We show that the range of a contractive projection on a Lebesgue-Bochner space of Hilbert valued functions Lp(H) is isometric to a p-direct sum of Hilbertvalued Lp-spaces. We explicit the structure of contractive projections. As a consequence for every 1 < p < ∞ the class Cp of p-direct sums of Hilbertvalued Lp-spaces is axiomatizable (in the class of all Banach spaces).

We introduce 2 parameter variants Lp,q of the Lebesgue spaces, to gain separate control on the asymptotic behaviour (p) and the local behaviour (q). Thus they behave w.r.t. p like the spaces lp and w.r.t. q like the spaces Lq on a probability space. Convolution behaves very well on those spaces.

We present an Itô formula for the Lp-norm of jump processes having stochastic differentials in Lp-spaces. The main results extend well-known theorems Krylov to case with jumps, which can be used prove existence and uniqueness Lp-spaces SPDEs driven by Lévy processes.

Overview of the field. NoncommutativeLp-spaces are at the heart of this conference. These spaces have a long history going back to pioneering works by von Neumann, Dixmier and Segal. They are the analogues of the classical Lebesgue spaces of pintegrable functions, where now functions are replaced by operators. These spaces have been investigated for operator algebras with a trace, and then arou...

We prove that for all 1 ≤ p ≤ ∞, p 6= 2, the Lp spaces associated to two von Neumann algebrasM, N are isometrically isomorphic if and only if M and N are Jordan *-isomorphic. This follows from a noncommutative Lp Banach-Stone theorem: a specific decomposition for surjective isometries of noncommutative Lp spaces.

we present some new results on the existence and the approximationof common fixed point of expansive mappings and semigroups in probabilisticmetric spaces.

We introduce the language LP that extends logic programs under the stable model semantics to allow weighted rules similar to the way Markov Logic considers weighted formulas. LP is a proper extension of the stable model semantics to enable probabilistic reasoning, providing a way to handle inconsistency in answer set programming. We also show that the recently established logical relationship b...

Some classical results due to Marcinkiewicz, Littlewood and Paley are proved for the Ciesielski-Fourier series. The Marcinkiewicz multiplier theorem is obtained for Lp spaces and extended to Hardy spaces. The boundedness of the Sunouchi operator on Lp and Hardy spaces is also investigated. 2000 AMS subject classifications: Primary 41A15, 42A45, 42B25, Secondary 42C10, 42B30.