Some permanence properties of C∗-unique groups

Leung, Chi-Wai; Ng, Chi–Keung

doi:10.1016/j.jfa.2003.11.003

Cited by 7 publications

(7 citation statements)

References 23 publications

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“…Indeed, on the one hand the relationship between * -regularity and C * -uniqueness has been studied at an in-depth level in [5]. On the other, it is not known whether every such group is automatically C * -unique, see [20] for partial positive results. The most natural candidate to disprove this conjecture is the so-called Grigorchuk group as remarked in [14].…”

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confidence: 99%

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

D’Andrea¹,

Pinzari²,

Rossi³

2017

Ann. inst. Fourier

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Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies * -regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C * -norm.

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confidence: 99%

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

D’Andrea¹,

Pinzari²,

Rossi³

2017

Ann. inst. Fourier

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“…Moreover, it is easy to see that one has the corresponding results for Theorem 9 and Corollary 11 for twisted covariant systems (in this case, almost-freeness means that t∈G\N Prim(A) t is nowhere dense in Prim(A)). Using this, one can show that if G is a separable group with an abelian closed normal subgroup N such that t∈G\NN t is nowhere dense in the dual groupN , then G is a C * -unique group (in the sense of [2]; see [10] for the details).…”

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confidence: 99%

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We will prove a result concerning the inclusion of non-trivial invariant ideals inside nontrivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on C * -algebras.The motivation of this study is our recent research on C * -unique groups (or Fourier groups) in [10]. In fact, one interesting question is when the semi-direct product of a C * -unique group with another group is again C * -unique.

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Invariant ideals of twisted crossed products

Leung

2003

Mathematische Zeitschrift

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We will prove a result concerning the inclusion of non-trivial invariant ideals inside non-trivial ideals of a twisted crossed product. We will also give results concerning the primeness and simplicity of crossed products of twisted actions of locally compact groups on C * -algebras. Mathematics Subject Classification (2000): 46L05, 46L55The motivation of this study is our recent research on C * -unique groups in [10]. In fact, one interesting question is when the semi-direct product of a C * -unique group with another group is again C * -unique. This turns out to be related to the following question: Given a C * -dynamical system (A, G, α), under what condition will it be true that any non-zero ideal of A × α G contains a non-zeroα-invariant ideal (α being the dual coaction)? The aim of this paper is to study this question.In fact, in the case of discrete amenable groups acting on compact spaces, Kawamura and Tomiyama gave (in [9]) a complete solution of the above question. In [1], Archbold and Spielberg generalised the main result in [9] to the case of discrete C * -dynamical system. In this article, we are going to present a weaker result but in the case of general locally compact groups. As a corollary, we obtain some equivalent conditions for the primeness of crossed products (in terms of the actions). Moreover, we will also give a brief discussion on the simplicity of crossed products (which is related but does not need the main theorem).

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“…Int(Prim α A) is dense in Prim A; see [6,Definition 4]), then any non-zero closed ideal of B will contain a non-zero invariant ideal (this result is a crucial step toward one of the main results in [7]). The aims of this article are to give another proof of [6, Theorem 9] using certain "topological arguments" and to study invariant ideals under similar situations using this topological method.…”

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Topological treatments of some results on invariant ideals

2005

Math. Z.

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In this short article, we will give several results concerning closed invariant ideals of crossed products using some "topological arguments". Mathematics Subject Classification (2000): 46L05, 46L55Suppose that A is a separable C * -algebra, G is a second countable locally compact amenable group and α is an action of G on A. Let B := A × α G and Prim α A := t∈G\{e} Prim A \ (Prim A) t (where (Prim A) t is the set of elements in Prim A that are fixed byα t andα is the induced action of G on Prim A). By an invariant ideal of A, we mean a closed ideal that is invariant under α while an invariant ideal of B is a closed ideal that is invariant under the dual coactionα. Note that an ideal of B is invariant if and only if it is of the form I × α G for an invariant ideal I of A.In [6, Theorem 9], it was proved that if α is almost free (i.e. Int(Prim α A) is dense in Prim A; see [6, Definition 4]), then any non-zero closed ideal of B will contain a non-zero invariant ideal (this result is a crucial step toward one of the main results in [7]). The aims of this article are to give another proof of [6, Theorem 9] using certain "topological arguments" and to study invariant ideals under similar situations using this topological method. Our main result is Theorem 7. Note that Theorem 7(a) can be regarded as a general form of [6, Theorem 9] because ker −1 (q G (Int(Prim α A))) = (0) when α is almost free (thus, this result give us a bit more information in the case when α is not almost free). On the other hand, notice that the assumption in Theorem 7(b) (i.e. Int(Prim α A) = ∅) is significantly weaker than almost freeness although in this case, we need to assume that a closed ideal J is essential in order to conclude that it contains an invariant ideal.For simplicity, we consider only ordinary actions in this article but the main theorem remains true for twisted actions because of the stabilisation trick of Packer-

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Some permanence properties of C∗-unique groups

Cited by 7 publications

References 23 publications

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

Invariant ideals of twisted crossed products

Topological treatments of some results on invariant ideals

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