scholarly journals Metric groups, unitary representations and continuous logic

2021 ◽  
Vol 29 (1) ◽  
pp. 35-48
Author(s):  
Aleksander Ivanov
Keyword(s):  
Unitary Representations ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Continuous Logic ◽  
Locally Compact ◽  
First Order ◽  
Property T ◽  
Metric Groups

Abstract We describe how properties of metric groups and of unitary representations of metric groups can be presented in continuous logic. In particular we find Lω 1 ω -axiomatization of amenability. We also show that in the case of locally compact groups some uniform version of the negation of Kazhdan’s property (T) can be viewed as a union of first-order axiomatizable classes. We will see when these properties are preserved under taking elementary substructures.

Contractive Representation Theory for the Unitary Group of C(X, M2)

1987 ◽  
Vol 39 (3) ◽  
pp. 612-624 ◽  
Author(s):  
Alan L. T. Paterson
Keyword(s):  
Representation Theory ◽  
Haar Measure ◽  
Unitary Group ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Theoretical Physics ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann

One motivation for studying representation theory for the unitary group of a unital C*-algebra arises from Theoretical Physics. (In the latter connection, Segal [9] and Arveson [1] have developed a representation theory for G. Their approach is in a different direction from ours.) Another motivation for studying the representation theory of G arises out of the desire to unify the theories of amenable von Neumann algebras and amenable locally compact groups.A serious problem for such a representation theory is the absence of Haar measure on G in general.In [7], the author introduced the class RepdG of contractive unitary representations of G, the strong metric condition involved compensating for the lack of Haar measure.

Property T for general C*-algebras

2013 ◽  
Vol 156 (2) ◽  
pp. 229-239 ◽  
Author(s):  
CHI–KEUNG NG
Keyword(s):  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
C Algebras ◽  
Strong Property ◽  
Property T ◽  
Definition Of ◽  
Locally Compact Hausdorff Space

AbstractIn this paper, we extend the definition of property T and strong property T to general C*-algebras (not necessarily unital). We show that if an inclusion pair of locally compact groups (G,H) has property T, then (C*(G), C*(H)) has property T. As a partial converse, if T is abelian and C*(G) has property T, then T is compact. We also show that if Ω is a first countable locally compact Hausdorff space, then C0(Ω) has (strong) property T if and only if Ω is discrete. Furthermore, the non-unital C*-algebra $c_0(\mathbb{Z}^n)\rtimes SL_n(\mathbb{Z})$ has strong property T when n ≥ 3. We also give some equivalent forms of strong property T, which are new even in the unital case.

Continuity of automorphic representations

1973 ◽  
Vol 74 (3) ◽  
pp. 461-465 ◽  
Author(s):  
J. Moffat
Keyword(s):  
Von Neumann Algebra ◽  
Locally Compact Group ◽  
Unitary Representations ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Strong Continuity ◽  
Locally Compact ◽  
Automorphic Representations ◽  
Von Neumann ◽  
Continuity Properties

Let ℛ be a von Neumann algebra, with predual ℛ*, acting on a Hilbert space ℋ; G a locally compact group with left Haar measure m, and α a representation of G on aut (ℛ), the group of all *-automorphisms of ℛ, i.e. α is a group homomorphism from G to aut (ℛ). We shall show that if ℋ is separable, then very weak measurability assumptions on the representation α produce strong continuity properties. This will be used to obtain results on the extension of representations from a C*-algebra to its weak closure, giving a much simpler proof of a result of Aarnes ((1), theorem 8, p. 31), and on continuity of tensor products of representations. The main result was suggested by the analogous theory concerning unitary representations of locally compact groups, and its proof employs ideas frequently used in that context. (See, for example, (5), theorem 22.20 (b), p. 347.)

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