Banach Spaces of Almost Universal Complemented Disposition

We introduce and study the notion of space of almost universal complemented disposition (a.u.c.d.) as a generalization of Kadec space. We show that every Banach space with separable dual is isometrically contained as a $1$-complemented subspace of a separable a.u.c.d. space and that all a.u.c.d. spaces with $1$-finite dimensional decomposition (FDD) are isometric and contain isometric $1$-complemented copies of every separable Banach space with $1$-FDD. We then study spaces of universal complemented disposition (u.c.d.) and provide different constructions for such spaces. We also consider spaces of u.c.d. with respect to separable spaces.

Lipschitz-free Spaces on Finite Metric Spaces

Main results of the paper are as follows:(1) For any finite metric space $M$ the Lipschitz-free space on $M$ contains a large well-complemented subspace that is close to $\ell _{1}^{n}$.(2) Lipschitz-free spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell _{1}^{n}$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.Interesting features of our approach are: (a) We consider averages over groups of cycle-preserving bijections of edge sets of graphs that are not necessarily graph automorphisms. (b) In the case of such recursive families of graphs as Laakso graphs, we use the well-known approach of Grünbaum (1960) and Rudin (1962) for estimating projection constants in the case where invariant projections are not unique.

On the Existence of a Basis in a Complemented Subspace of a Nuclear Kothe Space of Type $d_2$

В работе дано доказательство существования базиса в произвольном дополняемом подпространстве ядерного пространства Кете из класса $(d_2)$. Показано также, что в любом таком подпространстве существует базис, квазиэкивалентный части базиса ортов.

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.

NON-UNITARISABLE REPRESENTATIONS AND MAXIMAL SYMMETRY

We investigate questions of maximal symmetry in Banach spaces and the structure of certain bounded non-unitarisable groups on Hilbert space. In particular, we provide structural information about bounded groups with an essentially unique invariant complemented subspace. This is subsequently combined with rigidity results for the unitary representation of $\text{Aut}(T)$ on $\ell _{2}(T)$, where $T$ is the countably infinite regular tree, to describe the possible bounded subgroups of $\text{GL}({\mathcal{H}})$ extending a well-known non-unitarisable representation of $\mathbb{F}_{\infty }$.As a related result, we also show that a transitive norm on a separable Banach space must be strictly convex.

Von Neumann algebras as complemented subspaces of $B({\mathscr{H}})$

Let [Formula: see text] be a von Neumann algebra of type II1 which is also a complemented subspace of [Formula: see text]. We establish an algebraic criterion, which ensures that [Formula: see text] is an injective von Neumann algebra. As a corollary we show that if [Formula: see text] is a complemented factor of type II1 on a Hilbert space [Formula: see text], then [Formula: see text] is injective if its fundamental group is nontrivial.

On Complemented Subspaces of Non-Archimedean Power Series Spaces

The non-archimedean power series spaces, A1(a) and A∞(b), are the best known and most important examples of non-archimedean nuclear Fréchet spaces. We prove that the range of every continuous linear map from Ap(a) to Aq(b) has a Schauder basis if either p = 1 or p = ∞ and the set Mb,a of all bounded limit points of the double sequence (bi/aj )i, j∈ℕ is bounded. It follows that every complemented subspace of a power series space Ap(a) has a Schauder basis if either p = 1 or p = ∞ and the set Ma,a is bounded.