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...

Von Neumann conjectured that the countable chain condition and the weak (ω, ω)-distributive law characterize measurable algebras among Boolean σalgebras [Mau]. Consistent counter-examples have been obtained by Maharam [Mah], Jensen [J], Glówczyński [Gl], and Veličković [V]. However, whether von Neumann’s proposed characterization of measurable algebras fails within ZFC remains an open problem. ...

The von Neumann algebra free product of arbitary finite dimensional von Neumann algebras with respect to arbitrary faithful states, at least one of which is not a trace, is found to be a type III factor possibly direct sum a finite dimensional algebra. The free product state on the type III factor is what we call an extremal almost periodic state, and has centralizer isomorphic to L(F∞). This a...

Let M be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec-Pe lczynski subsequence decomposition of bounded sequences in L[0, 1] to the case of the Haagerup L-spaces (1 ≤ p < ∞). In particular, we prove that if (φn)n is a bounded sequence in the predual M∗ of M, then there exist a subsequence (φnk)k of (φn)n, a decomposition φnk = yk + zk su...

We describe a Markov dilation for any weak* continuous semigroup \((T_t)_t \geqslant 0\) of selfadjoint unital completely positive Fourier multipliers acting on the group von Neumann algebra \(\mathrm {VN}(G)\) locally compact G.