Strongly 1-Bounded Von Neumann Algebras

Abstract

Suppose F is a finite tuple of selfadjoint elements in a tracial von Neumann algebra M. For α > 0, F is α-bounded if $${\mathbb{P}}^\alpha (F) < \infty$$ where $${\mathbb{P}}^\alpha$$ is the free packing α-entropy of F introduced in [J3]. M is said to be strongly 1-bounded if M has a 1-bounded finite tuple of selfadjoint generators F such that there exists an $$x \in F$$ with $$\chi (x) > -\infty$$ . It is shown that if M is strongly 1-bounded, then any finite tuple of selfadjoint generators G for M is 1-bounded and δ0(G) ≤ 1; consequently, a strongly 1-bounded von Neumann algebra is not isomorphic to an interpolated free group factor and δ0 is an invariant for these algebras. Examples of strongly 1-bounded von Neumann algebras include (separable) II 1-factors which have property Γ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of $$SL_n({\mathbb{Z}}), n \geq 3$$ . If M and N are strongly 1-bounded and M ∩ N 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.