The Free Entropy Dimension of Hyperfinite Von Neumann Algebras

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 choice of generators for M . In the course of the argument we show that if {b1, . . . , bn} is a set of self-adjoint generators for a von Neumann algebra R with a tracial state and {b1, . . . , bn} has finite dimensional approximants, then for any b ∈ R δ0(b1, . . . , bn) ≥ δ0(b). Combined with a result by Voiculescu this implies that if R has a regular diffuse hyperfinite von Neumann subalgebra, then δ0(b1, . . . , bn) = 1.