Correspondences and Approximation Properties for von Neumann Algebras

2003 ◽  
Vol 14 (06) ◽  
pp. 619-665 ◽  
Author(s):  
Jon Kraus
Keyword(s):  
Approximation Property ◽  
Von Neumann Algebras ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Approximation Properties ◽  
Locally Compact ◽  
Von Neumann ◽  
Weak Amenability ◽  
Operator Approximation ◽  
Weak Operator

The notion of the amenability of a locally compact group has been extended in various ways. Two weaker versions of amenability, weak amenability and the approximation property, have been defined for locally compact groups (by Haagerup and Haagerup and Kraus, respectively) and Bekka has defined a notion of amenability for representations of locally compact groups. Correspondences can be viewed as a generalization of representations of such groups. Using this viewpoint, Ananthraman–Delaroche has defined a notion of (left) amenability for correspondences. In this paper, we define notions of weak amenability and the approximation property for correspondences (and representations of locally compact groups), and obtain various results concerning these notions. Ananthraman–Delaroche showed that if N ⊂ M is an inclusion of von Neumann algebras, and if the associated inclusion correspondence is left amenable, then various approximation properties of N (semidiscreteness, the weak* completely bounded approximation property, and the weak* operator approximation property) are shared by M. We show that if this correspondence has the (weaker) approximation property, then if N has the weak* operator approximation property, so does M. An application of this result to crossed products is also given.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

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.

scholarly journals APPROXIMATION PROPERTIES FOR CROSSED PRODUCTS BY ACTIONS AND COACTIONS

2001 ◽  
Vol 12 (05) ◽  
pp. 595-608 ◽  
Author(s):  
MAY M. NILSEN ◽  
ROGER R. SMITH
Keyword(s):  
Approximation Property ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Crossed Products ◽  
Approximation Properties ◽  
Locally Compact ◽  
Amenable Groups ◽  
C Algebras ◽  
Approximation Constant

We investigate approximation properties for C*-algebras and their crossed products by actions and coactions by locally compact groups. We show that Haagerup's approximation constant is preserved for crossed products by arbitrary amenable groups, and we show why this is not always true in the non-amenable case. We also examine similar questions for other forms of the approximation property.

scholarly journals A Duality of Locally Compact Groups That Does Not Involve the Haar Measure

2015 ◽  
Vol 116 (2) ◽  
pp. 250 ◽  
Author(s):  
Yulia Kuznetsova
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
C Algebras ◽  
Commutative Algebras

We present a simple and intuitive framework for duality of locally compacts groups, which is not based on the Haar measure. This is a map, functorial on a non-degenerate subcategory, on the category of coinvolutive Hopf $C^*$-algebras, and a similar map on the category of coinvolutive Hopf-von Neumann algebras. In the $C^*$-version, this functor sends $C_0(G)$ to $C^*(G)$ and vice versa, for every locally compact group $G$. As opposed to preceding approaches, there is an explicit description of commutative and co-commutative algebras in the range of this map (without assumption of being isomorphic to their bidual): these algebras have the form $C_0(G)$ or $C^*(G)$ respectively, where $G$ is a locally compact group. The von Neumann version of the functor puts into duality, in the group case, the enveloping von Neumann algebras of the algebras above: $C_0(G)^{**}$ and $C^*(G)^{**}$.

Ozawa's class 𝒮 for locally compact groups and unique prime factorization of group von Neumann algebras

2019 ◽  
Vol 150 (5) ◽  
pp. 2656-2681
Author(s):  
Tobe Deprez
Keyword(s):  
Von Neumann Algebras ◽  
Factorization Theorem ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Prime Factorization ◽  
Von Neumann ◽  
Amenable Action ◽  
Measure Equivalence ◽  
Equivalence Invariant

AbstractWe study class 𝒮 for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class 𝒮 and prove a unique prime factorization theorem for group von Neumann algebras of products of locally compact groups in this class. We also prove that class 𝒮 is a measure equivalence invariant.

Voiculescu amenable Hopf von Neumann algebras with applications

2016 ◽  
Vol 15 (06) ◽  
pp. 1650079 ◽  
Author(s):  
Fatemeh Akhtari ◽  
Rasoul Nasr-Isfahani
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Continuous Functions ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann

For a Hopf von Neumann algebra [Formula: see text], we give a fixed point characterization of Voiculescu amenability of [Formula: see text] in terms of modules over [Formula: see text]. As a consequence, we present some descriptions for amenability of locally compact groups in terms of certain associated Hopf von Neumann algebras. We finally apply this result to some modules of continuous functions on a multiplicative subsemigroup of [Formula: see text].

scholarly journals A Non-commutative Fejér Theorem for Crossed Products, the Approximation Property, and Applications

2020 ◽  
Author(s):  
Jason Crann ◽  
Matthias Neufang
Keyword(s):  
Dynamical Systems ◽  
Compact Group ◽  
Approximation Property ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Crossed Products ◽  
Locally Compact ◽  
Open Problems ◽  
Von Neumann

Abstract We prove that a locally compact group has the approximation property (AP), introduced by Haagerup–Kraus [ 21], if and only if a non-commutative Fejér theorem holds for its associated $C^*$- or von Neumann crossed products. As applications, we answer three open problems in the literature. Specifically, we show that any locally compact group with the AP is exact. This generalizes a result by Haagerup–Kraus [ 21] and answers a problem raised by Li [ 27]. We also answer a question of Bédos–Conti [ 4] on the Fejér property of discrete $C^*$-dynamical systems, as well as a question by Anoussis–Katavolos–Todorov [ 3] for all locally compact groups with the AP. In our approach, we develop a notion of Fubini crossed product for locally compact groups and a dynamical version of the slice map property.

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