[Paper Review] Generalized fixed point algebras for coactions of locally compact quantum groups

Alcides Buss|arXiv (Cornell University)|Nov 14, 2012

Advanced Operator Algebra Research4 citations

TL;DR

This paper generalizes Rieffel's theory of generalized fixed point algebras to coactions of locally compact quantum groups in the sense of Kustermans and Vaes, using Hilbert module theory and introducing a noncommutative analogue of continuous square-integrability. The key result establishes a Morita equivalence between the generalized fixed point algebra and an ideal in the reduced crossed product, with the fixed point algebra realized as the compact operators on a Hilbert module over the crossed product.

ABSTRACT

We extend the construction of generalized fixed point algebras to the setting of locally compact quantum groups - in the sense of Kustermans and Vaes - following the treatment of Marc Rieffel, Ruy Exel and Ralf Meyer in the group case. We mainly follow Meyer's approach analyzing the constructions in the realm of equivariant Hilbert modules. We generalize the notion of continuous square-integrability, which is exactly what one needs in order to define generalized fixed point algebras. As in the group case, we prove that there is a correspondence between continuously square-integrable Hilbert modules over an equivariant C*-algebra B and Hilbert modules over the reduced crossed product of B by the underlying quantum group. The generalized fixed point algebra always appears as the algebra of compact operators of the associated Hilbert module over the reduced crossed product.

Motivation & Objective

• To extend Rieffel's theory of generalized fixed point algebras from group actions to coactions of locally compact quantum groups.
• To generalize the notion of continuous square-integrability to the quantum group setting, enabling the construction of fixed point algebras in noncommutative dynamics.
• To establish a correspondence between continuously square-integrable Hilbert modules over a $C^*$-algebra with quantum group coaction and Hilbert modules over the reduced crossed product.
• To show that the generalized fixed point algebra arises naturally as the algebra of compact operators on the associated Hilbert module over the reduced crossed product.

Proposed method

• Adopts Meyer's framework of equivariant Hilbert modules to analyze coactions of locally compact quantum groups.
• Introduces a noncommutative generalization of continuous square-integrability for Hilbert modules over coacted $C^*$-algebras.
• Uses the generalized fixed point algebra construction via averaging along the coaction, modeled on Rieffel's approach in the group case.
• Applies the imprimitivity bimodule construction to link the fixed point algebra to an ideal in the reduced crossed product $A \rtimes_r \mathcal{G}$.
• Employs the canonical corepresentation $W$ of the quantum group and its associated operators to define and analyze the Hilbert module structure.
• Applies Cohen's factorization theorem and properties of the dual coaction to prove that saturated, s-complete subspaces yield full Morita equivalence.

Experimental results

Research questions

• RQ1 Can the theory of generalized fixed point algebras be extended from group actions to coactions of locally compact quantum groups?
• RQ2 What is the appropriate noncommutative analogue of continuous square-integrability in the context of quantum group coactions?
• RQ3 How is the generalized fixed point algebra related to the reduced crossed product in the quantum group setting?
• RQ4 Under what conditions does the generalized fixed point algebra arise as the compact operators on a Hilbert module over the crossed product?
• RQ5 Is there a characterization of freeness (saturation) in terms of the structure of the Hilbert module and the coaction?

Key findings

• The generalized fixed point algebra associated to a continuously square-integrable Hilbert module over a $C^*$-algebra with a coaction of a locally compact quantum group is isomorphic to the algebra of compact operators on a Hilbert module over the reduced crossed product.
• There is a Morita equivalence between the generalized fixed point algebra and an ideal in the reduced crossed product, induced by the completion of a dense subspace of square-integrable elements.
• The construction generalizes Rieffel's original framework from group actions to quantum group coactions, preserving the core duality between fixed point algebras and crossed products.
• Saturated, s-complete, relatively continuous subspaces of the Hilbert module correspond precisely to free coactions, generalizing the notion of freeness in noncommutative dynamics.
• The coaction of a quantum group on itself via its comultiplication is proper and free if and only if the quantum group is regular, providing a precise characterization of non-regular examples as non-proper.
• The Hilbert module $\mathcal{F}(\mathcal{G}, \mathcal{R})$ associated to a dense, s-complete, relatively continuous subspace $\mathcal{R}_0 \subseteq \mathcal{G}_{\mathrm{si}}$ is isomorphic to $L^2(\mathcal{G})^*$ as a Hilbert module over $\mathcal{K}(L^2(\mathcal{G}))$, establishing a canonical duality.