Linear maps preserving regularity in C∗-algebras

Abstract

Let $A$ and $B$ be unital $C^*$-algebras such that at least one of them is of real rank zero. We investigate surjective linear maps from $A$ to $B$ preserving the conorm, the (von Neumann) regularity, the generalized spectrum, and their essential versions. As a consequence, we recover results of Mbekhta, and Mbekhta and Šemrl for $\mathcal L(H)$ when $H$ is an infinite-dimensional complex Hilbert space.