Takai duality and crossed product hosts in C*-actions
Yusuke Nakae
TL;DR
The work tackles extending Takai duality to crossed product hosts for singular C*-actions and develops a Landstad-algebra framework to relate hosts to conventional crossed products. It proves a universal property for full crossed product hosts and establishes a Takai-type duality, yielding the isomorphism $fcph(A,G,α)⋊ hat{α} hatG ≃ A^# ⊗ K(L^2(G))$ with the double dual action matching $α^# ⊗ Ad ρ$. For amenable groups, the full and reduced hosts coincide, and the paper introduces a reduced host and a weak Landstad algebra to realize the host as a regular crossed product. It further characterizes ground-state existence via a Landstad-like ideal condition in fcph and illustrates the theory with the case $G=\mathbb{R}$ and $A=B(H)$. Overall, the results provide a bridge between singular C*-actions and traditional crossed product theory, with implications for representation theory and states in non-continuous dynamical systems.
Abstract
Crossed product algebras are fundamental in the study of C*-algebras, traditionally under the assumption of continuity of group actions. Recent work by Grundling and Neeb introduced the crossed product host, an analog of the crossed product for a singular action. In this paper, we investigate the structure of the crossed product host and its relation to the conventional crossed product. We examine the validity of Takai-type duality in this setting and establish connections via the Landstad algebra. Additionally, we provide a necessary and sufficient condition for the existence of ground states, and show that for amenable groups, if the full and reduced crossed product hosts exist, then they coincide.