Rank-preserving module maps

Abstract

In this paper, we characterize rank one preserving module maps on a Hilbert $C^\ast-$module and study its applications on free probability theory

Additive preserving rank one maps on Hilbert $C^\ast$-modules

“…The following Lemma 5, which will be used frequently, has been obtained in [8]. Nevertheless we give its proof for the sake of completeness.…”

“…Definition 1. [8] Let ε be a orthonormal basis in H A . x = 0 in H A will be called coordinatly invertible if for all e ∈ ε, e, x is invertible in A unless e, x = 0.…”

“…In our study of free probability theory, we find that module maps on Hilbert C * −module preserving certain properties are also important [6,7], and thus the study of the modules preserving problem becomes attractive. In [8], a complete description of modules maps which preserving rank one modular operators is given. Naturally in the present paper we consider additive maps on modules preserving rank one, which much more complicated than the module maps case.…”

“…[8] Supposing y ∈ CI and θ x,y = 0 then x = 0.Lemma 3. [8] Let M be Hilbert A−modules, where A is a unital C * −algebra, and let φ, σ : M → A be A−linear operators.…”