arXiv subjects

Andrzej Luczak

Publications and source records attributed to Andrzej Luczak.

3 recordsLinked to original sources

arXiv2006

Quantum stochatic integrals and Doob-Meyer decomposition

We show that for a quantum $L^p$-martingale $(X(t))$, $p>2$, there exists a Doob-Meyer decomposition of the submartingale $(|X(t)|^2)$. A noncommutative counterpart of a classical process continuous with probability one is introduced, and a quantum stochastic integral of such a process with respect to an $L^p$-martingale, $p>2$, is constructed. Using this construction, the uniqueness of the Doob-Meyer decomposition for a quantum martingale `continuous with probability one' is proved, and explicit forms of this decomposition and the quadratic variation process for such a martingale are obtained.

math.OA↗

arXiv2006

Frobenius theory fails for semigroups of positive maps on von Neumann algebras

The eigenvectors of an ergodic semigroup of linear normal positive unital maps on a von Neumann algebra are described. Moreover, it is shown by means of examples, that mere positivity of the maps in question is not sufficient for Frobenius theory as in S. Albeverio and R. Høegh-Krohn, \emph{Frobenius theory of positive maps of von Neumann algebras}, Comm. Math. Phys. \bf{64}(1978), 83--94, to hold.

math.OA↗

arXiv2002

Structure of the time projection for stopping times in von Neumann algebras

We give an explicit formula for the time projection in an arbitrary von Neumann algebra from which all its basic properties can be easily derived. The analysis of the situation when this time projection is a conditional expectation is also performed.

math.OA↗