Simplicity of reduced group Banach algebras

Phillips, N. Christopher

doi:10.48550/arxiv.1909.11278

Cited by 4 publications

(5 citation statements)

References 11 publications

(65 reference statements)

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“…Then the following statements are equivalent. This is not only of aesthetic value, but is the basis for transferring simplicity results to related operator algebras, as happened for example with Hecke C * -algebras in [CKL21;Kli21] and in Phillips' work on L p -simplicity [Phi19].…”

Section: Theorem F (See Theorem 61) Let G Be An éTale Groupoid With C...mentioning

confidence: 89%

The ideal intersection property for essential groupoid C*-algebras

Kennedy,

Kim,

et al. 2021

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We characterise, in several complementary ways, étale groupoids with locally compact Hausdorff space of units whose essential groupoid C * -algebra has the ideal intersection property, assuming that the groupoid is either Hausdorff or σ-compact. This leads directly to a characterisation of the simplicity of this C * -algebra which, for Hausdorff groupoids, agrees with the reduced groupoid C * -algebra. Specifically, we prove that the ideal intersection property is equivalent to the absence of essentially confined amenable sections of isotropy groups. For groupoids with compact space of units we moreover show that is equivalent to the uniqueness of equivariant pseudoexpectations and in the minimal case to an appropriate generalisation of Powers averaging property. A key technical idea underlying our results is a new notion of groupoid action on C * -algebras that includes the essential groupoid C * -algebra itself. By considering a relative version of Powers averaging property, we obtain new examples of C * -irreducible inclusions in the sense of Rørdam. These arise from the inclusion of the C * -algebra generated by a suitable group representation into a simple groupoid C * -algebra. This is illustrated by the example of the C * -algebra generated by the quasi-regular representation of Thompson's group T with respect to Thompson's group F, which is contained C * -irreducibly in the Cuntz algebra O 2 .

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Section: Theorem F (See Theorem 61) Let G Be An éTale Groupoid With C...mentioning

confidence: 89%

The ideal intersection property for essential groupoid C*-algebras

Kennedy,

Kim,

et al. 2021

Preprint

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“…The remark is due that some simplicity results for the normclosed r -analogues of Hecke operator algebras can be obtained from existing literature. Combining the averaging operators introduced in [CKL19] with the interpolating methods employed in [Phi19], it follows directly that for every Hecke C * -algebra C * q (W ) for which [CKL19] proves simplicity, all norm-closed r -analogues are simple for r from a neighbourhood of 2. The crucial difference to [Phi19], preventing further conclusions, is the lack of a good norm-estimate for the averaging operators on 1 (W ).…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, it is easy to see that the classical condition to have infinite (non-trivial) conjugacy classes characterises factoriality of group convolution algebras on arbitrary r -spaces. Moreover, the articles [HP15;Phi19] show that also on a C * -algebraic level, simplicity of a classical operator algebra associated with a group implies simplicity of its r -analogues. Our result shows that for Hecke operator algebras a similar implication holds no longer true.…”

Section: Introductionmentioning

confidence: 99%

Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings

Raum¹,

Skalski²

2020

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We obtain a complete characterisation of factorial multiparameter Hecke von Neumann algebras associated with right-angled Coxeter groups. Considering their p -convolution algebra analogues, we exhibit an interesting parameter dependence, contrasting phenomena observed earlier for group Banach algebras. Translated to Iwahori-Hecke von Neumann algebras, these results allow us to draw conclusions on spherical representation theory of groups acting on right-angled buildings, which are in strong contrast to behaviour of spherical representations in the affine case. We also investigate certain graph product representations of right-angled Coxeter groups and note that our von Neumann algebraic structure results show that these are finite factor representations. Further classifying them up to unitary equivalence allows us to reveal high-dimensional Euclidean subspaces of the space of extremal characters of right-angled Coxeter groups.

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“…For a discrete group Γ, if we consider the left regular representation of Γ on p -space p (Γ), then we can define the reduced group p -operator algebra B p r (Γ) of Γ for p ∈ [1, ∞] (see Definition 2.1). These p -operator algebras have been studied intensively; for example, the K-theory of some reduced group p -operator algebras [14,16], the rigidity results of reduced group p -operator algebras for p 2 [7], and the simplicity and tracial state of some reduced group p -operator algebras [7,9,17]. Phillips also posed a question concerning the existence of nontrivial idempotents in B p r (F n ) [18].…”

Section: Introductionmentioning

confidence: 99%