On type II0 E0-semigroups induced by q-pure maps on Mn C

Christopher Jankowski, University of Pennsylvania

Abstract

Using a dilation theorem of Bhat, Powers has shown that every non-trivial spatial E₀-semigroup can be obtained from a CP-flow acting on B(K ⊗ L²(0, ∞)), where K is a separable Hilbert space. In this thesis, we use boundary weight doubles (&phis;, ν) to define natural boundary weight maps which then induce CP-flows over K for 1 < dim(K) < ∞. Developing a comparison theory for the E ₀-semigroups arising from certain boundary weight doubles, we obtain examples of type II₀ E₀-semigroups and classify them up to cocycle conjugacy. We study the unital q-pure maps acting on M_n( C ), classifying all such maps which are invertible or have rank one. We find that the rank one unital q-pure maps arise from faithful states on M_n( C ) and, through a boundary weight construction using particular unbounded weights ν on B(L²(0, ∞)), yield uncountably many mutually non-cocycle conjugate type II₀ E₀-semigroups for each 1 < n ∈ N . We ask whether every unital q-pure map acting on M_n( C ) has rank one or is invertible, and show that this is the case when n = 2. In conclusion, we discuss future problems involving the examination and generalization of these results.^

Subject Area

Mathematics

Recommended Citation

Christopher Jankowski, "On type II0 E0-semigroups induced by q-pure maps on Mn C" (January 1, 2009). Dissertations available from ProQuest. Paper AAI3363373.
http://repository.upenn.edu/dissertations/AAI3363373