A certain class of Arens-Michael algebras having no non-zero injective topological ⨶-modules is introduced. This class is rather wide and contains, in particular, algebras of holomorphic functions on polydomains in ${ℂ}^n$, algebras of smooth functions on domains in ${ℝ}^n$, algebras of formal power series, and, more generally, any nuclear Fréchet-Arens-Michael algebra which has a free bimodule Koszul resolution.

A class of convex functions where the sets of subdifferentials behave like the unit ball of the dual space of an Asplund space is found. These functions, which we called Asplund functions also possess some stability properties. We also give a sufficient condition for a function to be an Asplund function in terms of the upper-semicontinuity of the subdifferential map.

A Banach space X is asymptotically symmetric (a.s.) if for some C < ∞, for all m ∈ ℕ, for all bounded sequences ${\left(x_j^i\right)}_{j=1}^{\infty }\subseteq X$, 1 ≤ i ≤ m, for all permutations σ of 1,...,m and all ultrafilters ₁,...,ₘ on ℕ, $li{m}_{n₁,₁}...li{m}_{nₘ,ₘ}\left|\right|{\sum }_{i=1}^m{x}_{n_i}^i\left|\right|\le Cli{m}_{n_{\sigma \left(1\right)}{,}_{\sigma \left(1\right)}}...li{m}_{n_{\sigma \left(m\right)}{,}_{\sigma \left(m\right)}}\left|\right|{\sum }_{i=1}^m{x}_{n_i}^i\left|\right|$. We investigate a.s. Banach spaces and several natural variations. X is weakly a.s. (w.a.s.) if the defining condition holds when restricted to weakly convergent sequences ${\left(x_j^i\right)}_{j=1}^{\infty }$. Moreover, X is w.n.a.s. if we restrict the condition further to normalized weakly null sequences. If X is a.s. then all spreading...

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_F$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F:\Omega \times X\to 2^Y$. It is shown that if $N_F$ maps a modular space $(N\left(L(\Omega ,\Sigma ,\mu ;X)\right),{\varrho }_{N,\mu })$ into subsets of a modular space $(M\left(L(\Omega ,\Sigma ,\mu ;Y)\right),{\varrho }_{M,\mu })$, then $N_F$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_K=sup{\varrho }_{N,\mu }\left(x\right):x\in K<\infty$ we have $sup{\varrho }_{M,\mu }\left(y\right):y\in N_F\left(K\right)<\infty$.

We investigate Banach space automorphisms $T:{\ell }_{\infty }/c₀\to {\ell }_{\infty }/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}:{\ell }_{\infty }\left(A\right)/c₀\left(A\right)\to {\ell }_{\infty }\left(B\right)/c₀\left(B\right)$ for A,B ⊆ ℕ infinite by means of linear operators from ${\ell }_{\infty }\left(A\right)$ into ${\ell }_{\infty }\left(B\right)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on ${\ell }_{\infty }/c₀$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for others give...

If ξ is a countable ordinal and (fk) a sequence of real-valued functions we define the repeated averages of order ξ of (fk). By using a partition theorem of Nash-Williams for families of finite subsets of positive integers it is proved that if ξ is a countable ordinal then every sequence (fk) of real-valued functions has a subsequence (f'k) such that either every sequence of repeated averages of order ξ of (f'k) converges uniformly to zero or no sequence of repeated averages of order ξ of (f'k)...

In this paper it is proved that if $\{E_n{\}}_{n=1}^{\infty }$ and $\{F_n{\}}_{n=1}^{\infty }$ are two sequences of infinite-dimensional Banach spaces then $H=\left({\oplus }_{n=1}^{\infty }{E}_n\right)\times {\prod }_{n=1}^{\infty }{F}_n$ is not $B_r$-complete. If $\{E_n{\}}_{n=1}^{\infty }$ and $\{F_n{\}}_{n=1}^{\infty }$ are also reflexive spaces there is on $H$ a separated locally convex topology $ℱ$, coarser than the initial one, such that $H\left[ℱ\right]$ is a bornological barrelled space which is not an inductive limit of Baire spaces. It is given also another results on $B_r$-completeness and bornological spaces.

We give a characterization of compact spaces K such that the Banach space C(K) is isomorphic to the space c₀(Γ) for some set Γ. As an application we show that there exists an Eberlein compact space K of weight ${\omega }_{\omega }$ and with the third derived set $K^{\left(3\right)}$ empty such that the space C(K) is not isomorphic to any c₀(Γ). For this compactum K, the spaces C(K) and $c₀\left({\omega }_{\omega }\right)$ are examples of weakly compactly generated (WCG) Banach spaces which are Lipschitz isomorphic but not isomorphic.

In this short note we give a negative answer to a question of Argyros, Castillo, Granero, Jiménez and Moreno concerning Banach spaces which contain complemented and uncomplemented subspaces isomorphic to c0.

A Banach space X is an M-ideal in its bidual if the relation ||f + w|| = ||f|| + ||w||holds for every f in X* and every w in X ⊥.The class of the Banach spaces which are M-ideals in their biduals, in short, the class of M-embedded spaces, has been carefully investigated, in particular by A. Lima, G. Godefroy and the West Berlin School. The spaces c0(I) -I any set- equipped with their canonical norm belong to this class, which also contains e.g. certain spaces K(E,F) of compact operators between...