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- W2951608015 abstract "We define an analog of Voiculescu's free entropy for n-tuples of unitaries (u_{1},...,u_{n}) in a tracial von Neumann algebra M, normalizing a unital diffuse abelian subalgebra B in M. Using this quantity, we define the free dimension delta_{0}(u_{1},..,u_{n}btw B). This number depends on (u_{1},... ,u_{n}) only up ``orbit equivalence'' over B. In particular, if R is an measirable equivalence relation on [0,1] generated by n automorphisms alpha_{1},...,alpha_{n}, let u_{1},..., u_{n} be the unitaries implementing alpha_{1},..,alpha_{n} in the Feldman-Moore crossed product algebra M=W^{*}([0,1],R), and let B be the canonical copy of L^infty functions on [0,1] inside M. In this way, we obtain an invariant delta (R)=delta _{0}(u_{1},...,u_{n},btw B) of the equivalence relation (R). If R is treeable, delta (R) coincides with the cost C(R) of R in the sense of Gaboriau. For a general equivalence relation R posessing a finite graphing, delta(R)leq C(R). Using the notion of free dimension, we define an dynamical entropy invariant for an automorphism of a measurable equivalence relation (or more generally of an r-discrete measure groupoid), and give examples." @default.
- W2951608015 created "2019-06-27" @default.
- W2951608015 creator A5085505149 @default.
- W2951608015 date "1999-12-29" @default.
- W2951608015 modified "2023-09-27" @default.
- W2951608015 title "Microstates free entropy and cost of equivalence relations" @default.
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