A Hyperfinite Inequality for Free Entropy Dimension

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 the free entropy dimension problem. Free entropy dimension ([6], [7]) associates to an n-tuple of selfadjoint operators, X = {x1, . . . , xn}, in a tracial von Neumann algebra M a number δ0(X) called the (modified) free entropy dimension of X. δ0(X) is an asymptotic Minkowski or packing dimension of sets of n-tuples of matrices which model the behavior of X. The free entropy dimension problem simply asks whether δ0(X) = δ0(Y ) for any other m-tuple of selfadjoint elements Y satisfying Y ′′ = X . It is known from [8] that δ0 is an algebraic invariant, i.e., δ0(X) = δ0(Y ) if X and Y generate the same algebra. The origin of this remark started with two extremely special and highly tractable cases of this problem, the first being: if X, Y and Z are finite sets of selfadjoint elements in M such that X ′′ = Z ′′ is hyperfinite, then is it true that δ0(X ∪ Y ) = δ0(Y ∪ Z)? The second problem concerns invariance of δ0 over the center: if Y is an arbitrary set of selfadjoint elements in M and y is any element in the center of Y , then is it true that δ0(Y ∪ {y}) = δ0(Y )? Both questions have affirmative answers and follow from a kind of hyperfinite inequality for δ0 : If X, Y, Z, are sets of selfadjoint elements in M and X generates a hyperfinite von Neumann algebra, then δ0(X ∪ Y ∪ Z) ≤ δ0(X ∪ Y ) + δ0(X ∪ Z)− δ0(X). Related inequalities of this nature can be found in Gaboriau’s work on the cost of equivalence relations [1]. The proof of the microstates inequality above is an application of the work in [3] paired with the packing formulation of δ0 in [4]. This remark has three sections. The first is a short list of assumptions. The second presents the hyperfinite inequality. The third and last section presents several corollaries. 1. PRELIMINARIES Throughout suppose M is a von Neumann algebra with a normal, tracial state φ. For any n ∈ N, | · |2 denotes the norm on (M k (C)) n given by |(x1, . . . , xn)|2 = ( ∑n j=1 trk(x 2 j )) 1 2 where trk is the tracial state on the k × k complex matrices, and | · |∞ denotes the operator norm. Uk denotes the k × k unitary matrices. For any k, n ∈ N, u ∈ Uk and x = (x1, . . . , xn) ∈ (M k (C)) , define 1991 Mathematics Subject Classification. Primary 46L54; Secondary 28A78. Research supported by the NSF Graduate Fellowship Program. 1