KMS States, Entropy and the Variational Principle¶in Full C*-Dynamical Systems

Abstract

Abstract. To any periodic and full C * -dynamical system (A, α, R), an invertible operator s acting on the Banach space of trace functionals of the fixed point algebra is canonically associated. KMS states correspond to positive eigenvectors of s. A Perron-Frobenius type theorem asserts the existence of KMS states at inverse temperatures equal the logarithms of the inner and outer spectral radii of s (extremal KMS states). Examples arising from subshifts in symbolic dynamics, self-similar sets in fractal geometry and noncommutative metric spaces are discussed.Certain subshifts are naturally associated to the system, and criteria for the equality of their topological entropy and inverse temperatures of extremal KMS states are given.Unital completely positive maps σ {x j } implemented by partitions of unity {x j } of grade 1 are considered, resembling the 'canonical endomorphism' of the Cuntz algebras. The relationship between the Voiculescu topological entropy of σ {x j } and the topological entropy of the associated subshift is studied. Examples where the equality holds are discussed among Matsumoto algebras associated to non finite type subshifts. In the general case ht(σ {x j } ) is bounded by the sum of the entropy of the subshift and a suitable entropic quantity of the homogeneous subalgebra. Both summands are necessary.The measure-theoretic entropy of σ {x j } , in the sense of Connes-NarnhoferThirring, is compared to the classical measure-theoretic entropy of the subshift.A noncommutative analogue of the classical variational principle for the entropy is obtained for the 'canonical endomorphism' of certain Matsumoto algebras. More generally, a necessary condition is discussed. In the case of CuntzKrieger algebras an explicit construction of the state with maximal entropy from the unique KMS state is done.

Topological entropy and AF subalgebras of graph 𝐶*-algebras

“…For a finite graph E, the topological entropy ht(Φ E ) has been obtained as follows (see [2], [5], [19], or [11]). …”

“…A typical one is the canonical which is isomorphic to C(X A ) in such a way that the restriction Φ A | D A corresponds to the shift map σ X A on the (compact) shift space X A associated with the incidence matrix A. The topological entropy of Φ A is then computed (see [5,2,11,19]) as ht(Φ A ) = log r(A) (r(A) is the spectral radius of A). But log r(A) = h top (X A ) is a well-known fact, so that one can deduce by [8] …”

Crossed Products by Endomorphisms, Vector Bundles and Group Duality, Ii

“…4.4. They naturally appear in the context of our crossed products (Prop.5.11), but not only there ( [35], [29], [43]). We characterize such noncommutative pullbacks in terms of properties of the associated semitensor C*-category of tensor powers (Prop.4.5).…”

Contextual emergence of mental states

“…By contrast, a sufficiently fine-grained partition of Y into mental states of different phenomenal content would have to be described by a high-dimensional irreducible transition matrix T since any such state should be connected to any other state by a symbolic trajectory of sufficient length. In this case the resulting symbolic dynamics is an ergodic, mixing Markov chain with a distinguished KMS equilibrium state (Pinzari et al 2000, Exel 2004.These stationary and structurally stable symbolic dynamical systems have strikingly different consequences (beim Graben and Atmanspacher, in press). While fixed points and limit tori do not possess generating partitions (beim Graben 2004), Markov chains can be obtained from Markov partitions which are generating.…”

“…By contrast, a sufficiently fine-grained partition of Y into mental states of different phenomenal content would have to be described by a high-dimensional irreducible transition matrix T since any such state should be connected to any other state by a symbolic trajectory of sufficient length. In this case the resulting symbolic dynamics is an ergodic, mixing Markov chain with a distinguished KMS equilibrium state (Pinzari et al 2000, Exel 2004). …”