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Existence of Feller Cocycles on a C * ‐Algebra
Author(s) -
Lindsay J. Martin,
Wills Stephen J.
Publication year - 2001
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609301008128
Subject(s) - mathematics , bounded function , separable space , commutative property , pure mathematics , dimension (graph theory) , algebra over a field , discrete mathematics , mathematical analysis
The quantum stochastic differential equation d k t = k t ^ θ rβ α d Λ α rβ ( t )is considered on a unital C * ‐algebra, with separable noise dimension space. Necessary conditions on the matrix of bounded linear maps θ for the existence of a completely positive contractive solution are shown to be sufficient. It is known that for completely positive contraction processes, k satisfies such an equation if and only if k is a regular Markovian cocycle. ‘Feller’ refers to an invariance condition analogous to probabilistic terminology if the algebra is thought of as a non‐commutative topological space. 2000 Mathematics Subject Classification 81S25, 46L07, 46L53, 47D06.