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 elementAssuming 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.

Linear maps preserving products equal to primitive idempotents of an incidence algebra

“…zero product preservers with respect to [−, −]), see [14,15]. In the setting of C * -algebras the problem of studying linear mappings that are * -homomorphisms at a fixed point has been considered in [3].…”

Linear Maps Which are Anti-derivable at Zero

“…We find in this way a natural link with the results on zero products preservers (see, for example, [1,2,8,10,28,29,32,33,[47][48][49][50][51] for additional details and results). Burgos, Cabello-Sánchez and the third author of this note explore in [6] those linear maps between C * -algebras which are * -homomorphisms at certain points of the domain, for example, at the unit element or at zero. We refer to [12,22,25,[52][53][54][55][56][57][58] and [60] for additional related results.…”