Free analog of pressure and its Legendre transform
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.
math.OA↗
arXiv subjects
Fumio Hiai
Publications and source records attributed to Fumio Hiai.
Inequalities related to free entropy derived from random matrix approximation
Biane proved the free analog of the logarithmic Sobolev inequality for probability measures on the real line by means of random matrix approximation procedure. We show that the same method can be applied to reprove Biane and Voiculescu's free analog of Talagrand's transportation cost inequality for measures on the real line. Furthermore, we prove the free analogs of the logarithmic Sobolev inequality and the transportation cost inequality for measures on 1-dimensional torus as well by extending the method to special unitary random matrices.
math.OA↗
Automorphisms of free product-type and their crossed-products
A continuous family of non-outer conjugate aperiodic automorphisms whose crossed-products are all isomorphic is given on every interpolated free group factor. An explicit "duality" relationship between compact co-commutative Kac algebra (minimal) free product actions and free shift actions is also discussed.
math.OA↗