Krein C*-categories

TL;DR

This paper extends the theory of C*-categories to Krein C*-categories, which are based on indefinite inner product spaces, and establishes a Gel'fand-Naimark representation theorem for them.

Contribution

It introduces axioms for Krein C*-categories and provides foundational examples and a representation theorem, broadening the scope of operator algebra theory.

Findings

Defined axioms for Krein C*-categories

Provided basic examples of Krein C*-categories

Established a Gel'fand-Naimark representation theorem

Abstract

C*-categories are essentially norm-closed *-categories of bounded linear operators between Hilbert spaces. The purpose of this work is to identify suitable axioms defining Krein C*-categories, i.e. those categories that play the role of C*-categories whenever Hilbert spaces are replaced by more general indefinite inner product Krein spaces, and provide some basic examples. Finally we provide a Gel'fand-Naimark representation theorem for Krein C*-algebras and Krein C*-categories.