Journal of Operator Theory

Volume 45, Issue 1, Winter 2001 pp. 195-208.

Conformal nets, maximal temperature and models from free probability

Authors: Claudio D'Antoni (1), Roberto Longo (2), and Florin Radulescu (3)
Author institution: (1) Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata'', Via della Ricerca Scientifica, I--00133 Roma, Italy
(2) Dipartimento di Matematica, Universit\`a di Roma ``Tor Vergata'', Via della Ricerca Scientifica, I--00133 Roma, Italy
(3) Department of Mathematics, University of Iowa, Iowa City, IA 52246, USA

Summary: We consider conformal nets on $S^1$ of von Neumann algebras, acting on the full Fock space, arising in Free Probability. These models are twisted local, but non-local. We extend to the non-local case the general analysis of the modular structure. The local algebras turn out to be III$_1$-factors associated with free groups. We use our setup to show examples exhibiting arbitrarily large maximal temperatures, but failing to satisfy the split property, then clarifying the relation between the latter property and the trace class conditions on ${\rm e}^{-\b L}$, where $L$ is the conformal Hamiltonian.

Keywords: von Neumann algebras, free probability, conformal quantum field theory, nuclearity, split

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