Let X be a Banach space, $B\subset B_{X*}$ a norming set and (X,B) the topology on X of pointwise convergence on B. We study the following question: given two (non-negative, countably additive and finite) measures μ₁ and μ₂ on Baire(X,w) which coincide on Baire(X,(X,B)), does it follow that μ₁ = μ₂? It turns out that this is not true in general, although the answer is affirmative provided that both μ₁ and μ₂ are convexly τ-additive (e.g. when X has the Pettis Integral Property). For a Banach space Y not containing...

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does $c_0\left(X\right)$. We also give some positive results including a simpler proof that $c_0\left({\ell }_1\right)$ has a unique unconditional basis and a proof that $c_0\left({\ell }_{p_n}^{N_n}\right)$ has a unique unconditional basis when $p_n￬1$, $N_{n+1}\ge 2N_n$ and $(p_n-p_{n+1})log{N}_n$ remains bounded.

We prove that if 0 < p < 1 then a normalized unconditional basis of a complemented subspace of $c_0\left(l_p\right)$ must be equivalent to a permutation of a subset of the canonical unit vector basis of $c_0\left(l_p\right)$. In particular, $c_0\left(l_p\right)$ has unique unconditional basis up to permutation. Bourgain, Casazza, Lindenstrauss, and Tzafriri have previously proved the same result for $c_0\left(l₁\right)$.

In a Banach algebra an invertible element which has norm one and whose inverse has norm one is called unitary. The algebra is unitary if the closed convex hull of the unitary elements is the closed unit ball. The main examples are the C*-algebras and the ℓ₁ group algebra of a group. In this paper, different characterizations of unitary algebras are obtained in terms of numerical ranges, dentability and holomorphy. In the process some new characterizations of C*-algebras are given.

We show that if a separable Banach space X contains an isometric copy of every strictly convex separable Banach space, then X contains an isometric copy of l1 equipped with its natural norm. In particular, the class of strictly convex separable Banach spaces has no universal element. This provides a negative answer to a question asked by J. Lindenstrauss.

Let X, Y be real Banach spaces and ε > 0. A standard ε-isometry f: X → Y is said to be (α,γ)-stable (with respect to $T:L\left(f\right)\equiv \overline{span}f\left(X\right)\to X$ for some α,γ > 0) if T is a linear operator with ||T|| ≤ α such that Tf- Id is uniformly bounded by γε on X. The pair (X,Y) is said to be stable if every standard ε-isometry f: X → Y is (α,γ)-stable for some α,γ > 0. The space X[Y] is said to be universally left [right]-stable if (X,Y) is always stable for every Y[X]. In this paper, we show that universally right-stable...

For $1<p\le \infty$, we show the existence of a Banach space which is both injectively and surjectively universal for the class of all separable Banach spaces with an equivalent $p$-asymptotically uniformly smooth norm. We prove that this class is analytic complete in the class of separable Banach spaces. These results extend previous works by N. J. Kalton, D. Werner and O. Kurka in the case $p=\infty$.

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite-dimensional decompositions of Banach spaces can be extended and modified to Schauder frames. We show as well that if a separable...

Here we study the existence of lower and upper ${\ell }_p$-estimates of sequences in some Banach sequence spaces. We also compute the sharp ${\ell }_p$ estimates in their basis. Finally, we give some applications to weak sequential continuity of polynomials.