We prove a game-theoretic dichotomy for sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis’ proof of Σ⁰₃ determinacy.
Soient une algèbre de Banach complexe, le groupe général linéaire stable de et sa composante connexe pour la topologie normique. Nous montrons que toute trace non nulle permet de définir un homomorphisme de sur le quotient du groupe additif par l’image du groupe de Grothendieck de . Si (respectivement si est un facteur fini continu) avec la trace usuelle, alors est le déterminant usuel (resp. est celui de Fuglede et Kadison). Dans le cas général, les déterminants permettent...
For a countable compact metric space and a seminormalized weakly null sequence (fₙ)ₙ in C() we provide some upper bounds for the norm of the vectors in the linear span of a subsequence of (fₙ)ₙ. These bounds depend on the complexity of and also on the sequence (fₙ)ₙ itself. Moreover, we introduce the class of c₀-hierarchies. We prove that for every α < ω₁, every normalized weakly null sequence (fₙ)ₙ in and every c₀-hierarchy generated by (fₙ)ₙ, there exists β ≤ α such that a sequence of β-blocks...
We introduce a seminorm for bounded linear operators between Banach spaces that shows the deviation from the weak Banach-Saks property. We prove that if (Xν) is a sequence of Banach spaces and a Banach sequence lattice E has the Banach-Saks property, then the deviation from the weak Banach-Saks property of an operator of a certain class between direct sums E(Xν) is equal to the supremum of such deviations attained on the coordinates Xν. This is a quantitative version for operators of the result...