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

Abstract

A real extension $\bar{\mathcal{A}}$ of a complex $C\ast$-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 $\bar{\mathcal{A}}{\ast }_n{\bar{\mathcal{A}}}_n$ can be split in two special parts. This property has been earlier known for sequences of linear operators.