Strength of convergence and multiplicities in the spectrum of a C*-dynamical system

We consider separable $C^*$-dynamical systems $(A,G,\alpha)$ for which the induced action of the group $G$ on the spectrum $\hat A$ of the $C^*$-algebra $A$ is free. We study how the representation theory of the associated crossed-product $C^*$-algebra $A\rtimes_\alpha G$ depends on the representation theory of $A$ and the properties of the action of $G$ on $\hat A$. Our main tools involve computations of upper and lower bounds on multiplicity numbers associated to irreducible representations of $A\rtimes_\alpha G$. We apply our techniques to give necessary and sufficient conditions, in terms of $A$ and the action of $G$ on $\hat A$, for $A\rtimes_{\alpha}G$ to be (i) a continuous-trace $C^*$-algebra, (ii) a Fell $C^*$-algebra and (iii) a bounded-trace $C^*$-algebra. When $G$ is amenable, we also give necessary and sufficient conditions for the crossed-product $C^*$-algebra $A\rtimes_{\alpha}G$ to be (iv) a liminal $C^*$-algebra and (v) a Type I $C^*$-algebra. The results in (i), (iii)--(v) extend some earlier special cases in which $A$ was assumed to have the corresponding property.Comment: Publication version, to appear in Proc. London Math. So

Strength of convergence in the orbit space of a transformation group

Let (G, X) be a second-countable transformation group with G acting freely on X. It is shown that measure-theoretic accumulation of the action and topological strength of convergence in the orbit space X/G provide equivalent ways of quantifying the extent of non-properness of the action. These notions are linked via the representation theory of the transformation-group C*-algebra.Comment: 27 page

Spectral synthesis in the multiplier algebra of a C_0(X)-algebra

We are grateful to the referee for a number of helpful comments and for pointing out an error in the original proof of Theorem 3.6.Peer reviewedPostprin

The Inner Corona Algebra of a C0(X)-Algebra

We are grateful to the referee for a number of helpful comments.Peer reviewedPostprin

Separation properties in the primitive ideal space of a multiplier algebra

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Norms of inner derivations for multiplier algebras of C ∗ -algebras and group C ∗ -algebras, II

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The centre-quotient property and weak centrality for C∗-algebras

Funding The second-named author was fully supported by the Croatian Science Foundation under the project IP-2016-06-1046.Peer reviewedPostprin

The Dixmier property and tracial states for C*-algebras

A.T. was partially supported by an NSERC Postdoctoral Fellowship and through the EPSRC grant EP/N00874X/1. Acknowledgements We are grateful to Luis Santiago for helpful discussions at an early stage of this investigation. We would also like to thank the referee for providing helpful comments, which have led to a number of improvements.Peer reviewedPublisher PD