Linear and multiplicative maps under spectral conditions
B. Amin,
R. Golla Indian Institute of Technology Hyderabad
Abstract:
The multiplicative version of the Gleason–Kahane–Żelazko theorem for
$C^*$-algebras given by
Brits et al. in [4] is extended to maps from
$C^*$-algebras to commutative semisimple Banach algebras.
In particular, it is proved that if a multiplicative
map
$\phi$ from a
$C^*$-algebra
$\mathcal{U}$ to a commutative semisimple
Banach algebra
$\mathcal{V}$ is continuous on the set of all noninvertible
elements of
$\mathcal{U}$ and
$\sigma(\phi(a)) \subseteq \sigma(a)$ for any
$a \in
\mathcal{U}$, then
$\phi$ is a linear map.
The multiplicative variation of the Kowalski–Słodkowski theorem
given by Touré et al. in
[14] is also generalized. Specifically, if
$\phi$ is a continuous map
from a
$C^*$-algebra
$\mathcal{U}$ to a commutative
semisimple Banach algebra
$\mathcal{V}$ satisfying the conditions
$\phi(1_\mathcal{U})=1_\mathcal{V}$ and
$\sigma(\phi(x)\phi(y)) \subseteq \sigma(xy)$ for all
$x,y \in \mathcal{U}$,
then
$\phi$ generates a linear
multiplicative map
$\gamma_\phi$ on
$\mathcal{U}$ which coincides with
$\phi$ on the principal component of the
invertible group of
$\mathcal{U}$. If
$\mathcal{U}$ is a Banach algebra such that each element of
$\mathcal{U}$
has totally disconnected spectrum, then
the map
$\phi$ itself is linear and multiplicative
on
$\mathcal{U}$. It is shown that a similar statement is valid for a map with
semisimple domain under a stricter spectral condition. Examples
which demonstrate that some hypothesis
in the results cannot be discarded.
Keywords:
Banach algebra, $C^*$-algebra, multiplicative map, linear map, semisimple algebra, spectrum, radical, GKŻ
theorem. Received: 23.06.2022
Revised: 18.03.2023
Accepted: 03.04.2023
DOI:
10.4213/faa4026
Fulltext:
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