ON TYPE II0 E0-SEMIGROUPS INDUCED BY q-PURE MAPS ON Mn(C)

ON TYPE II0 E0-SEMIGROUPS INDUCED BY q-PURE MAPS ON Mn(C) Christopher Jankowski Robert Powers, Advisor 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 ⊗ L(0,∞)), where K is a separable Hilbert space. In this thesis, we use boundary weight doubles (φ, ν) to define natural boundary weight maps which then induce CP -flows over K for 1 < dim(K) <∞. Developing a comparison theory for the E0-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 Mn(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 Mn(C) and, through a boundary weight construction using particular unbounded weights ν on B(L(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 Mn(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.