In this work we investigate the notion of action or coaction of a finite quantum groupoid in von Neumann algebras context. In particular we prove a double crossed product theorem and prove the existence of an universal von Neumann algebra on which any finite groupoids acts outerly. In previous works, N. Andruskiewitsch and S.Natale define for any match pair of groupoids two C∗-quantum groupoids...

ON THE DIAGONALS OF PROJECTIONS IN MATRIX ALGEBRAS OVER VON NEUMANN ALGEBRAS Soumyashant Nayak Richard V. Kadison The main focus of this dissertation is on exploring methods to characterize the diagonals of projections in matrix algebras over von Neumann algebras. This may be viewed as a non-commutative version of the generalized Pythagorean theorem and its converse (Carpenter’s Theorem) studie...

In 1967, when Kadison and Ringrose began the development of continuous cohomology theory for operator algebras, they conjectured that the cohomology groups Hn(M, M), n >/= 1, for a von Neumann algebra M, should all be zero. This conjecture, which has important structural implications for von Neumann algebras, has been solved affirmatively in the type I, IIinfinity, and III cases, leaving open o...

We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. The generalization yields new classes of quantum error correcting codes that have no finite-dimensional counterparts. The error correction theory we develop begins with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of...

We discuss information-theoretic concepts on infinite-dimensional quantum systems. In particular, we lift the smooth entropy formalism as introduced by Renner and collaborators for finite-dimensional systems to von Neumann algebras. For the smooth conditional minand max-entropy we recover similar characterizing properties and information-theoretic operational interpretations as in the finite-di...

An abelian self-commutator in a C*-algebra A is an element of the form A = XX−XX, with X ∈ A, such that XX and XX commute. It is shown that, given a finite AW*-factor A, there exists another finite AW*-factor M of same type as A, that contains A as an AW*-subfactor, such that any self-adjoint element X ∈ M of quasitrace zero is an abelian self-commutator in M. Introduction According to the Murr...

In this paper we study three aspects of (P(M)/ ∼), the set of Murray-von Neumann equivalence classes of projections in a von Neumann algebra M. First we determine the topological structure that (P(M)/ ∼) inherits from the operator topologies on M. Then we show that there is a version of the center-valued trace which extends the dimension function, even whenM is not σ-finite. Finally we prove th...

Let E, belonging to the spectrum of H, be a fixed energy level. We are interested in the states for which the expected value of the energy equals E; i.e., in the states ρ such that ρ(H) = E. The classical result says that the maximal value of the entropy for such states is attained for a so-called Gibbs state; that is, a state with the density matrix e βH tr eβH for some β ∈ R. In this note, we...

The theory of exact C∗-algebras was introduced by Kirchberg and has been influential in recent development of C∗-algebras. A fundamental result on exact C∗-algebras is a local characterization of exactness. The notion of weakly exact von Neumann algebras was also introduced by Kirchberg. In this paper, we give a local characterization of weak exactness. As a corollary, we prove that a discrete ...

We study the von Neumann algebra generated byq–deformed Gaussian elements li + l∗i where operators li fulfill the q–deformed canonical commutation relations lil∗j − ql ∗ j li = δij for −1 < q < 1. We show that if the number of generators is finite, greater than some constant depending on q, it is a II1 factor which does not have the property Γ . Our technique can be used for proving factorialit...