Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn a categorical framework for classifying C*-dynamics up to cocycle conjugacy
61
0
0.0
(
0
)
Authors
Gabor Szabo
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
We show that if (A,a) and (B,b) are automorphic multivariable C*-dynamical systems with isometrically isomorphic tensor algebras (or semi crossed products), then the systems are piecewise conjugate over their Jacobson spectrum. This answers a question of Kakariadis and the author.
Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $af$ and $bt$ are two automorphisms with the Rokhlin property that {induce the same action on the $K$-theoretical data of $A$.} We show that $af$ and $bt$ are strongly cocycle conjugate and uniformly approximately conjugate, that is, there exists a sequence of unitaries ${u_n}subset A$ and a sequence of strongly asymptotically inner automorphisms $sigma_n$ such that $$ af={rm Ad}, u_ncirc sigma_ncirc btcirc sigma_n^{-1}andeqn lim_{ntoinfty}|u_n-1|=0, $$ and that the converse holds. {We then give a $K$-theoretic description as to exactly when $af$ and $bt$ are cocycle conjugate, at least under a mild restriction. Moreover, we show that given any $K$-theoretical data, there exists an automorphism $af$ with the Rokhlin property which has the same $K$-theoretical data.
Fragment and glider representations (introduced by F. Caenepeel, S. Nawal, and F. Van Oystaeyen) form a generalization of filtered modules over a filtered ring. Given a $Gamma$-filtered ring $FR$ and a subset $Lambda subseteq Gamma$, we provide a category $operatorname{Glid}_Lambda FR$ of glider representations, and show that it is a complete and cocomplete deflation quasi-abelian category. We discuss its derived category, and its subcategories of natural gliders and Noetherian gliders. If $R$ is a bialgebra over a field $k$ and $FR$ is a filtration by bialgebras, we show that $operatorname{Glid}_Lambda FR$ is a monoidal category which is derived equivalent to the category of representations of a semi-Hopf category (in the sense of E. Batista, S. Caenepeel, and J. Vercruysse). We show that the monoidal category of glider representations associated to the one-step filtration $k cdot 1 subseteq R$ of a bialgebra $R$ is sufficient to recover the bialgebra $R$ by recovering the usual fiber functor from $operatorname{Glid}_Lambda FR.$ When applied to a group algebra $kG$, this shows that the monoidal category $operatorname{Glid}_Lambda F(kG)$ alone is sufficient to distinguish even isocategorical groups.
The purpose of this short note was to outline the current status, then in 2011, of some research programs aiming at a categorification of parts of A.Connes non-commutative geometry and to provide an outlook on some possible subsequent developments in categorical non-commutative geometry.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
