A Propagation Property of Free Entropy Dimension
نویسنده
چکیده
Let M be a tracial von Neumann algebra and A be a weakly dense unital C-subalgebra of M . We say that a set X is a W -generating set for M if the von Neumann algebra generated by X is M and that X is a C-generating set for A if the unital C-algebra generated by X is A. For any finite W generating set X for M we show that δ0(X) ≤ sup{δ0(Y ) : Y is a finite C-generating set for A}. It follows that if sup{δ0(Y ) : Y is a finite C-generating set for C∗ red(F2)} < ∞, then the free group factors are all nonisomorphic.
منابع مشابه
Free entropy and property T factors.
We show that a large class of finite factors has free entropy dimension less than or equal to one. This class includes all prime factors and many property T factors.
متن کاملFree Entropy Dimension and Property T
Suppose that N is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in ultraproducts of full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy...
متن کاملFree Entropy Dimension 1 for Some Generators ofProperty T Factors of Type II
The modified free entropy dimension of certain n-tuples of self-adjoint operators, satisfying sequential commutation, is shown to be ≤ 1. In particular the von Neumann algebras of type II1 of the groups SL(n,Z), n ≥ 3 have generators with free entropy dimension ≤ 1. The free entropy χ(X1, . . . , Xn) and the modified free entropy dimension δ0(X1, . . . , Xn) of an n-tuple of selfadjoint element...
متن کاملm at h . O A ] 2 9 M ar 2 00 6 ALL GENERATING SETS OF ALL PROPERTY T VON NEUMANN ALGEBRAS HAVE FREE ENTROPY DIMENSION ≤ 1
Suppose that N is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in ultraproducts of full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this description to show that the free entropy...
متن کاملOn Classical Analogues of Free Entropy Dimension
Abstract. We define a classical probability analog of Voiculescu’s free entropy dimension that we shall call the classical probability entropy dimension. We show that the classical probability entropy dimension is related with diverse other notions of dimension. First, it equals the fractal dimension. Second, if one extends Bochner’s inequalities to a measure by requiring that microstates aroun...
متن کامل