Crossed products of operator spaces and applications to Harmonic Analysis of non commutative groups

Doctoral Dissertation uoadl:2956762 294 Read counter

Department of Mathematics
Library of the School of Science
Deposit date:
Andreou Dimitrios
Dissertation committee:
Αριστείδης Κατάβολος, Ομότιμος Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Μιχαήλ Ανούσης, Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Αιγαίου
Απόστολος Γιαννόπουλος, Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Παντελής Δοδός, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Γεώργιος Ελευθεράκης, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, Πανεπιστήμιο Πατρών
Ηλίας Κατσούλης, Καθηγητής, Τμήμα Μαθηματικών, East Carolina University
Κωνσταντίνος Τύρος, Αναπληρωτής Καθηγητής, Τμήμα Μαθηματικών, ΕΚΠΑ
Original Title:
Crossed products of operator spaces and applications to Harmonic Analysis of non commutative groups
Translated title:
Crossed products of operator spaces and applications to Harmonic Analysis of non commutative groups
We study crossed products arising from actions of locally compact groups on dual operator spaces, which generalize the classical crossed product construction for group actions on von Neumann algebras. Our methods rely on the concepts of Hopf-von Neumann algebras and comodules, since they provide a very natural framework for the study of duality phenomena concerning actions of not necessarily abelian locally compact groups. Below, we give a brief summary of the main results of this thesis.

The first chapter is an introduction to the mathematical background which is necessary in order to develop the general theory later. In particular, we expose some basic definitions and properties regarding (dual) operator spaces and operator space tensor products, the notion of stable point-w*-convergence and the main von Neumann (and Banach) algebras associated with locally compact groups.

In the second chapter, we deal with Hopf-von Neumann algebras and comodules which are dual operator spaces. In particular, we study the concepts of saturated and non-degenerate comodules over a general Hopf-von Neumann algebra, as well as the interplay between the two notions. For instance, we prove that if a Hopf-von Neumann algebra admits only non-degenerate comodules, then it admits only saturated comodules. Also, we show that the latter, i.e. that a Hopf-von Neumann algebra admits only saturated comodules (which is by definition an algebraic concept), is equivalent to certain approximation conditions. As an application, we prove that a locally compact group G has the approximation property (AP) of Haagerup and Kraus if and only if every saturated comodule over the group von Neumann algebra L(G) is non-degenerate.

In the third chapter, we study spatial and Fubini crossed products for group actions on dual operator spaces and their natural comodule structure (dual actions). These two crossed products coincide for group actions on von Neumann algebras by the classical Digernes-Takesaki theorem. However, they may be different for arbitrary dual operator spaces. Using duality theory of actions and the general theory of comodules, we prove that the Fubini crossed product of an action is the smallest saturated comodule containing the respective spatial crossed product, whereas the latter is the largest non-degenerate subcomodule of the former. Therefore, by the previous characterization of groups with the AP, we obtain our main theorem, which states that a locally compact group G has the AP if and only if the Fubini and spatial crossed products coincide for any G-action on some dual operator space. This improves a recent result of Crann and Neufang.

Finally, in the last chapter, we apply the general theory in order to obtain a more conceptual perspective of certain classes of bimodules over group von Neumann algebras arising as crossed products of dual operator spaces which are not necessarily von Neumann algebras. As aresult, we obtain a less technical proof of a theorem of Anoussis-Katavolos-Todorov and we answer to a question raised by the same authors regarding the Fourier algebra and its ideals. Also, we extend a result of Crann and Neufang concerning L(G)-bimodules when the group G satisfies a condition a priori weaker than the AP.
Main subject category:
crossed products, operator spaces, approximation property, comodules
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