Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Determining elements in Banach algebras through spectral properties   Let $A$ be a Banach algebra. By $\sigma(x)$ and $r(x)$ we denote the spectrum
and the spectral radius of $x\in A$, respectively. We consider the relationship
between elements $a,b\in A$ that satisfy one of the following two conditions:
(1) $\sigma(ax) = \sigma(bx)$ for all $x\in A$, (2) $r(ax) \le r(bx)$ for all
$x\in A$. In particular we show that (1) implies $a=b$ if $A$ is a
$C^*$-algebra, and (2) implies $a\in \mathbb C b$ if $A$ is a prime
$C^*$-algebra. As an application of the results concerning the conditions (1)
and (2) we obtain some spectral characterizations of multiplicative maps.

5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
23
The categories of this paper are: solv-int
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
20
The categories of this paper are: acc-phys
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg