Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

Kasprzak, Paweł

doi:10.1142/s0129167x18500921

Cited by 6 publications

(3 citation statements)

References 23 publications

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“…Given some sequence of subspace E = (E π ) π∈Irr( G) , let P E = ⊕ π∈Irr( G) P π ∈ ℓ ∞ (G) be the orthogonal projection where each P π is the orthogonal projection onto E π . It turns out that P E is group-like if and only if E is the hull of a right coideal (see [1,20]). We identify the right coideal of ℓ ∞ (G),…”

Section: Coideals and Idempotent Statesmentioning

confidence: 99%

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Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups

Benjamin¹

2022

Preprint

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We find quantum group dynamic characterizations of the existence and uniqueness of tracial states on the reduced C * -algebra Cr( G) of an arbitrary discrete quantum group G. We prove that Cr( G) admits a tracial state if and only there exists a Ginvariant state if and only if the cokernel of the Furstenberg boundary of G, HF , is unimodular. We prove that there is a unique tracial state if and only if HF coincides with the canonical Kac quotient of G and a certain quantum group C * -algebra is exotic. Along the way, we obtain that Cr( G) is nuclear and has a tracial state if and only if G is amenable, which resolves an open problem due to C.-K Ng and Viselter, and Crann, in the discrete case. As applications of our work, we find that C * -simplicity implies the Haar state is the only possible tracial state for arbitrary G, and we answer two questions of Kalantar, Kasprzak, Skalski, and Vergnioux in the case where G is unimodular.

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Section: Coideals and Idempotent Statesmentioning

confidence: 99%

“…Of special interest are the coideals generated by idempotent states (see [13,20,21,30,31]). Given an idempotent state ω ∈ C u ( G) * , the adjoint of the map…”

Section: Coideals and Idempotent Statesmentioning

confidence: 99%

Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups

Benjamin¹

2022

Preprint

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“…We point out that the idempotent functional ω N • x −1 is easily seen to be a contractive idempotent. Contractive idempotents and their associated weak * closed right invariant subspaces were studied in [27,40] (at the level LCQGs). While given a contractive idempotent ω ∈ M u (G), M l ω (L ∞ (G)) is not an algebra, it is a ternary ring of operators (TRO), i.e., whenever x, y, z The following theorem is the statement that G is coamenable if and only if the preannihilator of an invariant quantum coset has a bai.…”

Section: Quantum Cosets Of Compact Quasi-subgroupsmentioning

confidence: 99%

On Ideals of $L^1$-algebras of Compact Quantum Groups

Benjamin¹

2021

Preprint

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We develop a notion of a non-commutative hull for a left ideal of the L 1 -algebra of a compact quantum group G. A notion of non-commutative spectral synthesis for compact quantum groups is achieved as well. It is shown that a certain Ditkin's property at infinity (which includes those G where G has the approximation property) is equivalent to every hull having synthesis. We use this work to extend recent work of White that characterizes the weak * closed ideals of a measure algebra of a compact group to those of the measure algebra of a coamenable compact quantum group. In the sequel, we use this work to study bounded right approximate identities of certain left ideals of L 1 (G) in relation to coamenability of G.Acknowledgements: I thank Nico Spronk, Brian Forrest, and Michael Brannan for their supervision and support over the fruition of this project. I would like to thank Nico especially for his careful reading of this article and helpful comments. I would also like to thank Jared White for initiating the Groups, Operators, and Banach Algebras (GOBA) webinar in the Winter of 2020. The ideas that built this work were seeded from Jared's talk at the onset of the GOBA webinar.

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Integrals in Left Coideal Subalgebras and Group-Like Projections

Chirvăsitu

Kasprzak

Szulim

2019

Algebr Represent Theor

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We develop a theory of right group-like projections in Hopf algebras linking them with the theory of left coideal subalgebras with two sided counital integrals. Every right grouplike projection is associated with a left coideal subalgebra, maximal among the ones containing the given group-like projection as an integral, and we show that that subalgebra is finite dimensional. We observe that in a semisimple Hopf algebra H every left coideal subalgebra has an integral and we prove a 1-1 correspondence between right group-like projections and left coideal subalgebras of H. We provide a number of equivalent conditions for a right group-like projections to be left group-like projection and prove a 1-1 correspondence between semisimple left coideal subalgebras preserved by the squared antipode and two sided group-like projections.We also classify left coideal subalgebras in Taft Hopf algebras H n 2 over a field , showing that the automorphism group splits them into ◮ a class of cardinality | | − 1 of semisimple ones which correspond to right group-like projections which are not two sided; ◮ finitely many semisimple singletons, each corresponding to two sided group-like projection;the number of those singletons for H n 2 is equal to the number of divisors of n; ◮ finitely many singletons, each non-semisimple and admitting no right group-like projection;the number of those singletons for H n 2 is equal to the number of divisors of n;In particular we answer the question of Landstad and Van Daele showing that there do exist right group-like projections which are not left group-like projections.2010 Mathematics Subject Classification. Primary: 46L65 Secondary: 43A05, 46L30, 60B15, 16T05, 16T15.

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Shifts of group-like projections and contractive idempotent functionals for locally compact quantum groups

Cited by 6 publications

References 23 publications

Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups

Tracial States and $\mathbb{G}$-Invariant States of Discrete Quantum Groups

On Ideals of $L^1$-algebras of Compact Quantum Groups

Integrals in Left Coideal Subalgebras and Group-Like Projections

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