Related papers: A new approach to induction and imprimitivity resu…

We first compare the mathematical structure of quantum and classical mechanics when both are formulated in a C*-algebraic framework. By using finite von Neumann algebras, a quantum mechanical analogue of Liouville's theorem is then…

We introduce the notion of confined subalgebras in the context of the group von Neumann algebra. We also define Uniformly Recurrent States -- an operator-algebraic analog of Uniformly Recurrent Subgroups. Using this framework, we show that…

We study $C^*$-algebras arising from $C^*$-correspondences, which was introduced by the author. We prove the gauge-invariant uniqueness theorem, and obtain conditions for our $C^*$-algebras to be nuclear, exact, or satisfy the Universal…

We investigate compact quantum group actions on unital $C^*$-algebras by analyzing invariant subsets and invariant states. In particular, we come up with the concept of compact quantum group orbits and use it to show that countable compact…

We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras…

In this paper we construct and study the representation theory of a Hopf C^*-algebra with approximate unit, which constitutes quantum analogue of a compact group C^*-algebra. The construction is done by first introducing a…

We construct a topos of quantum sets and embed into it the classical topos of sets. We show that the internal logic of the topos of sets, when interpreted in the topos of quantum sets, provides the Birkhoff-von Neumann quantum propositional…

We study actions of compact quantum groups on type I factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz' results concerning Peter-Weyl theory for…

Three properness conditions for actions of locally compact groups on C*-algebras are studied, as well as their dual analogues for coactions. To motivate the properness conditions for actions, the commutative cases (actions on spaces) are…

We study the $C^*$-algebra $C^*(\kappa)$ generated by the Koopman representation $\kappa=\kappa^\mu$ of a locally compact groupoid $G$ acting on a measure space $(X,\mu)$, where $\mu$ is quasi-invariant for the action. We interpret $\kappa$…

We construct ergodic actions of compact quantum groups on C^*-algebras and von Neumann algebras, and exhibit phenomena of such actions that are of a different nature from ergodic actions of compact Lie groups. In particular, we construct:…

Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous correpresentations of $C^*$-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is…

Spielberg's construction of C*-algebras from left cancellative small categories is a common generalization for most C*-algebras one would consider to come from ``combinatorial data,'' including graph and $k$-graph C*-algebras, Li's…

It is well-known that the maximalization of a coaction of a locally compact group on a C*-algebra enjoys a universal property. We show how this important property can be deduced from a categorical framework by exploiting certain properties…

In this work we construct a C*-algebra from an injective endomorphisms of some group G, allowing the endomorphism to have infinite cokernel. We generalize results obtained by I. Hirshberg and also by J. Cuntz and A. Vershik. In good cases…

We define a categorical framework in which we build a systematic construction that provides generic invariants for C*-algebras. The benefit is significant as we show that any invariant arising this way automatically enjoys nice properties…

We construct a general cohomological induction isomorphism from a uniform measure equivalence of locally compact, second countable, unimodular groups which, as a special case, yields that the graded cohomology rings of quasi-isometric,…

We present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert…

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed…

We introduce the Pythagorean C*-algebras and use the category/functor method to construct unitary representations of Thompson's groups from representations of them. We calculate several examples.