Continuity of ring *-homomorphisms between C*-algebras

TL;DR

This paper proves that unital *-preserving ring homomorphisms between unital C*-algebras are contractive, even without assuming linearity, and characterizes their form, providing insights for both C*-algebraists and algebraists.

Contribution

It establishes the contractiveness of unital *-preserving ring homomorphisms between C*-algebras without assuming linearity and characterizes their structure.

Findings

Unital *-preserving ring homomorphisms are contractive.

Characterization of the form of such homomorphisms.

Results applicable to noncommutative rings and algebras.

Abstract

The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $\phi : A \to B$ is a unital *-preserving ring homomorphism, then $\phi$ is contractive; i.e., $\| \phi (a) \| \leq \| a \|$ for all $a \in A$. (Note that we do not assume $\phi$ is linear.) We use this result to deduce a number of corollaries as well as characterize the form of such unital *-preserving ring homomorphisms. (This note may be of interest to C*-algebraists as well as algebraists who study noncommutative rings and algebras. It is meant to be accessible to a general mathematician and does not require any prior knowledge of C*-algebras.)