Linear orthogonality preservers of Hilbert $C^{\ast }$-modules over $C^{\ast }$-algebras with real rank zero

Abstract

Let $A$ be a $C^{\ast }$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $\theta :E\to F$ is a linear map preserving orthogonality, i.e., $<\theta (x),\theta (y)>=0$ whenever $<x,y>=0$. We show in this article that if, in addition, $A$ has real rank zero, and $\theta$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $u\in M(A)$ such that $<\theta (x),\theta (y)>=u<x,y>$ ($x,y\in E$). In the case when $A$ is a standard $C^{\ast }$-algebra, or when $A$ is a $W^{\ast }$-algebra containing no finite type II direct summand, we also obtain the same conclusion with the assumption of $\theta$ being an $A$-module map weakened to being a local map.

9 pages

References in corpus (1)

• Linear orthogonality preservers of Hilbert bundles

Cited by in corpus (1)

• Linear orthogonality preservers of Hilbert $C^{\ast }$-modules over general $C^{\ast }$-algebras