Strongly 1-Bounded Von Neumann Algebras |
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Authors: | Kenley Jung |
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Institution: | (1) Department of Mathematics, University of California, Los Angeles, CA 90095-1555, USA |
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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 where 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 with . 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 . 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.
Received: November 2005, Revision: March 2006, Accepted: March 2006 |
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Keywords: | Geometric measure theory free probability free entropy |
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