140 research outputs found
Regularity of Polynomials in Free Variables
We show that the spectral measure of any non-commutative polynomial of a
non-commutative -tuple cannot have atoms if the free entropy dimension of
that -tuple is (see also work of Mai, Speicher, and Weber). Under
stronger assumptions on the -tuple, we prove that the spectral measure is
not singular, and measures of intervals surrounding any point may not decay
slower than polynomially as a function of the interval's length.Comment: The second version (joint with I. Charlesworth) considerably improves
our previous results. The main new result is non-singularity of the spectral
measure of a non-commutative polynomial of n variables under assumptions of
existence of Voiculescu's dual syste
On Classical Analogues of Free Entropy Dimension
We define a classical probability analogue of Voiculescu's free entropy
dimension that we shall call the classical probability entropy dimension of a
probability measure on . We show that the classical probability
entropy dimension of a measure is related with diverse other notions of
dimension. First, it can be viewed as a kind of fractal dimension. Second, if
one extends Bochner's inequalities to a measure by requiring that microstates
around this measure asymptotically satisfy the classical Bochner's
inequalities, then we show that the classical probability entropy dimension
controls the rate of increase of optimal constants in Bochner's inequality for
a measure regularized by convolution with the Gaussian law as the
regularization is removed. We introduce a free analogue of the Bochner
inequality and study the related free entropy dimension quantity. We show that
it is greater or equal to the non-microstates free entropy dimension
- …