Strictly Outer Actions of Groups and Quantum Groups

By S. Vaes. 2002

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Abstract An action of a locally compact group or quantum group on a factor is said to be strictly outer when the relative commutant of the factor in the crossed product is trivial. We show that all locally compact quantum groups can act strictly outerly on a free Araki-Woods factor and that all locally compact groups can act strictly outerly on the hyperfinite II1 factor. We define a kind of Connes' T invariant for locally compact quantum groups and prove a link with the possibility of acting strictly outerly on a factor with a given T invariant. Necessary and sufficient conditions for the existence of strictly outer actions of compact Kac algebras on the hyperfinite II1 factor are given.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9dcd5f188076b654ec42b6d08ad3b5f705d4d23f

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In Algebra and Number Theory

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In Mathematical Physics

Citations in the corpus

Quantum permutation groups: a survey (2006, methodology): “A number of techniques coming from discrete groups are developed by Vaes, Vergnioux and their collaborators in [15], [21], [47], [48], [49], [50], [51], [52], for various types of discrete quantum groups.”
HOPF IMAGES AND INNER FAITHFUL REPRESENTATIONS (2008, background): “These notions appeared, under various degrees of generality, in a number of independent investigations: vertex models and related quantum groups [6], [11], [12], locally compact quantum groups and their outer actions [19], [26].”
The boundary of universal discrete quantum groups, exactness, and factoriality (2005, background): “Remark that the Poisson boundary has a natural interpretation as a relative commutant in the study of infinite tensor product actions [12, 21].”
Quantum automorphism groups of vertex-transitive graphs of order ≤ 11 (2006, background): “The study of Qn, and of free quantum groups in general, focuses now on more technical aspects: matrix models ([5], [6]), ergodic actions ([9], [11]), harmonic analysis ([12], [14]).”
Galois objects and cocycle twisting for locally compact quantum groups (2008, background): “On the other hand, this does not rule out the possibility that there exists an invariant nsf weight : for if the original coaction has an invariant nsf weight ψN (for example, the coactions occurring in [24]), then one checks that x ∈ (N ⋊M)+ → ψ M̂ ′ ((ψN ⊗ ι)(x)) ∈ [0,+∞] is a well-defined α̂-inva”