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


A real extension Aˉ\bar{\mathcal A} of a complex CC*-algebra A\mathcal A by some element mm 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\bar{\mathcal A}. After then we are able to study Moore-Penrose invertibility in Aˉ\bar{\mathcal A}. Because this notion strongly depends on the element mm, we study under what conditions different elements m produce just the same involution on Aˉ\bar{\mathcal A}.It is shown that the set of all additive continuous operators Ladd(H)\mathcal L_{add}(\mathcal H) acting in a complex Hilbert space H\mathcal 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\bar{\mathcal A}, and show that whenever an operator sequence {Aˉn}Ladd(H)\{ \bar{\mathcal A}_n \} \subset \mathcal L_{add}(\mathcal H) is weakly asymptotically Moore-Penrose invertible, then the real spectrum of AˉnAˉn\bar{\mathcal A}*_n \bar{\mathcal A}_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