Continuity of ring *-homomorphisms between C*-algebras

Details Continuity of ring *-homomorphisms between C*-algebras Continuity of ring *-homomorphisms between C*-algebras

2008-10-02

arxiv.org/abs/0810.0422v4

Mathematics

Operator Algebras

7 pages, Version IV changes: Some small typos corrected. This is the final version, to appear. Version III changes: Propositio

Scientific paper

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.)

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