Carlo Cecchini

Conditional Expectations in von Neumann Algebras and a Theorem of Takesaki

Conditional expectations play an important role in classical probability theory. In the general context of von Neumann algebras they were impliciteiy used by von Neumann [41, Chap. II] and by Dixmier [14]. Nakamura and Turumaru [27] and Umegaki [36-391 introduced an axiomatic definition of the concept of conditional expectation in the framework of von Neumann (or C*-) algebras and established many properties of these objects especially in the context of von Neumann algebras with a finite trace. Their starting point was the characterization, given by Moy [26], of the classical conditional expectations as operators on spaces of measurable functions. Tomiyama showed [33 ] that conditional expectations, in the sense of the above mentioned authors, can be characterized as norm one projection in C*algebras. The importance of norm one projection in the classification problem of von Neumann algebras was recognized by Hakeda and Tomiyama [23] and subsequent research on this argument confirmed the usefulness of these objects. This line of thought culminated in the fundamental work of Connes [ 121 in which approximately finite von 245 0022.1236/82/020245-29

Lacunary Fourier series on compact Lie groups

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Classes of conditional expectations over von Neumann algebras

We prove that if G is a compact Lie group, with irreducible unitary representations Dγ of degree dγ, ∝G ¦ Tr(Dγ(x))¦4 dx → ∞, as dγ → ∞. This result implies that if E is a set of inequivalent irreducible unitary representations of G which is a Λ(4) set, or even a central Λ(4) set, the degrees of the representations of E must be bounded. In particular, if G is semisimple E must be finite.

Stochastic coupling for von Neumann algebras

Let N ⊂ M be von Neumann algebras and Eω: M → N an ω-conditional expectation mapping. For a state ψ of N an extension \gyEω of ψ with respect to Eω is described. The relation Eω ~ Eϑ defined to hold if \gyEω = \gyEϑ for every ψ is an equivalence relation. The family of equivalence classes possesses an affine structure and shows analogy with the normal state space of a von Neumann algebra.

Magnetic Field as Pseudovector Entity in Physics Education

We consider even and odd stochastic transitions of von Neumann algebras when dual mappings intertwine (couple) modular groups of the corresponding states (with the occurrence of a sign exchange for the odd case). We show that one can define modular objects and cones associated to linear combinations of von Neumann algebras, which generalize objects and cones in the standard modular theory. In the odd case, we find sufficient conditions for the intertwining property and consider several applications to noncommutative Markov processes.

Markovian processes on mutually commuting von Neumann algebras

The analysis of the nature of the magnetic field offers the ideal framework in which students could address the mutual integration between mathematics and physical aspects facing experimentally the analysis of the phenomenology. Took look how students face experiments in which the phenomenology founds the theory and the mathematics offers its formalism as the best language to describe the explored phenomena, an activity concerning the pseudovectorial nature of the magnetic field was performed looking at the way in which the students’ reasoning evolve.

State extensions and a Radon-Nikodým theorem for conditional expectations on von Neumann algebras

1. The aim of this paper is to study markovianity for states on von Neumann algebras generated by the union of (not necessarily commutative) von Neumann subagebras which commute with each other. This study has been already begun in [2] using several a priori different notions of noncommutative markovianity. In this paper we assume to deal with the particular case of states which define odd stochastic couplings (as developed in [3]) for all couples of von Neumann algebras involved. In this situation these definitions are equivalent, and in this case it is possible to get the full noncommutative generalization of the basic classical Markov theory results. In particular we get a correspondence theorem, and an explicit structure theorem for Markov states.