Strongly 1-bounded von Neumann algebras

Abstract

Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M. For $\alpha >0$, F is $\alpha$-bounded if the free packing $\alpha$-entropy of F is bounded from above. We say that M is strongly 1-bounded if M has a 1-bounded finite set of selfadjoint generators F such that there exists an x in F with finite free entropy. It is shown that if M is strongly 1-bounded, then any finite set of selfadjoint generators G for M is 1-bounded and the microstates free entropy dimension of G is less than or equal to 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and the microstates free entropy dimension is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II_1-factors which have property Gamma, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of SL_n(Z), n >2. If M and N are strongly 1-bounded and their intersection is diffuse, then the von Neumann algebra generated by M and N is strongly 1-bounded. In particular, a free product of two strongly 1-bounded von Neumann algebras with amalgamation over a common, diffuse von Neumann subalgebra is strongly 1-bounded. It is also shown that a II_1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.