Improper filtrations for C*-algebras: spectra of unilateral tridiagonal operators

We extend the results of our previous paper "C*-algebras and numerical linear algebra" to cover the case of "unilateral" sections. This situation bears a close resemblance to the case of Toeplitz operators on Hardy spaces, in spite of the fact that the operators here are far from Toeplitz operators. In particular, there is a short exact sequence 0 --> K --> A --> B --> 0 whose properties are essential to the problem of computing the spectra of self adjoint operators.Comment: 12 pages, AMS-TeX 2.

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 asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map L: M --> M acting on a von Neumann algebra - or more generally a dual operator system - we show that there is a unique reversible system \alpha: N --> N (i.e., a complete order automorphism \alpha of a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*-dynamical system (N,\mathbb Z), whick we identify as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed point algebra of (N,\mathbb Z). In general, we show the action of the asymptotic lift is trivial iff L is {\em slowly oscillating} in the sense that $$\lim_{n\to\infty}\|\rho\circ L^{n+1}-\rho\circ L^n\|=0,\qquad \rho\in M_* .$$ Hence \alpha is often a nontrivial automorphism of N.Comment: New section added with an applicaton to the noncommutative Poisson boundary. Clarification of Sections 3 and 4. Additional references. 23 p

The index of a quantum dynamical semigroup

A numerical index is introduced for semigroups of completely positive maps of \Cal B(H) which generalizes the index of E_0-semigroups. It is shown that the index of a unital completely positive semigroup agrees with the index of its dilation to an E_0-semigroup, provided that the dilation is minimal.Comment: 26 pp. AMS-TeX 2.