Strongly 1-bounded Von Neumann Algebras
نویسنده
چکیده
Suppose F is a finite set of selfadjoint elements in a tracial von Neumann algebra M . For α > 0, F is α-bounded if Pα(F ) < ∞ where Pα is the α-packing entropy of F introduced in [7]. 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 ∈ F with χ(x) > −∞. It is shown that if M is strongly 1-bounded, then any finite set 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 those which have propertyΓ, have Cartan subalgebras, are non-prime, or the group von Neumann algebras of SLn(Z), n ≥ 3. If M and N are strongly 1-bounded and M ∩N = D 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 the normalizer of a strongly 1-bounded von Neumann algebra generates a von Neumann algebra with the property that any finite set of selfadjoint generators for the von Neumann algebra is 1-bounded.
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