Assume M is a type II1 von Neumann algebra, the algebra of n × n matrices over another von Neumann algebra N , with the tracial state τM. In this article, we discuss the properties of semicircular elements that are free with, with respect to τM, the algebra Dn consisting of scalar diagonal matrices in M. Then we define a concept “matricial distance” of two elements in M and compute it when the ...

By the Von Neumann regular graph of R, we mean the graph that its vertices are all elements of R such that there is an edge between vertices x,y if and only if x+y is a von Neumann regular element of R, denoted by G_Vnr (R). For a commutative ring R with unity, x in R is called Von Neumann regular if there exists x in R such that a=a2 x. We denote the set of Von Neumann regular elements by V nr...

This paper addresses a conjecture in the work by Kadison and Kastler [Kadison RV, Kastler D (1972) Am J Math 94:38-54] that a von Neumann algebra M on a Hilbert space H should be unitarily equivalent to each sufficiently close von Neumann algebra N, and, moreover, the implementing unitary can be chosen to be close to the identity operator. This conjecture is known to be true for amenable von Ne...

The Stone spectrum of a von Neumann algebra is a generalization of the Gelfand spectrum, as was shown by de Groote. In this article we clarify the structure of the Stone spectra of von Neumann algebras of type In.

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...