[1] introduced fractal geometric entropies and dimensions for Voiculescu’s microstate spaces ([3], [4]). One can associate to a finite set of selfadjoint elements X in a tracial von Neumann algebra and an α > 0 an extended real number H(X) ∈ [−∞,∞]. H(X) is a kind of asymptotic logarithmic α-Hausdorff measure of the microstate spaces of X. One can also define a free Hausdorff dimension of X, de...

The free analog of the pressure is introduced for multivariate noncommutative random variables and its Legendre transform is compared with Voiculescu’s microstate free entropy. Introduction The entropy and the pressure are two fundamental ingredients in both classical and quantum statistical mechanics, in particular, in classical and quantum lattice systems (see [11, 2] for example). They are t...

We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neumann algebras defined in [Po1] is equivalent to a weaker property, in which no “continuity constants” are required. The notion of relative property (T) (or rigidity) for inclusions of finite von Neumann algebras with countable decomposable center was introduced in ([P1]) by requiring that one of the ...

Let A ⊆ B(H) be a C∗-algebra of operators and let P1 ≤ P2 ≤ . . . be an increasing sequence of finite dimensional projections in B(H). In a previous paper [3] we developed methods for computing the spectrum of self adjoint operators T ∈ A in terms of the spectra of the associated sequence of finite dimensional compressions PnTPn. In a suitable context, we showed that this is possible when Pn in...

We study some general properties of tracial C∗-algebras. In the first part, we consider Dixmier type approximation theorem and characterize symmetric amenability for C∗-algebras. In the second part, we consider continuous bundles of tracial von Neumann algebras and classify some of them.