Idempotent States on Locally Compact Quantum Groups II

Salmi, Pekka; Skalski, Adam

doi:10.1093/qmath/haw045

Cited by 11 publications

(22 citation statements)

References 17 publications

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“…Conversely, if our E : H → A is positive in the sense that (1) holds in A u then it extends to a C * expectation. This type of extensibility is of interest, for instance, in the literature on idempotent states on (locally) compact quantum groups [7,6,8,19,20]. Idempotent states are, just as the name suggests, non-commutative analogues of idempotent measures on classical locally compact groups.…”

Section: And 35)mentioning

confidence: 95%

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Relative Fourier transforms and expectations on coideal subalgebras

Chirvăsitu

2018

Journal of Algebra

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For an algebraic compact quantum group H we establish a bijection between the set of right coideal * -subalgebras A → H and that of left module quotient * -coalgebras H → C. It turns out that the inclusion A → H always splits as a map of right A-modules and right H-comodules, and the resulting expectation E : H → A is positive (and lifts to a positive map on the full C * completion on H) if and only if A is invariant under the squared antipode of H.The proof proceeds by Tannaka-reconstructing the coalgebra C corresponding to A → H by means of a fiber functor from H-equivariant A-modules to Hilbert spaces, while the characterization of those A → H which admit positive expectations makes use of a Fourier transform turning elements of H into functions on C.

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Section: And 35)mentioning

confidence: 95%

“…Since for fixed v ∈ V and w ∈ W (with corresponding v * ∈ V * and w * ∈ W * obtained using the compatible inner products) the right hand sides of (20) and (21) are obtained from one another by applying the * operation of H, it follows that indeed (19) and (21) correspond to each other through the construction…”

Section: The Composition Of Maps F G In Hommentioning

confidence: 99%

Relative Fourier transforms and expectations on coideal subalgebras

Chirvăsitu

2018

Journal of Algebra

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“…Idempotent states on locally compact quantum groups have been investigated in a number of papers, e.g. [25,8,7,2,28,9,27,29,13,6]. The main idea behind these investigations was based on the classical result of Kawada and Itô which establishes a bijection between idempotent states on a classical locally compact group G and compact subgroups of G with the state given by integration with respect to the Haar measure of the subgroup.…”

Section: Introductionmentioning

confidence: 99%

Lattice of Idempotent States on a Locally Compact Quantum Group

Kasprzak

Sołtan

2020

Publ. Res. Inst. Math. Sci.

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We study lattice operations on the set of idempotent states on a locally compact quantum group corresponding to the operations of intersection of compact subgroups and forming the subgroup generated by two compact subgroups. Normal (σ-weakly continuous) idempotent states are investigated and a duality between normal idempotent states on a locally compact quantum group G and on its dual p G is established. Additionally we analyze the question when a left coideal corresponding canonically to an idempotent state is finite dimensional and give a characterization of normal idempotent states on compact quantum groups.2010 Mathematics Subject Classification. Primary: 46L89, Secondary: 43A05, 20G42. Key words and phrases. idempotent state, locally compact quantum group, open quantum subgroup.

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“…Idempotent states on quantum groups has been intensely studied by now, see e.g. [4], [21], [20], [11], [2]. Note that if µ ∈ C u 0 (G) * , µ = 0 (here µ is a functional, not necessarily a state) satisfies µ * µ = µ and µ ≤ 1 then µ = 1.…”

Section: Introductionmentioning

confidence: 99%

“…Let ω ∈ C u 0 (G) * be an idempotent state such that P = P ω . Remembering that ω is preserved by R u (see [21]) and using (…”

Section: Introductionmentioning

confidence: 99%

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

Kasprzak

2018

Int. J. Math.

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A one to one correspondence between shifts of group-like projections on a locally compact quantum group G which are preserved by the scaling group and contractive idempotent functionals on the dual G is established. This is a generalization of the Illie-Spronk's correspondence between contractive idempotents in the Fourier-Stieltjes algebra of a locally compact group G and cosets of open subgroups of G. We also establish a one to one correspondence between non-degenerate, integrable, G-invariant ternary rings of operators X ⊂ L ∞ (G), preserved by the scaling group and contractive idempotent functionals on G. Using our results we characterize coideals in L ∞ ( G) admitting an atom preserved by the scaling group in terms of idempotent states on G. We also establish a one to one correspondence between integrable coideals in L ∞ (G) and group-like projections in L ∞ ( G) satisfying an extra mild condition. Exploiting this correspondence we give examples of group like projections which are not preserved by the scaling group.

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Idempotent States on Locally Compact Quantum Groups II

Cited by 11 publications

References 17 publications

Relative Fourier transforms and expectations on coideal subalgebras

Relative Fourier transforms and expectations on coideal subalgebras

Lattice of Idempotent States on a Locally Compact Quantum Group

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

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