We study Besov spaces 5°(LP(Í2)), 0<p,q,a<oo,on domains Í2 in Rd . We show that there is an extension operator W which is a bounded mapping from B°(LP(Ü.)) onto B%(Lp(Rd)). This is then used to derive various properties of the Besov spaces such as interpolation theorems for a pair of B%(Lp(il)), atomic decompositions for the elements of 5°(Lp(f2)), and a description of the Besov spaces by means...

We study the construction of extrapolation spaces operator-valued multiplication operators on Bochner Lp-spaces by means Lp-fiber spaces.

The idea of replacing functions by linear operators, the process of quantization, goes back to the foundations of quantum mechanics and has a great impact in mathematics. This applies for instance to representation theory, operator algebra, noncommutative geometry, quantum and free probability or operator space theory. The quantization of measure theory leads to the theory of Lp spaces defined ...

We develop a general framework for various lattice-valued, probabilistic and approach uniform convergence spaces. To this end, we use the concept of $s$-stratified $LM$-filter, where $L$ and $M$ are suitable frames. A stratified $LMN$-uniform convergence tower is then a family of structures indexed by a quantale $N$. For different choices of $L,M$ and $N$ we obtain the lattice-valued, probabili...

We study some structural aspects of the subspaces of the non-commutative (Haagerup) Lp-spaces associated with a general (non necessarily semi-finite) von Neumann algebra a. If a subspace X of Lp(a) contains uniformly the spaces lnp , n ≥ 1, it contains an almost isometric, almost 1-complemented copy of lp. If X contains uniformly the finite dimensional Schatten classes S p , it contains their l...

We estimate the number of violator spaces of given dimension d and number of constraints n. We show that the number of nondegenerate regular violator spaces is at most n d , and that the number of all violator spaces is at least of order exp(n). This is related to the question whether the class of optimization problems described by violator spaces (or LP-type problems) is more general than line...

We investigate the norm of sums of independent vector-valued random variables in noncommutative Lp spaces. This allows us to obtain a uniform family of complete embeddings of the Schatten class S q in Sp(lmq ) with optimal order m ∼ n. Using these embeddings we show the surprising fact that the sharp type (cotype) index in the sense of operator spaces for Lp[0, 1] is min(p, p) (max(p, p)). Simi...

We study noncommutative maximal inequalities and ergodic theorems for group actions on von Neumann algebras. Consider a locally compact G of polynomial growth with symmetric subset V. Let ? be continuous action algebra M by trace-preserving automorphisms. then show that the operators defined Anx=1m(Vn)?Vn?gxdm(g),x?Lp(M),n?N,1?p??, are weak type (1,1) strong (p,p) 1<p<?. Consequently, sequence ...

In Banach space theory probabilistic techniques play a central role. For example in the local theory of Banach spaces, geometric properties of finite dimensional subspaces are proved from probabilistic inequalities. The probabilistic approach not only enriched Banach space theory, but also introduced Banach space techniques in other areas such as probability or convex geometry. A famous instanc...