A survey of noncommutative dynamical entropy

The paper is a survey of dynamical entropy of automorphisms of operator algebras. We describe the different entropies of Connes-Stormer, Connes-Narnhofer-Thirring, Sauvageot-Thouvenot, and Voiculescu, and discuss the main examples of the theory.Comment: 48 pages, late

Separable states and the SPA of a positive map

We introduce a nessecary condition for a state to be separable and apply this condition to the SPA of an optimal ositive map and give a proof of the fact that the SPA need not be the density ooperator for a separable state

Multiplicative properties of positive maps

Let $\phi$ be a positive unital normal map of a von Neumann algebra $M$ into itself, and assume there is a family of normal $\phi$-invariant states which is faithful on the von Neumann algebra generated by the image of $\phi$. It is shown that there exists a largest Jordan subalgebra $C_\phi$ of $M$ such that the restriction of $\phi$ to $C_\phi$ is a Jordan automorphhism, and each weak limit point of $(\phi^n (a))$ for $a\in M$ belongs to $C_\phi$.Comment: 8 page

Why Can’t Tyrone Write: Reconceptualizing Flower and Hayes for African-American Adolescent Male Writers

Using qualitative methods and a case study design, the perceptions and writing processes of three African-American eighth grade males were explored. Data were derived from semi-structured and informal interviews; and document analysis. The study concluded that the perceptions of the three participants’ writing processes did not adhere to the steps depicted by the cognitive process model of writing (Flower and Hayes, 1981) that has become a dominant model for describing the composing processes of students. Recommendations are made for altering the Flower and Hayes model to depict how these three, African-American eighth graders perceive school writing

Asymptotic lifts of positive linear maps

We show that the notion of asymptotic lift generalizes naturally to normal positive maps $\phi$ acting on von Neumann algebras M. We focus on cases in which the domain of the asymptotic lift can be embedded as an operator subsystem of M, and characterize when that subsystem is a Jordan subalgebra of M in terms of the asymptotic multiplicative properties of $\phi$.Comment: 13 page

The variational principle for a class of asymptotically abelian C*-algebras

Let (A,\alpha) be a C*-dynamical system. We introduce the notion of pressure P_\alpha(H) of the automorphism \alpha at a self-adjoint operator H\in A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense \alpha-invariant *-subalgebra \A of A such that for all pairs a,b\in\A the C*-algebra they generate is finite dimensional, and there is p=p(a,b)\in\N such that [\alpha^j(a),b]=0 for |j|\ge p. For systems in this class we prove the variational principle, i.e. show that P_\alpha(H) is the supremum of the quantities h_\phi(\alpha)-\phi(H), where h_\phi(\alpha) is the Connes-Narnhofer-Thirring dynamical entropy of \alpha with respect to the \alpha-invariant state \phi. If H\in\A, and P_\alpha(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h_\phi(\alpha), and any state of finite maximal entropy is a trace.Comment: LaTeX2e, 20 page

A reduction theorem for capacity of positive maps

We prove a reduction theorem for capacity of positive maps of finite dimensional C*-algebras, thus reducing the computation of capacity to the case when the image of a nonscalar projection is never a projection.Comment: 7 page

Entropy in type I algebras

It is shown that if (M,phi,alpha) is a W*-dynamical system with M a type I von Neumann algebra then the entropy of alpha w.r.t. phi equals the entropy of the restriction of alpha to the center of M. If furthermore (N,psi,beta) is a W*-dynamical system with N injective then the entropy of the tensor product system is the sum of the entropies.Comment: LaTeX2e, 7 page