Free Analog of Pressure and Its Legendre Transform

The free analog of the pressure is introduced for multivariate noncommutative random variables and its Legendre transform is compared with Voiculescu’s microstate free entropy. Introduction The entropy and the pressure are two fundamental ingredients in both classical and quantum statistical mechanics, in particular, in classical and quantum lattice systems (see [11, 2] for example). They are the dual concepts of each other; more precisely, the entropy function is the Legendre transform of the pressure function and vice versa under a certain duality between the state space and the potential space, and an equilibrium state associated with a potential is usually described by the so-called variational principle (the equality case of the Legendre transform). The free entropy introduced by D. Voiculescu [15, 16] has played a central role in free probability theory as the free analog of the Boltzmann-Gibbs entropy in classical theory. It then would be natural to consider the free probabilistic analog of the pressure. In [8] we indeed introduced the free pressure of real continuous functions on the interval [−R,R] and showed its properties like the statistical mechanical pressure (see Section 1 of this paper). The aim of the present paper is to introduce the notion of free pressure for multivariate noncommutative random variables and to investigate it in connection with the free entropy. (The contents of Sections 2 and 3 were indeed announced in [7, §§4.4].) In [8] we adopted the Legendre transform of the minus free entropy of probability measures to define the free pressure of real continuous functions. The idea here is opposite; we will first introduce the free pressure of noncommutative multivariables in the so-called microstate approach, and then we will examine what is the Legendre transform of the free pressure. In Section 2 we define, given N ∈ N and R > 0, the free pressure πR(h) for selfadjoint elements h of the N -fold full free product C∗-algebra A (N) R := C([−R,R]) ⋆N and give its basic properties. In Sections 3 and 4 we consider the Legendre transform ηR(μ) of πR for tracial states μ on A (N) R under the duality between the selfadjoint elements and the tracial states. For an N -tuple (a1, . . . , aN ) of selfadjoint noncommutative random variables in a W ∗-probability space (M, τ) such that ‖ai‖ ≤ R, a tracial state μ(a1,...,aN ) on A (N) R can be defined by μ(a1,...,aN )(h) := τ(h(a1, . . . , aN )) for h ∈ A (N) R where h(a1, . . . , aN ) is the noncommutative “functional calculus” of (a1, . . . , aN ). We then define the free entropy-like quantity ηR(a1, . . . , aN ) as ηR(μ(a1,...,aN )) and also η(a1, . . . , aN ) := supR>0 ηR(a1, . . . , aN ). The properties of ηR(a1, . . . , aN ) are similar to those of Voiculescu’s microstate free entropy χ(a1, . . . , aN ) while they do not generally coincide. But it is shown that ηR(a1, . . . , aN ) ≥ χ(a1, . . . , aN ) holds and equality arises when a1, . . . , aN are free. Also, we have ηR(a1, a2) = 1 Supported in part by Grant-in-Aid for Scientific Research (C)14540198 and by the program “R&D support scheme for funding selected IT proposals” of the Ministry of Public Management, Home Affairs, Posts and Telecommunications. 1 2 F. HIAI χ(a1, a2) if a1 + ia2 is R-diagonal (Section 5). In Section 6 we slightly modify πR to define the free pressure π (2) R (g) for selfadjoint elements g of A (N) R ⊗min A (N) R and prove that the quantity η̃(a1, . . . , aN ) induced from π (2) R via Legendre transform is equal to χ(a1, . . . , aN ). In this way, the free entropy can be understood as the Legendre transform of a certain free probabilistic pressure. Finally in Section 7 we consider the Gibbs probability measure on the N -fold product of { A ∈ M sa n : ‖A‖ ≤ R } associated with h0 ∈ ( A (N) R sa , and we examine the asymptotic behavior of its Boltzmann-Gibbs entropy as n → ∞ in relation to ηR(μ0) of an equilibrium tracial state μ0 associated with h0, i.e., a tracial state μ0 on A (N) R satisfying πR(h0) = −μ0(h0) + ηR(μ0). 1. Preliminaries Let (M, τ) be a tracial W ∗-probability space, that is, M is a von Neumann algebra with a faithful normal tracial state τ , and Msa be the set of selfadjoint elements in M. Let Mn be the algebra of n×n complex matrices and M sa n the set of selfadjoint matrices in Mn. The normalized trace of A ∈ Mn is denoted by trn(A) and the operator norm of A by ‖A‖. In [16] D. Voiculescu introduced the free entropy of an N -tuple (a1, . . . , aN ) of noncommutative random variables in Msa as follows: For each R > 0, ε > 0 and n, r ∈ N define ΓR(a1, . . . , aN ;n, r, ε) := { (A1, . . . , AN ) ∈ (M sa n ) N : ‖Ai‖ ≤ R, |trn(Ai1 · · ·Aik)− τ(ai1 · · · aik)| ≤ ε, 1 ≤ i1, . . . , ik ≤ N, k ≤ r } , χR(a1, . . . , aN ) := lim r→∞, ε→+0 lim sup n→∞ ( 1 n2 log Λ n ( ΓR(a1, . . . , aN ;n, r, ε) )