scholarly journals Property (T), finite-dimensional representations, and generic representations

2019 ◽  
Vol 22 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Michal Doucha ◽  
Maciej Malicki ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Unitary Representation ◽  
Unitary Group ◽  
Discrete Group ◽  
Coefficient Function ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Invariant Vectors ◽  
Property T

Abstract Let G be a discrete group with Property (T). It is a standard fact that, in a unitary representation of G on a Hilbert space {\mathcal{H}} , almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation σ, then the vector is close to a sub-representation isomorphic to σ: this makes quantitative a result of P. S. Wang. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot, that a group G with Property (T) and such that {C^{*}(G)} is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in {Rep(G,\mathcal{H})} under the unitary group {U(\mathcal{H})} is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in {\mathrm{Rep}(G,\mathcal{H})} .

scholarly journals Operator Amenability of the Fourier Algebra in the cb-Multiplier Norm

2007 ◽  
Vol 59 (5) ◽  
pp. 966-980 ◽  
Author(s):  
Brian E. Forrest ◽  
Volker Runde ◽  
Nico Spronk
Keyword(s):  
Compact Group ◽  
Free Group ◽  
Discrete Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Residually Finite ◽  
Finite Dimensional

AbstractLet G be a locally compact group, and let Acb(G) denote the closure of A(G), the Fourier algebra of G, in the space of completely boundedmultipliers of A(G). If G is a weakly amenable, discrete group such that C*(G) is residually finite-dimensional, we show that Acb(G) is operator amenable. In particular, Acb() is operator amenable even though , the free group in two generators, is not an amenable group. Moreover, we show that if G is a discrete group such that Acb(G) is operator amenable, a closed ideal of A(G) is weakly completely complemented in A(G) if and only if it has an approximate identity bounded in the cb-multiplier norm.

HAAGERUP PROPERTY FOR -CROSSED PRODUCTS

2016 ◽  
Vol 95 (1) ◽  
pp. 144-148 ◽  
Author(s):  
QING MENG
Keyword(s):  
Conditional Expectation ◽  
Discrete Group ◽  
Crossed Products ◽  
Haagerup Property ◽  
Residually Finite ◽  
Tracial State ◽  
Finite Dimensional ◽  
Countable Discrete Group ◽  
Strong Property

Let $\unicode[STIX]{x1D6E4}$ be a countable discrete group that acts on a unital $C^{\ast }$-algebra $A$ through an action $\unicode[STIX]{x1D6FC}$. If $A$ has a faithful $\unicode[STIX]{x1D6FC}$-invariant tracial state $\unicode[STIX]{x1D70F}$, then $\unicode[STIX]{x1D70F}^{\prime }=\unicode[STIX]{x1D70F}\circ {\mathcal{E}}$ is a faithful tracial state of $A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}$ where ${\mathcal{E}}:A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4}\rightarrow A$ is the canonical faithful conditional expectation. We show that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property if and only if both $(A,\unicode[STIX]{x1D70F})$ and $\unicode[STIX]{x1D6E4}$ have the Haagerup property. As a consequence, suppose that $(A\rtimes _{\unicode[STIX]{x1D6FC},r}\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D70F}^{\prime })$ has the Haagerup property where $\unicode[STIX]{x1D6E4}$ has property $T$ and $A$ has strong property $T$. Then $\unicode[STIX]{x1D6E4}$ is finite and $A$ is residually finite-dimensional.

scholarly journals On compact group extension of Bernoulli shifts

2000 ◽  
Vol 61 (2) ◽  
pp. 277-288
Author(s):  
Youngho Ahn
Keyword(s):  
Hilbert Space ◽  
Compact Group ◽  
Unitary Representation ◽  
Bernoulli Shift ◽  
Irreducible Unitary Representation ◽  
Group Extension ◽  
Step Function ◽  
Unitary Operators ◽  
Finite Dimensional ◽  
Finite Dimensional Hilbert Space

Let ρ : G →  (H) be an irreducible unitary representation of a compact group G where  (H) is a set of unitary operators of finite dimensional Hilbert space H. For the (p1, …, PL)-Bernoulli shift, the solvability of ρ(φ(x)) g (Tx) = g (x) is investigated, where φ(x) is a step function.

scholarly journals Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type

2020 ◽  
Vol 2020 (758) ◽  
pp. 223-251
Author(s):  
Hiroshi Ando ◽  
Yasumichi Matsuzawa ◽  
Andreas Thom ◽  
Asger Törnquist
Keyword(s):  
Hilbert Space ◽  
Finite Type ◽  
Unitary Group ◽  
Discrete Group ◽  
Separable Hilbert Space ◽  
Invariant Metric ◽  
Polish Groups ◽  
Continuous Maps ◽  
Equivariant Version ◽  
Uniformly Bounded

AbstractLet Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{GL}}(H)} be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group {G=H\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{N}))}) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a {\mathrm{II}_{1}}-factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space {L^{0}(X,m)} of all measurable maps on a probability space.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals When are maps preserving semi-inner products linear?

2021 ◽  
Author(s):  
Paweł Wójcik
Keyword(s):  
Hilbert Space ◽  
Infinite Dimensions ◽  
Normed Spaces ◽  
Linear Isometry ◽  
Inner Products ◽  
Finite Dimensional ◽  
Uniformly Smooth ◽  
Non Linear ◽  
Extra Assumption ◽  
Continuous Injection

AbstractWe observe that every map between finite-dimensional normed spaces of the same dimension that respects fixed semi-inner products must be automatically a linear isometry. Moreover, we construct a uniformly smooth renorming of the Hilbert space $$\ell _2$$ ℓ 2 and a continuous injection acting thereon that respects the semi-inner products, yet it is non-linear. This demonstrates that there is no immediate extension of the former result to infinite dimensions, even under an extra assumption of uniform smoothness.

Spectral synthesis and residually finite-dimensional algebras

2017 ◽  
Vol 16 (10) ◽  
pp. 1750200 ◽  
Author(s):  
László Székelyhidi ◽  
Bettina Wilkens
Keyword(s):  
Spectral Analysis ◽  
Abelian Group ◽  
Theoretical Approach ◽  
Abelian Groups ◽  
Spectral Synthesis ◽  
Residually Finite ◽  
Finite Dimensional ◽  
Analysis And Synthesis ◽  
Spectral Analysis And Synthesis

In 2004, a counterexample was given for a 1965 result of R. J. Elliott claiming that discrete spectral synthesis holds on every Abelian group. Since then the investigation of discrete spectral analysis and synthesis has gained traction. Characterizations of the Abelian groups that possess spectral analysis and spectral synthesis, respectively, were published in 2005. A characterization of the varieties on discrete Abelian groups enjoying spectral synthesis is still missing. We present a ring theoretical approach to the issue. In particular, we provide a generalization of the Principal Ideal Theorem on discrete Abelian groups.

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