Le problème de Dirichlet dans les C∗-algèbres

AbstractA C∗-algebra A, a closed ideal I, and the generator Δ of a Markov semigroup of completely positive contractions from A into itself, are given. Under the same kind of assumptions as in the classical case, a completely positive lifting Λ from the quotient algebra AI into the algebra M(A) of multipliers of A is constructed, such that, for any α in AI, its lifting Λ(α) is harmonic on I, i.e., belongs to the kernel of the generator Δ localized on I