Continuity of ring *-homomorphisms between C*-algebras
Abstract
Description
The purpose of this short note is to prove that if $A$ and $B$ are unital C*-algebras and $ϕ: A \to B$ is a unital *-preserving ring homomorphism, then $ϕ$ is contractive; i.e., $\| ϕ(a) \| \leq \| a \|$ for all $a \in A$. (Note that we do not assume $ϕ$ 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.)
7 pages, Version IV changes: Some small typos corrected. This is the final version, to appear. Version III changes: Proposition 3.9 is strengthened, and an alternate proof of Theorem 3.6 is described in the Acknowledgements. Version II changes: A few comments added
7 pages, Version IV changes: Some small typos corrected. This is the final version, to appear. Version III changes: Proposition 3.9 is strengthened, and an alternate proof of Theorem 3.6 is described in the Acknowledgements. Version II changes: A few comments added