Spectral Theory on Commutative Krein C*-algebras

TL;DR

This paper develops a spectral theory for commutative unital Krein C*-algebras, focusing on their decomposition into even and odd parts and the role of symmetries, advancing understanding of their algebraic structure.

Contribution

It introduces a spectral framework for commutative Krein C*-algebras with symmetric bimodules and exchange symmetries, extending classical spectral theory to this setting.

Findings

Decomposition of Krein C*-algebras into even and odd parts.

Spectral theory formulated for commutative unital Krein C*-algebras.

Identification of conditions for exchange symmetries.

Abstract

A Banach involutive algebra is called a Krein C*-algebra if there is a fundamental symmetry (an involutive automorphism of period 2) such that the C*-property is satisfied when the original involution is replaced with the new one obtained by composing the automorphism with the old involution. For a given fundamental symmetry, a Krein C*-algebra decomposes as a direct sum of an even part (a C*-algebra) and an odd part (a Hilbert C*-bimodule on the even part). Our goal here is to develop a spectral theory for commutative unital Krein C*-algebras when the odd part is a symmetric imprimitivity C*-bimodule over the even part and there exists an additional suitable "exchange symmetry" between the odd and even parts.