Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Asymptotically unitary equivalence and asymptotically inner
  automorphisms   Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA
with tracial rank zero. Suppose that $\phi_1, \phi_2: C\to A$ are two unital
monomorphisms. We show that there is a continuous path of unitaries $\{u_t:
t\in [0, \infty)\}$ of $A$ such that $$
\lim_{t\to\infty}u_t^*\phi_1(a)u_t=\phi_2(a)\tforal a\in C $$ if and only if
$[\phi_1]=[\phi_2]$ in $KK(C,A),$ $\tau\circ \phi_1=\tau\circ \phi_2$ for all
$\tau\in T(A)$ and the rotation map ${\tilde\eta}_{\phi_1,\phi_2}$ associated
with $\phi_1$ and $\phi_2$ is zero. In particular, an automorphism $\af$ on a
unital separable simple \CA $A$ in ${\cal N}$ with tracial rank zero is
asymptotically inner if and only if $$ [\af]=[{\rm id}_A] \text{in} KK(A,A) $$
and the rotation map ${\tilde\eta}_{\phi_1, \phi_2}$ is zero.
  Let $A$ be a unital AH-algebra (not necessarily simple) and let $\af\in
Aut(A)$ be an automorphism. As an application, we show that the associated
crossed product $A\rtimes_{\af}\Z$ can be embedded into a unital simple
AF-algebra if and only if $A$ admits a strictly positive $\af$-invariant
tracial state.

8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
31
The categories of this paper are: ao-sci
32
The categories of this paper are: atom-ph
12
The categories of this paper are: nucl-th
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
27
The categories of this paper are: comp-gas
21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
26
The categories of this paper are: bayes-an
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph