Square-integrable representations and multipliers

2019 ◽  
Vol 26 (4) ◽  
pp. 505-513
Author(s):  
Gerhard Racher
Keyword(s):  
Compact Group ◽  
Von Neumann Algebra ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Von Neumann

Abstract We observe a connection between the existence of square-integrable representations of a locally compact group G and the existence of nonzero translation invariant operators from its Fourier–Stieltjes algebra {B(G)} into {L^{2}(G)} or, equivalently, from {L^{2}(G)} into its enveloping von Neumann algebra {C^{*}(G)^{**}} .

Some Results on the Center of an Algebra of Operators on VN(G) for the Heisenberg group

1981 ◽  
Vol 33 (6) ◽  
pp. 1469-1486 ◽  
Author(s):  
C. Cecchini ◽  
A. Zappa
Keyword(s):  
Compact Group ◽  
Heisenberg Group ◽  
Von Neumann Algebra ◽  
Regular Representation ◽  
Locally Compact Group ◽  
Linear Operators ◽  
Locally Compact ◽  
Bounded Linear ◽  
Von Neumann ◽  
Algebra Of Operators

Let G be an amenable locally compact group. We will use the terminology of [3] and denote by VN(G) the Von Neumann algebra of the regular representation and by A(G) its predual, which is the algebra of the coefficients of the regular representation. The Von Neumann algebra VN(G) is, in a natural fashion, a module with respect to A(G) [3].The algebra of bounded linear operators on VN(G), which commute with the action of A(G), has been studied in [6] and in [1]. If UCB(Ĝ) is the space of the elements of VN(G) of the form vT, for some v in A(G) and some T in VN(G) (see for instance [4]), in [6] and in [1] it is proved that, for any amenable locally compact group there exists an isometric bijection between and UCB(Ĝ)*.

The covariance algebra of an extended covariant system

1979 ◽  
Vol 85 (2) ◽  
pp. 271-280 ◽  
Author(s):  
Ronny Rousseau
Keyword(s):  
Hilbert Space ◽  
Compact Group ◽  
Unitary Group ◽  
Von Neumann Algebra ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Additional Axiom

Let M be a von Neumann algebra acting on a Hilbert space , and let G be a locally compact group. We consider an extension of G by , the unitary group of M. If the triple satisfies an additional axiom, we say that it is an extended covariant system. We define a Hilbert space and operators , acting on . The von Neumann algebra is then the covariance algebra of the extended covariant system , denoted by .

The Von Neumann Algebra VN(G) of a Locally Compact Group and Quotients of Its Subspaces

1997 ◽  
Vol 49 (6) ◽  
pp. 1117-1138
Author(s):  
Zhiguo Hu
Keyword(s):  
Compact Group ◽  
Von Neumann Algebra ◽  
Quotient Space ◽  
Linear Mapping ◽  
Locally Compact Group ◽  
Continuous Linear ◽  
Locally Compact ◽  
Von Neumann ◽  
Quotient Spaces ◽  
Invariant Means

AbstractLet VN(G) be the von Neumann algebra of a locally compact group G. We denote by μ the initial ordinal with |μ| equal to the smallest cardinality of an open basis at the unit of G and X = ﹛α ; α < μ﹜.We show that if G is nondiscrete then there exist an isometric *-isomorphism of l∞(X) into VN(G) and a positive linear mapping π of VN(G) onto l∞(X) such that π o = idl∞(X) and and π have certain additional properties. Let UCB((Ĝ)) be the C*–algebra generated by operators in VN(G) with compact support and F(Ĝ) the space of all T∈ VN(G) such that all topologically invariant means on VN(G) attain the same value at T. The construction of the mapping π leads to the conclusion that the quotient space UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) has l∞(X) as a continuous linear image if G is nondiscrete. When G is further assumed to be non-metrizable, it is shown that UCB((Ĝ))/F((Ĝ)) ∪UCB((Ĝ)) contains a linear isomorphic copy of l∞(X). Similar results are also obtained for other quotient spaces.

Decomposability of von Neumann Algebras and the Mazur Property of Higher Level

2006 ◽  
Vol 58 (4) ◽  
pp. 768-795 ◽  
Author(s):  
Zhiguo Hu ◽  
Matthias Neufang
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Group Structure ◽  
Cardinal Number ◽  
Banach Algebras ◽  
Locally Compact Group ◽  
Measure Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Topological Centre

AbstractThe decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.

Translation-Invariant Operators On Lp(G), 0 < p < 1 (II)

1977 ◽  
Vol 29 (3) ◽  
pp. 626-630 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Lebesgue Space ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Translation Invariant ◽  
Translation Operators ◽  
Left And Right

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

scholarly journals On Group Von Neumann Algebras with Vector-Valued Functions

2018 ◽  
Vol 14 (1) ◽  
pp. 7596-7614
Author(s):  
Julien Esse Atto ◽  
Victor Kofi Assiamoua
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Regular Representation ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Von Neumann ◽  
Left Regular Representation ◽  
Left Regular ◽  
Vector Valued Functions ◽  
Vector Valued

Let G be a locally compact group equipped with a normalized Haar measure , A(G) the Fourier algebraof G and V N(G) the von Neumann algebra generated by the left regular representation of G. In this paper, we introduce the space V N(G;A) associated with the Fourier algebra A(G;A) for vector-valued functions on G, where A is a H-algebra. Some basic properties are discussed in the category of Banach space, and alsoin the category of operator space.

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.)

On invariant convex subsets in algebras defined on a locally compact group G

2012 ◽  
Vol 49 (3) ◽  
pp. 301-314
Author(s):  
Ali Ghaffari
Keyword(s):  
Compact Group ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Left Translation ◽  
Measure Algebra ◽  
Locally Compact ◽  
Translation Invariant ◽  
Second Dual ◽  
Necessary And Sufficient ◽  
Convex Subsets

Suppose that A is either the Banach algebra L1(G) of a locally compact group G, or measure algebra M(G), or other algebras (usually larger than L1(G) and M(G)) such as the second dual, L1(G)**, of L1(G) with an Arens product, or LUC(G)* with an Arenstype product. The left translation invariant closed convex subsets of A are studied. Finally, we obtain necessary and sufficient conditions for LUC(G)* to have 1-dimensional left ideals.

Invariant Means on a Class of von Neumann Algebras Related to Ultraspherical Hypergroups II

2017 ◽  
Vol 60 (2) ◽  
pp. 402-410
Author(s):  
N. Shravan Kumar
Keyword(s):  
Compact Group ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Von Neumann ◽  
Invariant Means

AbstractLet K be an ultraspherical hypergroup associated with a locally compact group G and a spherical projector π and let VN(K) denote the dual of the Fourier algebra A(K) corresponding to K. In this note, we show that the set of invariant means on VN(K) is singleton if and only if K is discrete. Here K need not be second countable. We also study invariant means on the dual of the Fourier algebra A0(K), the closure of A(K) in the cb-multiplier norm. Finally, we consider generalized translations and generalized invariant means.

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