A propagation property of free entropy dimension

Abstract

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 ${\delta }_0(X)\le sup{\delta }_0(Y):YisafiniteC\ast -generatingsetforA$ where ${\delta }_0$ denotes the microstates free entropy dimension. It follows that if $sup{\delta }_0(Y):YisafiniteC\ast -generatingsetforC{\ast }_{red}({\mathbb{F}}_2)<\infty$, then the free group factors are all nonisomorphic.

4 pages, minor corrections

References in corpus (1)

• Fractal entropies and dimensions for microstate spaces