On Conditional Expectations Arising from Group Actions

Frank, Michael; Manuilov, Vladimir; Troitsky, Evgenij

doi:10.4171/zaa/794

Cited by 14 publications

(17 citation statements)

References 10 publications

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“…Let {e α } be an approximate unit of A. Then ρ(e α )x = ρ(e α a i )y i converges to x, and the supremum in (7) achieves the norm x on the approximate unit {ρ(e α )} of ρ(A).…”

Section: Quasi-multipliers Of Hilbert C * -Bimodules In Kasparov's Sensementioning

confidence: 99%

Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

2011

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Quasi-multipliers for a Hilbert C * -bimodule V were introduced by L. G. Brown, J. A. Mingo, as a certain subset of the Banach bidual module V * * . We give another (equivalent) definition of quasi-multipliers for Hilbert C * -bimodules using the centralizer approach and then show that quasi-multipliers are, in fact, universal (maximal) objects of a certain category. We also introduce quasi-multipliers for bimodules in Kasparov's sense and even for Banach bimodules over C * -algebras, provided these C * -algebras act non-degenerately. A topological picture of quasimultipliers via the quasi-strict topology is given. Finally, we describe quasi-multipliers in two main situations: for the standard Hilbert bimodule l 2 (A) and for bimodules of sections of Hilbert C * -bimodule bundles over locally compact spaces.

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“…Let {e α } be an approximate unit of A. Then ρ(e α )x = ρ(e α a i )y i converges to x, and the supremum in (7) achieves the norm x on the approximate unit {ρ(e α )} of ρ(A).…”

Section: Quasi-multipliers Of Hilbert C * -Bimodules In Kasparov's Sensementioning

confidence: 99%

Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

2011

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“…, s, form an ε-net for the set {L g ϕ x : g ∈ Γ}, with respect to the uniform norm. Indeed, for any g ∈ Γ, there is an index i such that (5) holds. Then, by (4), we have…”

Section: Lyapunov Stability and Continuity Of Averagingmentioning

confidence: 99%

“…the appendix). The link to results in [5,18] is given by constructing a suitable conditional expectation E Γ : C(X) → C Γ (X) by the rule 1') For any ϕ ∈ C(X) the function E Γ (φ)(x) := M(φ x ) is continuous in x. Properties 1') and 2') provide that the formula (2) makes C(X) a pre-Hilbert C * -module over C Γ (X).…”

Section: Introductionmentioning

confidence: 99%

Hilbert C^*-modules from group actions: beyond the finite orbits case

2010

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Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C * -algebra. The unique invariant mean on the group resulting from averaging allows to derive a C * -valued inner product and a Hilbert C * -module which serve as an environment to describe characteristics of the group action. For uniformly continuous, Lyapunov stable actions the derived invariant mean M (φ x ) is continuous on X for any element φ ∈ C(X), and the induced C * -valued inner product corresponds to a conditional expectation from C(X) onto the fixed point algebra of the action defined by averaging on orbits. In the case of selfduality of the Hilbert C * -module all orbits are shown to have the same cardinality. Stable actions on compact metric spaces give rise to C * -reflexive Hilbert C * -modules. The same is true if the cardinality of finite orbits is uniformly bounded and the number of closures of infinite orbits is finite. A number of examples illustrate typical situations appearing beyond the classified cases.

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“…Let us reformulate some definitions and statements from [5]. Actions will always be assumed to satisfy the following condition.…”

Section: Preliminaries and Remindingmentioning

confidence: 99%

“…The study of the relations between these properties was started systematically in [5] (for preliminary and related research see also [1,2,4,7,8,10,14,15,18] and the overview in [5]). …”

Section: Introductionmentioning

confidence: 99%

Discrete groups actions and corresponding modules

Troitsky

2003

Proc. Amer. Math. Soc.

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Abstract. We address the problem of interrelations between the properties of an action of a discrete group Γ on a compact Hausdorff space X and the algebraic and analytical properties of the module of all continuous functions C(X) over the algebra of invariant continuous functions C Γ (X). The present paper is a continuation of our joint paper with M. Frank and V. Manuilov. Here we prove some statements inverse to the ones obtained in that paper: we deduce properties of actions from properties of modules. In particular, it is proved that if for a uniformly continuous action the module C(X) is finitely generated projective over C Γ (X), then the cardinality of orbits of the action is finite and fixed. Sufficient conditions for existence of natural conditional expectations C(X) → C Γ (X) are obtained.

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On Conditional Expectations Arising from Group Actions

Cited by 14 publications

References 10 publications

Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

Hilbert C^*-modules from group actions: beyond the finite orbits case

Discrete groups actions and corresponding modules

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On Conditional Expectations Arising from Group Actions

Cited by 14 publications

References 10 publications

Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

Quasi-multipliers of Hilbert and Banach $C^*$-bimodules

Hilbert C*-modules from group actions: beyond the finite orbits case

Discrete groups actions and corresponding modules

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Product

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Hilbert C^*-modules from group actions: beyond the finite orbits case