arXiv · 0907.0618
This thesis contains the formulation and computation of quantum isometry groups.
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Jyotishman Bhowmick. 2009-07-03. Quantum isometry groups. https://arxiv.org/abs/0907.0618
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In this paper, we establish a quantum Cheeger inequality for primitive KMS-symmetric quantum Markov semigroups in terms of projection conductance. We discuss both projection conductance and classical conductance for graph-based KMS-symmetric quantum Markov semigroups. We show that hypercontractivity and the logarithmic Sobolev inequality hold for primitive KMS-symmetric quantum Markov semigroups. We also present applications of the quantum Cheeger inequality to logarithmic Sobolev inequalities, hypercontractivity, and complete modified logarithmic Sobolev inequalities.
We show that, for a second-countable locally compact Hausdorff étale minimal groupoid with compact unit space, simplicity of the reduced groupoid C*-algebra implies the existence of a comeager set of unit points with C*-simple isotropy group. Combining this result with work of Christensen and Neshveyev on exotic completions of isotropy group algebras, we show that the converse implication is also true. Finally, we construct a Hausdorff étale minimal groupoid with an isotropy group whose induced exotic completion differs from its reduced group C*-algebra, answering a question of Christensen and Neshveyev.
A three-functor formalism is the half of a six-functor formalism that supports the projection and base change formulas. In this paper, we provide a three-functor formalism for commutative von Neumann algebras and their modules. Using the Gelfand-Naimark theorem, this gives rise to a three-functor formalism for measure spaces and measurable bundles of Hilbert spaces. We use this to prove Fell absorption for unitary representations of measure groupoids. The three-functor formalism for commutative von Neumann algebras takes values in W*-categories, and we discuss in what sense it is a unitary three-functor formalism.