C ∗ -Algebras Associated with Groups with Kazhdan's Property T

Wassermann, Simon

doi:10.2307/2944351

Cited by 66 publications

(42 citation statements)

References 5 publications

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“…The order of growth of our lower bound exp βnN 2 in Theorem1.3 is roughly optimal because of the upper bound given explicitly in [10] (and implicitly in [37]). The latter upper bound can be proved as follows.…”

Section: Quantum Expandersmentioning

confidence: 99%

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Quantum expanders and geometry of operator spaces

Pisier¹

2014

J. Eur. Math. Soc.

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We show that there are well separated families of quantum expanders with asymptotically the maximal cardinality allowed by a known upper bound. This has applications to the "growth" of certain operator spaces: It implies asymptotically sharp estimates for the growth of the multiplicity of M N -spaces needed to represent (up to a constant C > 1) the M N -version of the n-dimensional operator Hilbert space OH n as a direct sum of copies of M N . We show that, when C is close to 1, this multiplicity grows as exp βnN 2 for some constant β > 0. The main idea is to relate quantum expanders with "smooth" points on the matricial analogue of the Euclidean unit sphere. This generalizes to operator spaces a classical geometric result on n-dimensional Hilbert space (corresponding to N=1). In an appendix, we give a quick proof of an inequality (related to Hastings's previous work) on random unitary matrices that is crucial for this paper.The term "Quantum Expander" is used by Hastings in [11] and by Ben-Aroya and Ta-Shma in [2] to designate a sequencen ) of N × N unitary matrices such that there is an ε > 0 satisfying the following "spectral gap" condition:where . 2 denotes the Hilbert-Schmidt norm on M N . More generally, the term is extended to the case when this is only defined for infinitely many N 's, and also to n-tuples of matrices satisfying merely= nI. We will say that an n-tuple U (N ) satisfying (0.1) is a ε-quantum expander. We refer the reader to the survey [3] for more information and references on quantum expanders.In analogy with the classical expanders (see below), one seeks to exhibit (and hopefully to construct explicitly ) sequences {U (Nm) | m ≥ 1} of n-tuples of N m × N m unitary matrices that are ε-quantum expanders with N m → ∞ while n and ε > 0 remain fixed.When G is a finite group generated by S = {t 1 , · · · , t n } the associated Cayley graph G(G, S) is said to have a spectral gap if the left regular representation λ G satisfies (0.2) λ G (t j ) |I ⊥ < n(1 − ε) * Partially supported by ANR-2011-BS01-008-01.

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Section: Quantum Expandersmentioning

confidence: 99%

“…In fact, in some sense it is as large as can be. Indeed, it is known ( [37,10], see also Remark 1.5) that the maximal m is at most exp β ′ nN 2 for some constant β ′ .…”

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confidence: 99%

Quantum expanders and geometry of operator spaces

Pisier¹

2014

J. Eur. Math. Soc.

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“…Considérons l'exemple suivant dont la construction m'a été indiquée par S. Wassermann [49]. Soit F le groupe discret SL(3,Z) et {^n} l'ensemble de ses classes de représentations irréductibles de dimension finie où 71-0 est la représentation triviale.…”

Section: Question -éTant Données Deux C(x)-algèbres Stellaires Ai Etunclassified

Déformations de $C\sp*$-algèbres de Hopf

Blanchard

1996

Bul. Soc. Math. France

109

183

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“…In section 4 we consider the number of the unitary d−dimensional representations of G and its connection with the Kazhdan constants of a generating set S ⊂ G. Wasserman [Wa91] showed that r d (G) = r d (G; C) can be bounded in terms of d , |S| andκ G (S) alone. An explicit form of his argument due to de la Harpe, Robertson and Vallete [HRV93] gives the following…”

Section: Definitions and Resultsmentioning

confidence: 99%

Expanders In Group Algebras

Meshulam

Wigderson

2004

Combinatorica

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Let G be a finite group and let p be a prime such that (p, |G|) = 1. We study conditions under which the Abelian group F p [G] has a few G-orbits whose union generate it as an expander (equivalently, all the discrete Fourier coefficients (in absolute value) of this generating set are bounded away uniformly from one).We prove a (nearly sharp) bound on the distribution of dimensions of irreducible representations of G which implies the existence of such expanding orbits. We further show a class of groups for which such a bound follows from the expansion properties of G. Together, these lead to a new iterative construction of expanding Cayley graphs of nearly constant degree.

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C ∗ -Algebras Associated with Groups with Kazhdan's Property T

Cited by 66 publications

References 5 publications

Quantum expanders and geometry of operator spaces

Quantum expanders and geometry of operator spaces

Déformations de $C\sp*$-algèbres de Hopf

Expanders In Group Algebras

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