Extensions of c0c_0

If $X$ is a closed subspace of a Banach space $L$ which embeds into a Banach lattice not containing $\ell_\infty^n$'s uniformly and $L/X$ contains $\ell_\infty^n$'s uniformly, then $X$ cannot have local unconditional structure in the sense of Gordon-Lewis (GL-{\sl l.u.st.})

A Schauder basis for L1(0,∞)L_1(0,\infty) consisting of non-negative functions

We construct a Schauder basis for $L_1$ consisting of non-negative functions and investigate unconditionally basic and quasibasic sequences of non-negative functions in $L_p$, $1\le p < \infty$

Universal Non-Completely-Continuous Operators

A bounded linear operator between Banach spaces is called {\it completely continuous} if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from $L_1$ into an arbitrary Banach space, namely, the operator from $L_1$ into $\ell_\infty$ defined by $$T_0 (f) =\left( \int r_n f \, d\mu \right)_{n\ge 0} \ ,$$ where $r_n$ is the $n^{\text{th}}$ Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach space. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space

Extension of Operators from Weak∗^*-closed Subspaces of ℓ1\ell_1

It is proved that every operator from a weak$^*$-closed subspace of $\ell_1$ into a space $C(K)$ of continuous functions on a compact Hausdorff space $K$ can be extended to an operator from $\ell_1$ to $C(K)$

The Complete Continuity Property and Finite Dimensional Decompositions

A Banach space \X has the complete continuity property (CCP) if each bounded linear operator from $L_1$ into \X is completely continuous (i.e., maps weakly convergent sequences to norm convergent sequences). The main theorem shows that a Banach space failing the CCP (resp., failing the CCP and failing cotype) has a subspace with a finite dimensional decomposition (resp., basis) which fails the CCP

Subspaces of LpL_p that embed into Lp(μ)L_p(\mu) with μ\mu finite

Enflo and Rosenthal proved that $\ell_p(\aleph_1)$, $1 < p < 2$, does not (isomorphically) embed into $L_p(\mu)$ with $\mu$ a finite measure. We prove that if $X$ is a subspace of an $L_p$ space, $1< p < 2$, and $\ell_p(\aleph_1)$ does not embed into $X$, then $X$ embeds into $L_p(\mu)$ for some finite measure $\mu$

Computing p-summing norms with few vectors

It is shown that the p-summing norm of any operator with n-dimensional domain can be well-aproximated using only ``few" vectors in the definition of the p-summing norm. Except for constants independent of n and log n factors, ``few" means n if 1<p<2 and n^{p/2} if 2<p<infinity

Polynomial Schur and Polynomial Dunford-Pettis Properties

A Banach space is {\it polynomially Schur} if sequential convergence against analytic polynomials implies norm convergence. Carne, Cole and Gamelin show that a space has this property and the Dunford-Pettis property if and only if it is Schur. Herein is defined a reasonable generalization of the Dunford--Pettis property using polynomials of a fixed homogeneity. It is shown, for example, that a Banach space will has the $P_N$ Dunford--Pettis property if and only if every weakly compact $N-$homogeneous polynomial (in the sense of Ryan) on the space is completely continuous. A certain geometric condition, involving estimates on spreading models and implied by nontrivial type, is shown to be sufficient to imply that a space is polynomially Schur

The proportional UAP characterizes weak Hilbert spaces

We prove that a Banach space has the uniform approximation property with proportional growth of the uniformity function iff it is a weak Hilbert space

Commutators on ℓ∞\ell_{\infty}

The operators on $\ell_{\infty}$ which are commutators are those not of the form $\lambda I + S$ with $\lambda\neq 0$ and $S$ strictly singular.Comment: 15 pages. Submitted to the Bulletin of the London Mathematical Society The proof of Theorem 3.3 was corrected. Other minor changes were mad