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

V. Didenko and B. Silbermann

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

Keywords: real $C^*$-algebras, Moore-Penrose invertibility, singular integral equations with conjugation

Classification (MSC2000): 65R20

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