Orbital approach to microstate free entropy

Abstract

Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy ${\chi }_{o}rb$ for non-commutative self-adjoint random variables (also for "hyperfinite random multi-variables"). Besides its basic properties the relation of ${\chi }_{o}rb$ with the usual free entropy $\chi$ is shown. Moreover, the dimension counterpart ${\delta }_{0,orb}$ of ${\chi }_{o}rb$ is discussed, and we obtain the relation of ${\delta }_{0,orb}$ with the original free entropy dimension ${\delta }_0$ with applications to ${\delta }_0$ itself.

38 pages; Section 5 was largely improved and Section 6 was added

References in corpus (2)

• Free Entropy Dimension in Amalgamated Free Products
• Notes on microstate free entropy of projections

Cited by in corpus (9)

• Liberation of Projections
• Strong Convergence of Unitary Brownian Motion
• Remarks on free mutual information and orbital free entropy
• Liberation, free mutual information and orbital free entropy
• A remark on orbital free entropy
• Orbital free pressure and its Legendre transform
• Matrix liberation process I: Large deviation upper bound and almost sure convergence
• Analogues of Entropy in Bi-Free Probability Theory: Microstates
• Matrix liberation process II: Relation to orbital free entropy