Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Socle and finite dimensionality of some Banach algebras   The purpose of this note is to describe some algebraic conditions on a Banach
algebra which force it to be finite dimensional. One of the main results in
Theorem~2 which states that for a locally compact group $G$, $G$ is compact if
there exists a measure $\mu$ in $\hbox{Soc}(L^{1}(G))$ such that $\mu(G) \neq
0$. We also prove that $G$ is finite if $\hbox{Soc}(M(G))$ is closed and every
nonzero left ideal in $M(G)$ contains a minimal left ideal.

21
The categories of this paper are: adap-org
4
The categories of this paper are: cond-mat
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
36
The categories of this paper are: cmp-lg
19
The categories of this paper are: eess
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn