Non-microstates free entropy dimension for groups
نویسندگان
چکیده
منابع مشابه
Non-microstates Free Entropy Dimension for Groups
We show that for any discrete finitely-generated group G and any self-adjoint n-tuple X1, . . . , Xn of generators of the group algebra CG, Voiculescu’s non-microstates free entropy dimension δ(X1, . . . , Xn) is exactly equal to β1(G) − β0(G) + 1, where βi are the L Betti numbers of G.
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We give an general estimate for the non-microstates free entropy dimension δ∗(X1, . . . , Xn). If X1, . . . , Xn generate a diffuse von Neumann algebra, we prove that δ∗(X1, . . . , Xn) ≥ 1. In the case that X1, . . . , Xn are q-semicircular variables as introduced by Bozejko and Speicher and qn < 1, we show that δ∗(X1, . . . , Xn) > 1. We also show that for |q| < √ 2−1, the von Neumann algebra...
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His approach involved non-commutative Hilbert transform and is algebraic in nature. In the case that B = C, this quantity is denoted χ(X1, . . . , Xn), and its properties are very similar to those of the free entropy χ(X1, . . . , Xn) introduced by Voiculescu in [4] using microstates; in fact, it may very well be that the two quantities coinside. Using the microstates approach to free entropy, ...
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ژورنال
عنوان ژورنال: GAFA Geometric And Functional Analysis
سال: 2005
ISSN: 1016-443X,1420-8970
DOI: 10.1007/s00039-005-0513-z