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SupposeM is a von Neumann algebra with normal, tracial state φ and {a1, . . . , an} is a set of self-adjoint elements in M. We provide an alternative uniform packing description of δ0(a1, . . . , an), the modified free entropy dimension of {a1, . . . , an}. For Arlan Ramsay In the attempt to understand the free group factors Voiculescu created a type of(More)
ABSTRACT. Suppose M is a hyperfinite von Neumann algebra with a tracial state φ and {a1, . . . , an} is a set of self-adjoint generators for M. We calculate δ0(a1, . . . , an), the modified free entropy dimension of {a1, . . . , an}. Moreover we show that δ0(a1, . . . , an) depends only on M and φ. Consequently δ0(a1, . . . , an) is independent of the(More)
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(More)
Suppose that N is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in ultraproducts of full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this(More)
If X,Y, Z are finite sets of selfadjoint elements in a tracial von Neumann algebra and X generates a hyperfinite von Neumann algebra, then δ0(X ∪Y ∪Z) ≤ δ0(X ∪Y )+δ0(X ∪Z)−δ0(X). We draw several corollaries from this inequality. In [9] Voiculescu describes the role of entropy in free probability. He discusses several problems in the area, one of which is(More)
[1] introduced fractal geometric entropies and dimensions for Voiculescu’s microstate spaces ([3], [4]). One can associate to a finite set of selfadjoint elements X in a tracial von Neumann algebra and an α > 0 an extended real number H(X) ∈ [−∞,∞]. H(X) is a kind of asymptotic logarithmic α-Hausdorff measure of the microstate spaces of X. One can also(More)
Suppose N ⊂ M is an inclusion of II1-factors of finite index. If N can be generated by a finite set of elements, then there exist finite generating sets X for N and Y for M such that δ0(X) ≥ δ0(Y ), where δ0 denotes Voiculescu’s microstates (modified) free entropy dimension. Moreover, given ǫ > 0 one has δ0(F ) ≥ δ0(G) ≥ ([M : N ] −2 − ǫ) · (δ0(F ) − 1) + 1(More)
Using Voiculescu’s notion of a matricial microstate we introduce fractal dimensions and entropy for finite sets of self-adjoint operators in a tracial von Neumann algebra. We show that they possess properties similar to their classical predecessors. We relate the new quantities to free entropy and free entropy dimension and show that a modified version of(More)
Suppose that N is a diffuse, property T von Neumann algebra and X is an arbitrary finite generating set of selfadjoint elements for N. By using rigidity/deformation arguments applied to representations of N in ultraproducts of full matrix algebras, we deduce that the microstate spaces of X are asymptotically discrete up to unitary conjugacy. We use this(More)