The structure of triple homomorphisms onto prime algebras
Cheng-Kai Liu
Department of Mathematics
National Changhua University of Education
ckliu@cc.ncue.edu.tw
A well-known result of Kaup states that a linear bijection between two `tt"JB"^\ast`-triples is an isometry if and only if it is a triple isomorphism. Fundamental examples of `tt"JB"^\ast`-triples are `C^\ast`-algebras and `tt"JB"^\ast`-algebras. From the viewpoint of associative algebras, we characterize the structure of triple homomorphisms from an arbitrary `\star`-algebra onto a prime `\ast`-algebra. The analogous results for prime `C^\ast`-algebras factor von Neumann algebras and standard operator `\ast`-algebras on Hilbert spaces are also described. As an application, we show that every triple homomorphism from a Banach `\star`-algebra onto a prime semisimple idempotent Banach `\ast`-algebra or a prime `C^\ast`-algebra is automatically continuous.Keyword: Triple homomorphism, prime algebra, Banach algebra, `C^\ast`-algebra, standard operator algebra.