Unitary Representations of Inductive Limits of Commutative Locally Compact Groups

1991 ◽  
pp. 41-66
Author(s):  
Y. S. Samoilenko
Keyword(s):  
Unitary Representations ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Inductive Limits ◽  
Locally Compact

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.

Studies in harmonic analysis

1964 ◽  
Vol 60 (3) ◽  
pp. 465-516 ◽  
Author(s):  
N. Th. Varopoulos
Keyword(s):  
Harmonic Analysis ◽  
Complete Characterization ◽  
Locally Compact Groups ◽  
Topological Spaces ◽  
Compact Groups ◽  
Measure Algebra ◽  
Topological Groups ◽  
Inductive Limits ◽  
Locally Compact ◽  
Character Group

In this paper we shall be mainly concerned with the following three apparently widely differing questions.(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?

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.

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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