Introduction Let Γ be a fuchsian subgroup of P SL(2, R). In this paper we consider the Γ-equivariant form of the Berezin's quantization of the upper half plane which will correspond to a deformation quantization of the (singular) space H/Γ. Our main result is that the von Neumann algebra associated to the Γ− equivariant form of the quantization is stable isomorphic with the von Neumann algebra ...

in this article we study two different generalizations of von neumann regularity, namely strong topological regularity and weak regularity, in the banach algebra context. we show that both are hereditary properties and under certain assumptions, weak regularity implies strong topological regularity. then we consider strong topological regularity of certain concrete algebras. moreover we obtain ...

In these notes we develop a link between the Kadison-Singer problem and questions about certain dynamical systems. We conjecture that whether or not a given state has a unique extension is related to certain dynamical properties of the state. We prove that if any state corresponding to a minimal idempotent point extends uniquely to the von Neumann algebra of the group, then every state extends ...

Suppose X = {x1, . . . , xn} is an n-tuple of selfadjoint elements in a tracial von Neumann algebra M . If z = z∗ ∈ M and for some y = y∗ in the von Neumann algebra generated by X δ0(y, z) < δ0(y) + δ0(z), then χ(X ∪ {z}) = −∞ (here χ and δ0 denote the microstates free entropy and free entropy dimension, respectively). In particular, if z lies in the von Neumann algebra generated by X , then χ(...

We investigate various characterizations of the Haagerup property (H) for a second countable locally compact group G, in terms of orthogonal representations of G on Lp(M), the non-commutative Lp-space associated with the von Neumann algebra M. For a semi-finite von Neumann algebra M, we introduce a variant of property (H), namely property HLp(M), defined in terms of orthogonal representations o...

It is known that every semigroup of normal completely positive maps of a von Neumann can be “dilated” in a particular way to an E0-semigroup acting on a larger von Neumann algebra. The E0-semigroup is not uniquely determined by the completely positive semigroup; however, it is unique (up to conjugacy) provided that certain conditions of minimality are met. Minimality is a subtle property, and i...

We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L 2-homology for tracial algebras. As an application, we also give a new proof of a Theorem of Gaboriau on invaria...

Normal endomorphisms of von Neumann algebras need not be extendable to automorphisms of a larger von Neumann algebra, but they always have asymptotic lifts. We describe the structure of endomorphisms and their asymptotic lifts in some detail, and apply those results to complete the identification of asymptotic lifts of unital completely positive linear maps on von Neumann algebras in terms of t...

For von Neumann algebras M,N not isomorphic to C C and without type I2 summands, we show that for an order-isomorphism f : AbSub M! AbSub N between the posets of abelian von Neumann subalgebras of M and N , there is a unique Jordan ⇤-isomorphism g : M! N with the image g[S] equal to f(S) for each abelian von Neumann subalgebra S of M. The converse also holds. This shows the Jordan structure of ...

λ(s)δt = δst and ρ(s)δt = δts−1 for s, t ∈ Γ. The reduced group C∗-algebra C∗ λΓ is the C∗-algebra generated by λ, likewise for C∗ ρΓ; C∗ λΓ = λ(CΓ) ‖ ‖ ⊂ B(`2Γ) and C∗ ρΓ = ρ(CΓ) ‖ ‖ ⊂ B(`2Γ). The group von Neumann algebra LΓ is the von Neumann algebra generated by λ; LΓ = λ(CΓ)′′ = ρ(CΓ)′. We sometime use M instead of LΓ, and L2M instead of `2Γ. We denote by τ the canonical tracial state on L...