# On Real and Complex Spectra in some Real C*-Algebras and Applications

### Victor D. Didenko

University of Brunei Darussalam, Brunei Darussalam### Bernd Silbermann

Technische Universität Chemnitz, Germany

## Abstract

A real extension $Aˉ$ of a complex $C∗$-algebra $A$ by some element $m$ which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on $Aˉ$. After then we are able to study Moore-Penrose invertibility in $Aˉ$. Because this notion strongly depends on the element $m$, we study under what conditions different elements m produce just the same involution on $Aˉ$.It is shown that the set of all additive continuous operators $L_{add}(H)$ acting in a complex Hilbert space $H$ possesses unique involution only (in the sense defined below). In addition, we consider some properties of the real and complex spectra of elements belonging to $Aˉ$, and show that whenever an operator sequence ${Aˉ_{n}}⊂L_{add}(H)$ is weakly asymptotically Moore-Penrose invertible, then the real spectrum of $Aˉ∗_{n}Aˉ_{n}$ can be split in two special parts. This property has been earlier known for sequences of linear operators.

## Cite this article

Victor D. Didenko, Bernd Silbermann, On Real and Complex Spectra in some Real C*-Algebras and Applications. Z. Anal. Anwend. 18 (1999), no. 3, pp. 669–686

DOI 10.4171/ZAA/905