Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Arici, Francesca; Kaad, Jens

doi:10.1112/tlm3.12038

Cited by 4 publications

(5 citation statements)

References 40 publications

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“…More recently, efforts have begun to extend this work to a theory of noncommutative fiber bundles. This has focused on the study of noncommutative sphere bundles from both a Hopf algebraic [11] and a C * -algebraic point of view [2]. In this paper we propose a simple but effective new framework for producing examples of noncommutative fibrations, both principal and non-principal, from a nested pair of quantum homogeneous spaces.…”

Section: Introductionmentioning

confidence: 99%

Principal Pairs of Quantum Homogeneous Spaces

Carotenuto¹,

Buachalla²

2021

Preprint

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We propose a simple but effective framework for producing examples of covariant faithfully flat (generalised) Hopf-Galois extensions from a nested pair of quantum homogeneous spaces. Our construction is modelled on the classical situation of a homogeneous fibration G/N → G/M , for G a group, and N ⊆ M ⊆ G subgroups. Variations on Takeuchi's equivalence and Schneider's descent theorem are presented in this context. Quantum flag manifolds and their associated quantum Poisson homogeneous spaces are taken as motivating examples. Moreover, a large collection of noncommutative fibrations (in the spirit of Brzezinski and Szymanski) are constructed.

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Section: Introductionmentioning

confidence: 99%

Principal Pairs of Quantum Homogeneous Spaces

Carotenuto¹,

Buachalla²

2021

Preprint

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“…Definition 4. 4 Let E, F be Hilbert spaces, and let A ⊆ E be a homogeneous set. A map F ∶ A → F is said to be holomorphic if for all x ∈ A/{0} and all y ∈ F, the function D → C, t ↦ ⟨F(t x ∥x∥ ), y⟩, Theorem 4.8 Let A X and A Y be (isometrically/completely boundedly) isomorphic tensor algebras of subproduct systems.…”

Section: The Disk Trickmentioning

confidence: 99%

Tensor algebras of subproduct systems and noncommutative function theory

Hartz,

Shalit

2023

Can. J. Math.-J. Can. Math.

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We revisit tensor algebras of subproduct systems with Hilbert space fibers, resolving some open questions in the case of infinite-dimensional fibers. We characterize when a tensor algebra can be identified as the algebra of uniformly continuous noncommutative functions on a noncommutative homogeneous variety or, equivalently, when it is residually finite-dimensional: this happens precisely when the closed homogeneous ideal associated with the subproduct system satisfies a Nullstellensatz with respect to the algebra of uniformly continuous noncommutative functions on the noncommutative closed unit ball. We show that – in contrast to the finite-dimensional case – in the case of infinite-dimensional fibers, this Nullstellensatz may fail. Finally, we also resolve the isomorphism problem for tensor algebras of subproduct systems: two such tensor algebras are (isometrically) isomorphic if and only if their subproduct systems are isomorphic in an appropriate sense.

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“…For instance, the question of co-universality does not lend itself available to non-commutative boundary techniques (see [25,Corollary 3.16]), and the existence of a co-universal quotient usually requires adding additional symmetries (see [64,Example 2.3]). Thus, completely new techniques are often necessary in order to prove the existence of a natural co-universal quotient in various scenarios (see for instance [3,42]). By using deep results from the theory of random walks [38,68], the first author established the existence of a natural co-universal quotient for Toeplitz C*-algebras arising from random walks, when the random walks are symmetric, aperiodic, and on non-elementary hyperbolic groups [21,Corollary 5.2].…”

Section: Introductionmentioning

confidence: 99%

Ratio-limit boundaries of random walks on relatively hyperbolic groups

Dor-On¹,

Dussaule²,

Gekhtman³

2023

Preprint

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We study boundaries arising from limits of ratios of transition probabilities for random walks on relatively hyperbolic groups. We extend, as well as determine significant limitations of, a strategy employed by Woess for computing ratio-limit boundaries for the class of hyperbolic groups. On the one hand we employ results of the second and third authors to adapt this strategy to spectrally non-degenerate random walks, and show that the closure of minimal points in R-Martin boundary is the unique smallest invariant subspace in ratio-limit boundary. On the other hand we show that the general strategy can fail when the random walk is spectrally degenerate and adapted on a free product. Using our results, we are able to extend a theorem of the first author beyond the hyperbolic case and establish the existence of a co-universal quotient for Toeplitz C*-algebras arising from random walks which are spectrally non-degenerate on relatively hyperbolic groups.

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Gysin sequences and SU(2)‐symmetries of C∗‐algebras

Cited by 4 publications

References 40 publications

Principal Pairs of Quantum Homogeneous Spaces

Principal Pairs of Quantum Homogeneous Spaces

Tensor algebras of subproduct systems and noncommutative function theory

Ratio-limit boundaries of random walks on relatively hyperbolic groups

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