It is proved that every continuous derivation on the *-algebra S(M, τ) of all τ-measurable operators affiliated with a von Neumann algebra M is inner. For every properly infinite von Neumann algebra M, any derivation on the *-algebra S(M, τ) is inner.

methods

Theory of Operator Algebras II

Continuous derivations on algebras of locally measurable operators are inner

These notes give an introduction to some aspects of quantum Markov processes Quantum Markov processes come into play whenever a mathematical description of irreversible time behaviour of quantum systems is aimed at Indeed, there is hardly a book on quantum optics without having at least a chapter on quantum Markov processes However, it is not always easy to recognize the basic concepts of probability theory in families of creation and annihilation operators on Fock space Therefore, in these lecture notes much emphasis is put on explaining the intuition behind the mathematical machinery of classical and quantum probability The lectures start with describing how probabilistic intuition is cast into the mathematical language of classical probability (Sects 41-43) Later on, we show how this formulation can be extended such as to incorporate the Hilbert space formulation of quantum mechanics (Sects 44,45) Quantum Markov processes are constructed and discussed in Sects 46,47, and we add some further discussions and examples in Sects 48-411

Quantum Markov Processes

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Banach principle in the space of τ-measurable operators

Fixed point property for Banach algebras associated to locally compact groups

In this article, we give a definition for measured quantum groupoids. We want to get objects with duality extending both quantum groups and groupoids. We base ourselves on J. Kustermans and S. Vaes' works about locally compact quantum groups that we generalize thanks to formalism introduced by M. Enock and J.M. Vallin in the case of inclusion of von Neumann algebras. From a structure of Hopf-bimodule with left and right invariant operator-valued weights, we define a fundamental pseudo-multiplicative unitary. To get a satisfying duality in the general case, we assume the existence of an antipode given by its polar decomposition. This theory is illustrated with many examples among others inclusion of von Neumann algebras (M. Enock) and a sub family of measured quantum groupoids with easier axiomatic.

Theory of Operator Algebras II

Citations

Continuous derivations on algebras of locally measurable operators are inner

Quantum Markov Processes

Banach principle in the space of τ-measurable operators

Fixed point property for Banach algebras associated to locally compact groups

Measured quantum groupoids

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Citations

Continuous derivations on algebras of locally measurable operators are inner

Quantum Markov Processes

Banach principle in the space of τ-measurable operators

Fixed point property for Banach algebras associated to locally compact groups

Measured quantum groupoids

Related Papers (5)

Fundamentals of the Theory of Operator Algebras

Operator algebras and quantum statistical mechanics

C-Algebras and Their Automorphism Groups

C[*]-algebras and W[*]-algebras

Completely Bounded Maps and Operator Algebras

C[]-algebras and W[]-algebras