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Strongly 1-Bounded Von Neumann Algebras

Research paper by Kenley Jung

Indexed on: 24 Sep '07Published on: 24 Sep '07Published in: Geometric and Functional Analysis



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) II1-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 II1-factor generated by the normalizer of a strongly 1-bounded von Neumann subalgebra is strongly 1-bounded.