Interactions in noncommutative dynamics

A mathematical notion of interaction is introduced for noncommutative dynamical systems, i.e., for one parameter groups of *-automorphisms of \Cal B(H) endowed with a certain causal structure. With any interaction there is a well-defined "state of the past" and a well-defined "state of the future". We describe the construction of many interactions involving cocycle perturbations of the CAR/CCR flows and show that they are nontrivial. The proof of nontriviality is based on a new inequality, relating the eigenvalue lists of the "past" and "future" states to the norm of a linear functional on a certain C^*-algebra.Comment: 22 pages. Replacement corrects misnumbering of formulas in section 4. No change in mathematical conten

The noncommutative Choquet boundary

Let S be an operator system -- a self-adjoint linear subspace of a unital C*-algebra A such that contains 1 and A=C*(S) is generated by S. A boundary representation for S is an irreducible representation \pi of C*(S) on a Hilbert space with the property that $\pi\restriction_S$ has a unique completely positive extension to C*(S). The set $\partial_S$ of all (unitary equivalence classes of) boundary representations is the noncommutative counterpart of the Choquet boundary of a function system $S\subseteq C(X)$ that separates points of X. It is known that the closure of the Choquet boundary of a function system S is the Silov boundary of X relative to S. The corresponding noncommutative problem of whether every operator system has "sufficiently many" boundary representations was formulated in 1969, but has remained unsolved despite progress on related issues. In particular, it was unknown if $\partial_S$ is nonempty for generic S. In this paper we show that every separable operator system has sufficiently many boundary representations. Our methods use separability in an essential way.Comment: 22 pages. A significant revision, including a new section and many clarifications. No change in the basic mathematic

Pure E_0-semigroups and absorbing states

An E_0-semigroup acting on B(H) is called pure if the intersection of the ranges $\alpha_t(B(H))$, $t>0$, is the algebra of scalars. We determine all pure E_0-semigroups which have a weakly continuous invariant state $\omega$ and which are minimal in an appropriate sense. In such cases, for every normal state $\rho$ there is convergence to equilibrium in the trace norm $\lim_{t\to\infty}\| \rho\circ\alpha_t - \omega \| = 0.$ A normal state $\omega$ with this property is called an absorbing state. Such E_0-semigroups mest be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed finite eigenvalue lists.Comment: 27 pp. AMS-TeX 2.