Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Gelfand theory for non-commutative Banach algebras   Let $A$ be a Banach algebra. We call a pair $(G, B)$ a Gelfand theory for $A$
if the following axioms are satisfied: (G 1) $B$ is a $C^\ast$-algebra, and $G
: A \to B$ is a homomorphism; (G 2) the assignment $L \mapsto G^{-1}(L)$ is a
bijection between the sets of maximal modularleft ideals of $B$ and $A$,
respectively; (G 3) for each maximal modular left ideal $L$ of $B$, the linear
map $G_L : A / G^{-1}(L) \to B /L $ induced by $B$ has dense range. The Gelfand
theory of a commutative Banach algebra is easily seen to be characterized by
these axioms. Gelfand theories of arbitrary Banach algebras enjoy many of the
properties of commutative Gelfand theory. We show that unital, homogeneous
Banach algebras always have a Gelfand theory. For liminal $C^\ast$-algebras
with discrete spectrum, we show that the identity is the only Gelfand theory
(up to an appropriate notion of equivalence).

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The categories of this paper are: hep-th
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
37
The categories of this paper are: supr-con
20
The categories of this paper are: acc-phys
26
The categories of this paper are: bayes-an
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The categories of this paper are: astro-ph
31
The categories of this paper are: ao-sci
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
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The categories of this paper are: hep-lat