Generalized Fixed-Point Algebras for Twisted $ C^{\ast} $-Dynamical Systems

TL;DR

This paper generalizes fixed-point algebra constructions to twisted $C^*$-dynamical systems, introducing twisted Hilbert modules and classifying all modules over non-commutative tori via twisted actions.

Contribution

It extends Meyer's and Rieffel's fixed-point algebra frameworks to twisted systems, defining twisted Hilbert modules and a classification approach for modules over non-commutative tori.

Findings

Constructed generalized fixed-point algebras for twisted systems.

Introduced the concept of twisted Hilbert $C^*$-modules.

Classified all Hilbert modules over non-commutative tori.

Abstract

In his seminal paper "Generalized Fixed Point Algebras and Square-Integrable Group Actions", Ralf Meyer showed how to construct generalized fixed-point algebras for $ C^{\ast} $-dynamical systems via their square-integrable representations on Hilbert $ C^{\ast} $-modules. His method extends Marc Rieffel's construction of generalized fixed-point algebras from proper group actions on $ C^{\ast} $-algebras. This dissertation seeks to generalize Meyer's work to construct generalized fixed-point algebras for twisted $ C^{\ast} $-dynamical systems. To accomplish this, we must introduce some new concepts, the foremost being that of a twisted Hilbert $ C^{\ast} $-module, which is a Hilbert $ C^{\ast} $-module equipped with a twisted group action by bijective linear isometries that is compatible with the module's right $ C^{\ast} $-algebra action and its $ C^{\ast} $-algebra-valued inner product. Twisted Hilbert $ C^{\ast} $-modules over a fixed twisted $ C^{\ast} $-dynamical system form a category, where morphisms are twisted-equivariant adjointable operators. Given a twisted $ C^{\ast} $-dynamical system, we provide a definition of a relatively continuous subspace of a twisted Hilbert $ C^{\ast} $-module (inspired by an idea due to Ruy Exel), and then prescribe a method of constructing, from such a subspace, a generalized fixed-point algebra that is Morita-Rieffel equivalent to an ideal of the corresponding reduced twisted crossed product. Our construction thus generalizes that of Meyer and, by extension, that of Rieffel. Our main result is the description of a classifying category for the class of all Hilbert modules over a reduced twisted crossed product. This implies that every Hilbert module over a $ d $-dimensional non-commutative torus can be constructed from a Hilbert space endowed with a twisted $ \mathbb{Z}^{d} $-action by unitary operators and a relatively continuous subspace.