Generalized Fixed-Point Algebras for Twisted C*-Dynamical Systems

Abstract

In his paper “Generalized Fixed Point Algebras and Square-Integrable Group Actions,” Ralf Meyer showed how to construct generalized fixed-point algebras for C*-dynamical systems via their square-integrable representations on Hilbert C*-modules. His method extends Marc Rieffel’s construction of generalized fixed-point algebras from proper group actions. This dissertation seeks to generalize Meyer’s work to construct generalized fixed-point algebras for twisted C*-dynamical systems. To accomplish this, we must introduce some brand-new concepts, the foremost being that of a twisted Hilbert C*-module. A twisted Hilbert C*-module is basically a Hilbert C*-module equipped with a twisted group action that is compatible with the module’s right C*-algebra action and its C*-algebra-valued inner product. Twisted Hilbert C*-modules form a category, where morphisms are twisted-equivariant adjointable operators, and we will establish that Meyer’s bra-ket operators are morphisms between certain objects in this category. A by-product of our work is a twisted-equivariant version of Kasparov’s Stabilization Theorem, which states that every countably generated twisted Hilbert C*-module is isomorphic to an invariant orthogonal summand of the countable direct sum of a standard one if and only if the module is square-integrable. Given a twisted C*-dynamical system, we provide a definition of a relatively continuous subspace of a twisted Hilbert C*-module (inspired by Ruy Exel) and then prescribe a new method of constructing generalized fixed-point algebras that are Morita-Rieffel equivalent to an ideal of the corresponding reduced twisted crossed product. Our construction 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 and a relatively continuous subspace.