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

Abstract

A real extension $\bar{\mathcal A}$ of a complex $C*$-algebra $\mathcal A$ 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 $\bar{\mathcal A}$. After then we are able to study Moore-Penrose invertibility in $\bar{\mathcal A}$. Because this notion strongly depends on the element $m$, we study under what conditions different elements m produce just the same involution on $\bar{\mathcal A}$.It is shown that the set of all additive continuous operators $\mathcal L_{add}(\mathcal H)$ acting in a complex Hilbert space $\mathcal H$ 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 $\bar{\mathcal A}$, and show that whenever an operator sequence $\{ \bar{\mathcal A}_n \} \subset \mathcal L_{add}(\mathcal H)$ is weakly asymptotically Moore-Penrose invertible, then the real spectrum of $\bar{\mathcal A}*_n \bar{\mathcal A}_n$ can be split in two special parts. This property has been earlier known for sequences of linear operators.