A real extension of a complex -algebra by some element which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on . After then we are able to study Moore-Penrose invertibility in . Because this notion strongly depends on the element , we study under what conditions different elements m produce just the same involution on .It is shown that the set of all additive continuous operators acting in a complex Hilbert space 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 , and show that whenever an operator sequence is weakly asymptotically Moore-Penrose invertible, then the real spectrum of 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. 669–686DOI 10.4171/ZAA/905