13,649 research outputs found

    The asymptotic lift of a completely positive map

    Get PDF
    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 limnρLn+1ρLn=0,ρM. \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

    Bounded Point Evaluations and Local Spectral Theory

    Get PDF
    We study in this paper the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams {\it Dynamic Systems and Apllications} 3(1994) 103-112. Furthermore, we generalize some results of Williams and give a simple proof of theorem 2.5 of L.R. Williams (The Local Spectra of Pure Quasinormal Operators J. Math. anal. Appl. 187(1994) 842-850) that non normal hyponormal weighted shifts have fat local spectra.Comment: 44pp, diploma thesi
    corecore