On the Covariance of Moore-Penrose Inverses in Rings with Involution

Hesam Mahzoon

doi:10.1155/2014/309708

Abstract

We consider the so-called covariance set of Moore-Penrose inverses in rings with an involution. We deduce some new results concerning covariance set. We will show that ifais a regular element in aC∗-algebra, then the covariance set ofais closed in the set of invertible elements (with relative topology) ofC∗-algebra and is a cone in theC∗-algebra.

Highlights

Suppose that R is a ring with unity 1 ≠ 0
An element a ∈ R is called regular if it has a generalized inverse in R; that is, there exists b ∈ R such that aba = a
We show that if {an} is a sequence of MP-invertible elements of a C∗algebra such that their MP-inverses norm is bounded and an converges to a, there is some kind of convergence of C(an) to C(a)

Summary

Introduction

Suppose that R is a ring with unity 1 ≠ 0. An element a ∈ R is called regular if it has a generalized inverse (in the sense of von Neumann) in R; that is, there exists b ∈ R such that aba = a. Note that such b is not unique [1, 2]. (i) a is called Moore-Penrose invertible if there exists b ∈ R such that aba = a, bab = b,. (ii) a is called Drazin invertible if there exists b ∈ R such that bab = b, ab = ba, ak+1b = ak (4).

Abstract and Applied Analysis

Now if we set y