k^{-1}=1$, $C_{\hat H_k,H_k}(0)\hat H_k^{-1}=1$, and so
\[U_F(\tilde a,\pi/2,h,\mathsf v)=
U(a_s,\pi/2,\mathsf v)\ldots U(a_1,\pi/2,\mathsf v)=U(a,\pi/2,\mathsf v).\]

iv.) $U(a_k,0,\mathsf v)=\mathsf U(a_k)$.
Then $H_k=\Red_{(M_k,N_k)}( \mathsf U(a_k)  H_{k-1} \mathsf U(a_k)^{-1} )$,
which shows by induction that all that $\hat H_k$ and $H_k$ must be linear loops generated by involutions.
So, $\hat H_k(\tilde a,0,\mathsf v)=\Lambda(\mathsf v,\hat Q_k(\tilde a))$,
$H_k(\tilde a,0,\mathsf v)=\Lambda(\mathsf v,Q_k(\tilde a))$.
Then
\begin{multline}\notag
C_{\hat H_k(\tilde a,0,\mathsf v), H_k(\tilde a,0,\mathsf v)}(h)=
C_{\Lambda(\mathsf v,\hat Q_k(\tilde a)), \Lambda(\mathsf v,Q_k(\tilde a))}(h)=
\Lambda(\mathsf v,C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h))=\\
=\Lambda(\mathsf v,C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h/2)\hat Q_k(\tilde a)
\hat Q_k(\tilde a)\hat Q_k(\tilde a)C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h/2))=\\
=C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h/2)\hat Q_k(\tilde a)
\Lambda(\mathsf v,\hat Q_k(\tilde a))\hat Q_k(\tilde a)C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h/2)=\\
=C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h/2)\hat Q_k(\tilde a)
H_k(\tilde a,0,\mathsf v)\hat H_k(\tilde a,0,-1)
C_{\hat H_k(\tilde a,0,-1), H_k(\tilde a,0,-1)}(h/2).
\end{multline}
This implies
\begin{multline}\notag
C_{\hat H_k(\tilde a,0,\mathsf v), H_k(\tilde a,0,\mathsf v)}(h)
C_{\hat H_k(\tilde a,0,-1), H_k(\tilde a,0,-1)}(h/2)^{-1}
\cdot\\\cdot
\hat H_k(\tilde a,0,-1)\hat H_k(\tilde a,0,\mathsf v)^{-1} U(a_k,0,\mathsf v)=
C_{\hat Q_k(\tilde a), Q_k(\tilde a)}(h/2)\hat Q_k(\tilde a)\mathsf U(a_k).
\end{multline}
Multiplying these identities (iv) is immediate.
\end{proof}
\end{lemma}
\begin{prop}[$\rightarrow$ Statement \ref{stat:subfinite}]
The continuous map
\[K_F:\mathfrak A_F\times [0,\pi/2]\times[0,1]\rightarrow \unit(\mathcal K^{-\infty}_{\mathbb Z}
(\mathfrak A)[\mathsf v^{-1},\mathsf v]^{\infty,\mathrm{po}})\]
defined by
\[K_F(\tilde a,\theta,h)=\Lambda(\mathsf v,\mathsf Q)^{-1}U_F(\tilde a,\theta,h,\mathsf v)
\Lambda(\mathsf v,\mathsf Q) U_F(\tilde a,\theta,h,1)^{-1}\]
is smooth in $[0,\pi/2]\times[0,1]$, and it has the following properties:

i.) $K_F(\tilde a,\theta,0)=K(a,\theta)$;

ii.) $K_F(\tilde a,\theta,1)$ is equal to
\begin{multline}\notag
\Lambda(\mathsf v,\mathsf Q)^{-1}
\Red_{(M_s,N_s)}(U(a_s,\theta,\mathsf v)\ldots \Red_{(M_1,N_1)}(U(a_1,\theta,\mathsf v)\\
\Lambda(\mathsf v,\mathsf Q)U(a_1,\theta,1)^{-1})\ldots U(a_s,\theta,1)^{-1}),
\end{multline}
which is an $(M_s,N_s)$-perturbation of $1_{\mathbb Z}$;

iii.) $K_F(\tilde a,\pi/2,h)=\mathsf E_{\mathbb Z}(a(\mathsf z)a(1)^{-1})$ is constant;

iv.)
$K_F(\tilde a,0,h)=\Lambda(\mathsf v,\mathsf Q)^{-1}{\mathsf U}_{F}(\tilde a,h)\Lambda(\mathsf v,\mathsf Q)
{\mathsf U}_{F}(\tilde a,h)^{-1}$.
Here ${\mathsf U}_{F}(\tilde a,h)$ differs from $\mathsf U(a)$ in
a rapidly decreasing matrix and ${\mathsf U}_{F}(\tilde a,h)=\mathsf U(a)$.

This yields:

The linearizing homotopy $K(a,\theta)$  of  $a(\mathsf v)a(1)^{-1}$
can be deformed into
the finite perturbation linearizing homotopy $K_F(\tilde a,\theta,1)$  of  $a(\mathsf v)a(1)^{-1}$.
The Bott involution $\mathsf B(a)=\mathsf U(a)\mathsf Q\mathsf U(a)^{-1}$
gets replaced by the involution
\begin{multline}\notag
\mathsf B_F(\tilde a)=\mathsf U_F(\tilde a,1)\mathsf U(a)^{-1}\mathsf B(a)
(\mathsf U_F(\tilde a,1)\mathsf U(a)^{-1})^{-1}=\\
=\Red_{(M_s,N_s)}(\mathsf U(a_s)\ldots \Red_{(M_1,N_1)}(\mathsf U(a_1)\mathsf Q
\mathsf U(a_1)^{-1})\ldots    \mathsf U(a_s)^{-1}),
\end{multline}
which is an $(M_s,N_s)$-perturbation of $\mathsf Q$.
\begin{proof}
This follows from Lemma \ref{lem:caplemma}.
\end{proof}
\end{prop}

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math0506059.extracted_bib
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