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.

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.

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 .

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)^{**}} .

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)**.

The Extensions of an Invariant Mean and the Set LIM ∽ TLIM

1994 ◽  
Vol 46 (4) ◽  
pp. 808-817
Author(s):  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Discrete Group ◽  
Locally Compact Group ◽  
Invariant Mean ◽  
Locally Compact ◽  
Invariant Means

AbstractLet with . If G is a nondiscrete locally compact group which is amenable as a discrete group and m ∈ LIM(CB(G)), then we can embed into the set of all extensions of m to left invariant means on L∞(G) which are mutually singular to every element of TLIM(L∞(G)), where LIM(S) and TLIM(S) are the sets of left invariant means and topologically left invariant means on S with S = CB(G) or L∞(G). It follows that the cardinalities of LIM(L∞(G)) ̴ TLIM(L∞(G)) and LIM(L∞(G)) are equal. Note that which contains is a very big set. We also embed into the set of all left invariant means on CB(G) which are mutually singular to every element of TLIM(CB(G)) for G = G1 ⨯ G2, where G1 is nondiscrete, non–compact, σ–compact and amenable as a discrete group and G2 is any amenable locally compact group. The extension of any left invariant mean on UCB(G) to CB(G) is discussed. We also provide an answer to a problem raised by Rosenblatt.

Uniformly Continuous Functionals and M-Weakly Amenable Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1005-1019 ◽  
Author(s):  
Brian Forrest ◽  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Amenable Groups ◽  
Invariant Means ◽  
Continuous Functionals ◽  
Uniformly Continuous

AbstractLet G be a locally compact group. Let AM(G) (A0(G))denote the closure of A(G), the Fourier algebra of G in the space of bounded (completely bounded) multipliers of A(G). We call a locally compact group M-weakly amenable if AM(G) has a bounded approximate identity. We will show that when G is M-weakly amenable, the algebras AM(G) and A0(G) have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of topologically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra.

Topological inner invariant means

2003 ◽  
Vol 40 (3) ◽  
pp. 293-299
Author(s):  
R. Memarbashi ◽  
A. Riazi
Keyword(s):  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact ◽  
Invariant Means ◽  
Inner Amenability

For a locally compact group G, we investigate topological inner invariant means on L8(G) and its subspaces. In particular, we characterize strict inner amenability of L1(G) to study the relation between this notion and strict inner amenability of G.

Sign in / Sign up

Export Citation Format

Share Document