Dual spaces and translation invariant means on group von Neumann algebras

Michael Yin-Hei Cheng

Displaying similar documents to “Dual spaces and translation invariant means on group von Neumann algebras”

Invariant means on a class of von Neumann algebras related to ultraspherical hypergroups

Nageswaran Shravan Kumar (2014)

Studia Mathematica

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Let K be an ultraspherical hypergroup associated to 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, invariant means on VN(K) are defined and studied. We show that the set of invariant means on VN(K) is nonempty. Also, we prove that, if H is an open subhypergroup of K, then the number of invariant means on VN(H) is equal to the number of invariant means on VN(K). We also show that a unique...

Shift-modulation invariant spaces on LCA groups

Carlos Cabrelli, Victoria Paternostro (2012)

Studia Mathematica

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A (K,Λ) shift-modulation invariant space is a subspace of L²(G) that is invariant under translations along elements in K and modulations by elements in Λ. Here G is a locally compact abelian group, and K and Λ are closed subgroups of G and the dual group Ĝ, respectively. We provide a characterization of shift-modulation invariant spaces when K and Λ are uniform lattices. This extends previous results known for $L ² (ℝ^{d})$ . We develop fiberization techniques and suitable range functions adapted...

Moebius-invariant algebras in balls

Walter Rudin (1983)

Annales de l'institut Fourier

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It is proved that the Fréchet algebra $C (B)$ has exactly three closed subalgebras $Y$ which contain nonconstant functions and which are invariant, in the sense that $f \circ Ψ \in Y$ whenever $f \in Y$ and $Ψ$ is a biholomorphic map of the open unit ball $B$ of $C^{n}$ onto $B$ . One of these consists of the holomorphic functions in $B$ , the second consists of those whose complex conjugates are holomorphic, and the third is $C (B)$ .

Amenability properties of Fourier algebras and Fourier-Stieltjes algebras: a survey

Nico Spronk (2010)

Banach Center Publications

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Let G be a locally compact group, and let A(G) and B(G) denote its Fourier and Fourier-Stieltjes algebras. These algebras are dual objects of the group and measure algebras, $L^{- 1} (G)$ and M(G), in a sense which generalizes the Pontryagin duality theorem on abelian groups. We wish to consider the amenability properties of A(G) and B(G) and compare them to such properties for $L^{- 1} (G)$ and M(G). For us, “amenability properties” refers to amenability, weak amenability, and biflatness, as well as some properties...

A geometric approach to full Colombeau algebras

R. Steinbauer (2010)

Banach Center Publications

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We present a geometric approach to diffeomorphism invariant full Colombeau algebras which allows a particularly clear view of the construction of the intrinsically defined algebra $\hat{} (M)$ on the manifold M given in [gksv].

Beurling-Figà-Talamanca-Herz algebras

Serap Öztop, Volker Runde, Nico Spronk (2012)

Studia Mathematica

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For a locally compact group G and p ∈ (1,∞), we define and study the Beurling-Figà-Talamanca-Herz algebras $A_{p} (G, ω)$ . For p = 2 and abelian G, these are precisely the Beurling algebras on the dual group Ĝ. For p = 2 and compact G, our approach subsumes an earlier one by H. H. Lee and E. Samei. The key to our approach is not to define Beurling algebras through weights, i.e., possibly unbounded continuous functions, but rather through their inverses, which are bounded continuous functions. We...

A new invariant and parametric connected sum of embeddings

A. Skopenkov (2007)

Fundamenta Mathematicae

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We define an isotopy invariant of embeddings $N \to ℝ^{m}$ of manifolds into Euclidean space. This invariant together with the α-invariant of Haefliger-Wu is complete in the dimension range where the α-invariant could be incomplete. We also define parametric connected sum of certain embeddings (analogous to surgery). This allows us to obtain new completeness results for the α-invariant and the following estimation of isotopy classes of embeddings. In the piecewise-linear category, for a (3n-2m+2)-connected...

Linear combinations of generators in multiplicatively invariant spaces

Victoria Paternostro (2015)

Studia Mathematica

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Multiplicatively invariant (MI) spaces are closed subspaces of L²(Ω, ) that are invariant under multiplication by (some) functions in $L^{\infty} (Ω)$ ; they were first introduced by Bownik and Ross (2014). In this paper we work with MI spaces that are finitely generated. We prove that almost every set of functions constructed by taking linear combinations of the generators of a finitely generated MI space is a new set of generators for the same space, and we give necessary and sufficient conditions...

Amenability properties of Figà-Talamanca-Herz algebras on inverse semigroups

Hasan Pourmahmood-Aghababa (2016)

Studia Mathematica

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This paper continues the joint work with A. R. Medghalchi (2012) and the author’s recent work (2015). For an inverse semigroup S, it is shown that $A_{p} (S)$ has a bounded approximate identity if and only if l¹(S) is amenable (a generalization of Leptin’s theorem) and that A(S), the Fourier algebra of S, is operator amenable if and only if l¹(S) is amenable (a generalization of Ruan’s theorem).

Inductive extreme non-Arens regularity of the Fourier algebra A(G)

Zhiguo Hu (2002)

Studia Mathematica

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Let G be a non-discrete locally compact group, A(G) the Fourier algebra of G, VN(G) the von Neumann algebra generated by the left regular representation of G which is identified with A(G)*, and WAP(Ĝ) the space of all weakly almost periodic functionals on A(G). We show that there exists a directed family ℋ of open subgroups of G such that: (1) for each H ∈ ℋ, A(H) is extremely non-Arens regular; (2) $V N (G) = ⋃_{H \in ℋ} V N (H)$ and $V N (G) / W A P (G ̂) = ⋃_{H \in ℋ} [V N (H) / W A P (H ̂)]$ ; (3) $A (G) = ⋃_{H \in ℋ} A (H)$ and it is a WAP-strong inductive union in the sense that the unions in...

Infinite dimensional linear groups with a large family of $G$ -invariant subspaces

L. A. Kurdachenko, A. V. Sadovnichenko, I. Ya. Subbotin (2010)

Commentationes Mathematicae Universitatis Carolinae

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Let $F$ be a field, $A$ be a vector space over $F$ , $GL (F, A)$ be the group of all automorphisms of the vector space $A$ . A subspace $B$ is called almost $G$ -invariant, if ${dim}_{F} (B / {Core}_{G} (B))$ is finite. In the current article, we begin the study of those subgroups $G$ of $GL (F, A)$ for which every subspace of $A$ is almost $G$ -invariant. More precisely, we consider the case when $G$ is a periodic group. We prove that in this case $A$ includes a $G$ -invariant subspace $B$ of finite codimension whose subspaces are $G$ -invariant.

On a secondary invariant of the hyperelliptic mapping class group

Takayuki Morifuji (2009)

Banach Center Publications

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We discuss relations among several invariants of 3-manifolds including Meyer's function, the η-invariant, the von Neumann ρ-invariant and the Casson invariant from the viewpoint of the mapping class group of a surface.

Approximate amenability for Banach sequence algebras

H. G. Dales, R. J. Loy, Y. Zhang (2006)

Studia Mathematica

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We consider when certain Banach sequence algebras A on the set ℕ are approximately amenable. Some general results are obtained, and we resolve the special cases where $A = ℓ^{p}$ for 1 ≤ p < ∞, showing that these algebras are not approximately amenable. The same result holds for the weighted algebras $ℓ^{p} (ω)$ .

$C^{*}$ algebras associated with von Neumann algebras

Tullio G. Ceccherini-Silberstein (1999)

Bollettino dell'Unione Matematica Italiana

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Ad un'algebra di von Neumann separabile $M$ , in forma standard su di uno spazio di Hilbert $H$ , si associa la $C^{*}$ algebra $O_{M}$ definita come la $C^{*}$ algebra $O_{U (M)}$ costituita dai punti fissi dell'algebra di Cuntz generalizzata $O_{H}$ mediante l'azione canonica del gruppo $U (M)$ degli unitari di $M$ . Si dà una caratterizzazione di $O_{M}$ nel caso in cui $M$ è un fattore iniettivo. In seguito, come applicazione della teoria dei sistemi asintoticamente abeliani, si mostra che, se $ω$ è uno stato vettoriale normale e fedele...

A finite multiplicity Helson-Lowdenslager-de Branges theorem

Sneh Lata, Meghna Mittal, Dinesh Singh (2010)

Studia Mathematica

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We prove two theorems. The first theorem reduces to a scalar situation the well known vector-valued generalization of the Helson-Lowdenslager theorem that characterizes the invariant subspaces of the operator of multiplication by the coordinate function z on the vector-valued Lebesgue space L²(;ℂⁿ). Our approach allows us to prove an equivalent version of the vector-valued Helson-Lowdenslager theorem in a completely scalar setting, thereby eliminating the use of range functions and partial...

On projections of $L^{\infty} (G)$ onto translation-invariant subspaces

W. Chojnacki (1978)

Colloquium Mathematicae

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Noncommutative function theory and unique extensions

David P. Blecher, Louis E. Labuschagne (2007)

Studia Mathematica

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We generalize, to the setting of Arveson’s maximal subdiagonal subalgebras of finite von Neumann algebras, the Szegő $L^{p}$ -distance estimate and classical theorems of F. and M. Riesz, Gleason and Whitney, and Kolmogorov. As a byproduct, this completes the noncommutative analog of the famous cycle of theorems characterizing the function algebraic generalizations of $H^{\infty}$ from the 1960’s. A sample of our other results: we prove a Kaplansky density result for a large class of these algebras, and...

Factoriality of von Neumann algebras connected with general commutation relations-finite dimensional case

Ilona Królak (2006)

Banach Center Publications

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We study a certain class of von Neumann algebras generated by selfadjoint elements $ω_{i} = a_{i} + a ⁺_{i}$ , where $a_{i}, a ⁺_{i}$ satisfy the general commutation relations: $a_{i} a ⁺_{j} = \sum_{r, s} t_{j}^{i} r_{s} a ⁺_{r} a_{s} + δ_{i j} I d$ . We assume that the operator T for which the constants $t_{j}^{i} r_{s}$ are matrix coefficients satisfies the braid relation. Such algebras were investigated in [BSp] and [K] where the positivity of the Fock representation and factoriality in the case of infinite dimensional underlying space were shown. In this paper we prove that under certain conditions on the...

Schwartz kernel theorem in algebras of generalized functions

Vincent Valmorin (2010)

Banach Center Publications

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A new approach to the generalization of Schwartz’s kernel theorem to Colombeau algebras of generalized functions is given. It is based on linear maps from algebras of classical functions to algebras of generalized ones. In particular, this approach enables one to give a meaning to certain hypotheses in preceding similar work on this theorem. Results based on the properties of $G^{\infty}$ -generalized functions class are given. A straightforward relationship between the classical and the generalized...

On invariant, dual invariant and absolute formulas

Andrzej Mostowski

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CONTENTS Introduction..............................................................................................................................................................3 1. Lemmas concerning first order formulas.....................................................................................................5 2. Representability of recursively enumerable sets........................................................................................9 3. Simple theory of types.......................................................................................................................................10...

Operator Segal algebras in Fourier algebras

Brian E. Forrest, Nico Spronk, Peter J. Wood (2007)

Studia Mathematica

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Let G be a locally compact group, A(G) its Fourier algebra and L¹(G) the space of Haar integrable functions on G. We study the Segal algebra S¹A(G) = A(G) ∩ L¹(G) in A(G). It admits an operator space structure which makes it a completely contractive Banach algebra. We compute the dual space of S¹A(G). We use it to show that the restriction operator ${u \mapsto u |}_{H} : S ¹ A (G) \to A (H)$ , for some non-open closed subgroups H, is a surjective complete quotient map. We also show that if N is a non-compact closed subgroup,...