Schur Multipliers and Spherical Functions on Homogeneous Trees

Haagerup, Uffe; Steenstrup, Troels; Szwarc, Ryszard

doi:10.1142/s0129167x10006537

Cited by 19 publications

(50 citation statements)

References 14 publications

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“…Then, by Lemma 9, letting the space L be Cζ, we infer (see Remark 42) that t defines a˚-representation of the operator system OpK, Gq with values in the von Neumann algebra LpGq (the complex conjugation in formula (20) is due to the fact that, via the canonical anti-isomorphism, we are switching from the algebra RpGq to the algebra LpGq). The Hecke algebra˚-representation Ψ, introduced in formula (19), takes now values in LpΓq and is given by the formula:…”

Section: Lemmamentioning

confidence: 98%

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Rădulescu¹

2016

Russ. Math. Surv.

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ABSTRACT. Consider a unitary representation π of a discrete group G, which, when restricted to an almost normal subgroup Γ Ď G, is of type II. We analyze the associated unitary representation π p of G on the Hilbert space of "virtual" Γ 0 -invariant vectors, where Γ 0 runs over a suitable class of finite index subgroups of Γ. The unitary representation π p of G is uniquely determined by the requirement that the Hecke operators, for all Γ 0 , are the "block matrix coefficients" of π p .If π| Γ is an integer multiple of the regular representation, there exists a subspace L of the Hilbert space of the representation π, acting as a fundamental domain for Γ. In this case, the space of Γ-invariant vectors is identified with L. When π| Γ is not an integer multiple of the regular representation, (e.g. if G " P GLp2, Zr 1 p sq, Γ is the modular group, π belongs to the discrete series of representations of PSLp2, Rq, and the Γ-invariant vectors are the cusp forms) we assume that π is the restriction to a subspace H 0 of a larger unitary representation having a subspace L as above.The operator angle between the projection P L onto L (typically the characteristic function of the fundamental domain) and the projection P 0 onto the subspace H 0 (typically a Bergman projection onto a space of analytic functions), is the analogue of the space of Γ-invariant vectors.We prove that the character of the unitary representation π p is uniquely determined by the character of the representation π.

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Section: Lemmamentioning

confidence: 98%

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Rădulescu¹

2016

Russ. Math. Surv.

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“…We do not make any assumptions on the degrees. We follow the same strategy as in [7]. For each i ∈ {1, ..., N }, fix an infinite geodesic ω (i) 0 : N → X i .…”

Section: Multi-radial Multipliers On Products Of Treesmentioning

confidence: 99%

“…Now we will deal with the only if part of Proposition 2.1. Once again, we follow the strategy of [7]. Their argument is based on the study of a certain isometry on the ℓ 2 space of a homogeneous tree.…”

Section: 2mentioning

confidence: 99%

Radial Schur multipliers on some generalisations of trees

Vergara¹

2019

Studia Math.

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We give a characterisation of radial Schur multipliers on finite products of trees. The equivalent condition is that a certain generalised Hankel matrix involving the discrete derivatives of the radial function is a trace class operator. This extends Haagerup, Steenstrup and Szwarc's result for trees. The same condition can be expressed in terms of Besov spaces on the torus. We also prove a similar result for products of hyperbolic graphs and provide a sufficient condition for a function to define a radial Schur multiplier on a finite dimensional CAT(0) cube complex.

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“…An exact formula for the norm of radial Schur multipliers on free groups was given by Haagerup and Szwarc in 1987. The result was published in [11] with Steenstrup, where they extended the study to homogenous trees. A similar characterization for radial Fourier multipliers with respect to the block length on free products of groups is proved by Wysoczański in 1995 ([24]).…”

Section: Radial Multipliers On Treesmentioning

confidence: 99%

“…A characterization of the completely bounded radial multipliers on free groups was given by Haagerup and Szwarc in terms of some Hankel matrix being of trace class (this was published in [11], see also [24]). On the other hand a famous result of V. V. Peller ([20]) states that a Hankel matrix belongs to the trace class iff the symbol function belongs to the Besov space B 1 1 (see Section 3).…”

Section: Introductionmentioning

confidence: 99%

Complete boundedness of heat semigroups on the von Neumann algebra of hyperbolic groups

Mei

2017

Trans. Amer. Math. Soc.

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We prove that (λg → e −t|g| r λg)t>0 defines a completely bounded semigroup of multipliers on the von Neuman algebra of hyperbolic groups for all real number r. One ingredient in the proof is the observation that a construction of Ozawa allows to characterize the radial multipliers that are bounded on every hyperbolic graph, partially generalizing results of Haagerup-Steenstrup-Szwarc and Wysoczański. Another ingredient is an upper estimate of trace class norms for Hankel matrices, which is based on Peller's characterization of such norms.

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Schur Multipliers and Spherical Functions on Homogeneous Trees

Cited by 19 publications

References 14 publications

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Endomorphisms of spaces of virtual vectors fixed by a discrete group

Radial Schur multipliers on some generalisations of trees

Complete boundedness of heat semigroups on the von Neumann algebra of hyperbolic groups

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