Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero

Details Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero

2009-10-13

arxiv.org/abs/0910.2335v1

Mathematics

Operator Algebras

9 pages

Scientific paper

Let $A$ be a $C^*$-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 $ = 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 $C^*$-algebra, or when $A$ is a $W^*$-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.

Affiliated with

Also associated with

No associations

Rating

Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Linear orthogonality preservers of Hilbert $C^*$-modules over $C^*$-algebras with real rank zero will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-640629