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Register a new userA Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems
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Siegfried Echterhoff
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Imprimitivity theorems provide a fundamental tool for studying the representation theory and structure of crossed-product C*-algebras. In this work, we show that the Imprimitivity Theorem for induced algebras, Greens Imprimitivity Theorem for actions of groups, and Mansfields Imprimitivity Theorem for coactions of groups can all be viewed as natural equivalences between various crossed-product functors among certain equivariant categories. The categories involved have C*-algebras with actions or coactions (or both) of a fixed locally compact group G as their objects, and equivariant equivalence classes of right-Hilbert bimodules as their morphisms. Composition is given by the balanced tensor product of bimodules. The functors involved arise from taking crossed products; restricting, inflating, and decomposing actions and coactions; inducing actions; and various combinations of these. Several applications of this categorical approach are also presented, including some intriguing relationships between the Green and Mansfield bimodules, and between restriction and induction of representations.
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After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories.
We provide the rigorous foundations for a categorical approach to the classification of C*-dynamics up to cocycle conjugacy. Given a locally compact group $G$, we consider a category of (twisted) $G$-C*-algebras, where morphisms between two objects are allowed to be equivariant maps or exterior equivalences, which leads to the concept of so-called cocycle morphisms. An isomorphism in this category is precisely a cocycle conjugacy in the known sense. We show that this category allows sequential inductive limits, and that some known functors on the usual category of $G$-C*-algebras extend. After observing that this setup allows a natural notion of (approximate) unitary equivalence, the main aim of the paper is to generalize the fundamental intertwining results commonly employed in the Elliott program for classifying C*-algebras. This reduces a given classification problem for C*-dynamics to the prevalence of certain uniqueness and existence theorems, and may provide a useful alternative to the Evans--Kishimoto intertwining argument in future research.
In this paper, we accomplish two objectives. Firstly, we extend and improve some results in the theory of (semi-)strongly self-absorbing C*-dynamical systems, which was introduced and studied in previous work. In particular, this concerns the theory when restricted to the case where all the semi-strongly self-absorbing actions are assumed to be unitarily regular, which is a mild technical condition. The central result in the first part is a strengthened version of the equivariant McDuff-type theorem, where equivariant tensorial absorption can be achieved with respect to so-called very strong cocycle conjugacy. Secondly, we establish completely new results within the theory. This mainly concerns how equivariantly $cal Z$-stable absorption can be reduced to equivariantly UHF-stable absorption with respect to a given semi-strongly self-absorbing action. Combining these abstract results with known uniqueness theorems due to Matui and Izumi-Matui, we obtain the following main result. If $G$ is a torsion-free abelian group and $cal D$ is one of the known strongly self-absorbing C*-algebras, then strongly outer $G$-actions on $cal D$ are unique up to (very strong) cocycle conjugacy. This is new even for $mathbb{Z}^3$-actions on the Jiang-Su algebra.
A base $Delta$ generating the topology of a space $M$ becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over $Delta$ of fixed-point spaces (typically C*-algebras) under the action of a group $G$, in general one cannot find a precosheaf of $G$-spaces having it as fixed-point precosheaf. Rather one gets a gerbe over $Delta$, that is, a twisted precosheaf whose twisting is encoded by a cocycle with coefficients in a suitable 2-group. We give a notion of holonomy for a gerbe, in terms of a non-abelian cocycle over the fundamental group $pi_1(M)$. At the C*-algebraic level, holonomy leads to a general notion of twisted C*-dynamical system, based on a generic 2-group instead of the usual adjoint action on the underlying C*-algebra. As an application of these notions, we study presheaves of group duals (DR-presheaves) and prove that the dual object of a DR-presheaf is a group gerbe over $Delta$. It is also shown that any section of a DR-presheaf defines a twisted action of $pi_1(M)$ on a Cuntz algebra.
We investigate some ergodic and spectral properties of general (discrete) $C^*$-dynamical systems $({mathfrak A},Phi)$ made of a unital $C^*$-algebra and a multiplicative, identity-preserving $*$-map $Phi:{mathfrak A}to{mathfrak A}$, particularising the situation when $({mathfrak A},Phi)$ enjoys the property of unique ergodicity with respect to the fixed-point subalgebra. For $C^*$-dynamical systems enjoying or not the strong ergodic property mentioned above, we provide conditions on $lambda$ in the unit circle ${zin{mathbb C}mid |z|=1}$ and the corresponding eigenspace ${mathfrak A}_lambdasubset{mathfrak A}$ for which the sequence of Cesaro averages $left(frac1{n}sum_{k=0}^{n-1}lambda^{-k}Phi^kright)_{n>0}$, converges point-wise in norm. We also describe some pivotal examples coming from quantum probability, to which the obtained results can be applied.
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