Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Riesz representation theorems for positive algebra homomorphisms   Let $X$ be a locally compact Hausdorff space, let $\mathrm A$ be a partially
ordered algebra, and let $T:\mathrm{C}_{\mathrm c}(X)\to \mathrm A$ be a
positive algebra homomorphism. Under conditions on $\mathrm A$ that are
satisfied in a good number of cases of practical interest, we show that $T$ is
represented by a (unique regular) measure $\mu$ on the Borel $\sigma$-algebra
of $X$ that takes it values in the positive cone of $\mathrm A$, and with the
property that $\mu(A_1\cap A_2)=\mu(A_1)\mu(A_2)$ for Borel subsets $A_1,A_2$
of $X$.
  The positive algebra homomorphism $T$ can be extended from ${\mathrm
C}_{\mathrm c}(X)$ to the accompanying $\mathcal L^1$-space of $\mu$. We show
that, quite often, this $\mathcal L^1$-space is closed under multiplication, so
that it is a Riesz algebra, and that the extended map $T:\mathcal L^1\to\mathrm
A$ is not only an algebra homomorphism, but, even when $\mathrm A$ is not a
Riesz space, also a vector lattice homomorphism in a sense that is explained in
the paper. The latter property enables one to describe images of the extended
map in terms of sequential up-downs and down-ups of the image of (the positive
cone of ) ${\mathrm C}_{\mathrm c}(X)$ when $\mathrm A$ has the countable sup
property.
  We apply the main results, which are obtained by purely order-theoretical
methods, to positive algebra homomorphisms from ${\mathrm C}_0(X)$ into the
order continuous operators on a Banach lattice, and to representations of
${\mathrm C}_0(X,\mathbb C)$ on Hilbert spaces.
  It is thus seen that, for representations on Banach lattices and on Hilbert
spaces, although situated in rather different contexts, spectral theorems can
be established that are both rooted in the same order-theoretical Riesz
representation theorem for positive algebra homomorphisms from ${\mathrm
C_\mathrm c}(X)$ into partially ordered algebras.

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The categories of this paper are: math-ph
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The categories of this paper are: nlin
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The categories of this paper are: chao-dyn
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The categories of this paper are: eess
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: chem-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: atom-ph
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The categories of this paper are: stat
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc