Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Abstract

Let A and B be C*-algebras. A linear map T : A → B is said to be a*-homomorphism at an element z ∈ A if ab*= z in A implies T(ab*) = T(a) T(b)*= T(z), and c*d = z in A gives T(c * d) = T(c) * T(d) = T(z): Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a* -homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C* -algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a *-homomorphism at p and at 1 - p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.

Mathematics Subject Classification (2010): 47B49, 46L05, 46L40, 46T20, 47L99.

Keywords: Multiplicative mapping at a point, Hua's theorem, linear preservers,*-homomorphism at a point