A Spectral Characterization of Isomorphisms on $C^\star$-Algebras

TL;DR

This paper provides a spectral characterization of isomorphisms between $C^*$-algebras and Banach algebras, establishing conditions under which such an isomorphism exists based on spectral properties and continuity.

Contribution

It introduces a new spectral criterion for isomorphisms of $C^*$-algebras onto Banach algebras, extending previous results and clarifying the role of spectral conditions.

Findings

Spectral characterization of isomorphisms from $C^*$-algebras to Banach algebras.

Equivalence of $C^*$-algebra isomorphism to existence of a surjective spectral-preserving map.

Illustration that the spectral condition cannot be simplified to two-element products.

Abstract

Following a result of Hatori, Miura and Tagaki ([4]) we give here a spectral characterization of an isomorphism from a $C^\star$-algebra onto a Banach algebra. We then use this result to show that a $C^\star$-algebra $A$ is isomorphic to a Banach algebra $B$ if and only if there exists a surjective function $\phi:A\rightarrow B$ satisfying (i) $\sigma\left(\phi(x)\phi(y)\phi(z)\right)=\sigma\left(xyz\right)$ for all $x,y,z\in A$ (where $\sigma$ denotes the spectrum), and (ii) $\phi$ is continuous at $\mathbf 1$. A simple example shows that (i) cannot be relaxed to products of two elements, as is the case with commutative Banach algebras. Our results also elaborate on a paper ([3]) of Bre\v{s}ar and \v{S}penko.