On wide-(s)(s) sequences and their applications to certain classes of operators

A basic sequence in a Banach space is called wide-$(s)$ if it is bounded and dominates the summing basis. (Wide-$(s)$ sequences were originally introduced by I.~Singer, who termed them $P^*$-sequences). These sequences and their quantified versions, termed $\lambda$-wide-$(s)$ sequences, are used to characterize various classes of operators between Banach spaces, such as the weakly compact, Tauberian, and super-Tauberian operators, as well as a new intermediate class introduced here, the strongly Tauberian operators. This is a nonlocalizable class which nevertheless forms an open semigroup and is closed under natural operations such as taking double adjoints. It is proved for example that an operator is non-weakly compact iff for every $\varepsilon >0$, it maps some $(1+\varepsilon)$-wide-$(s)$-sequence to a wide-$(s)$ sequence. This yields the quantitative triangular arrays result characterizing reflexivity, due to R.C.~James. It is shown that an operator is non-Tauberian (resp. non-strongly Tauberian) iff for every $\varepsilon>0$, it maps some $(1+\varepsilon)$-wide-$(s)$ sequence into a norm-convergent sequence (resp. a sequence whose image has diameter less than $\varepsilon$). This is applied to obtain a direct ``finite'' characterization of super-Tauberian operators, as well as the following characterization, which strengthens a recent result of M.~Gonz\'alez and A.~Mart{\'\i}nez-Abej\'on: An operator is non-super-Tauberian iff there are for every $\varepsilon>0$, finite $(1+\varepsilon)$-wide-$(s)$ sequences of arbitrary length whose images have norm at most $\varepsilon$

Some new characterizations of Banach spaces containing ℓ1\ell^1

Several new characterizations of Banach spaces containing a subspace isomorphic to $\ell^1$, are obtained. These are applied to the question of when $\ell^1$ embeds in the injective tensor product of two Banach spaces.Comment: 27 pages, AMSLaTe

On an inequality of A.~Grothendieck concerning operators on L1L^1

In 1955, A.~Grothendieck proved a basic inequality which shows that any bounded linear operator between $L^1(\mu)$-spaces maps (Lebesgue-) dominated sequences to dominated sequences. An elementary proof of this inequality is obtained via a new decomposition principle for the lattice of measurable functions. An exposition is also given of the M.~L\'evy extension theorem for operators defined on subspaces of $L^1(\mu)$-spaces

A characterization of Banach spaces containing c0c_0

A subsequence principle is obtained, characterizing Banach spaces containing $c_0$, in the spirit of the author's 1974 characterization of Banach spaces containing $\ell^1$. Definition: A sequence $(b_j)$ in a Banach space is called {\it strongly summing\/} (s.s.) if $(b_j)$ is a weak-Cauchy basic sequence so that whenever scalars $(c_j)$ satisfy $\sup_n \|\sum_{j=1}^n c_j b_j\| <\infty$, then $\sum c_j$ converges. A simple permanence property: if $(b_j)$ is an (s.s.) basis for a Banach space $B$ and $(b_j^*)$ are its biorthogonal functionals in $B^*$, then $(\sum_{j=1}^n b_j^*)_{n=1}^ \infty$ is a non-trivial weak-Cauchy sequence in $B^*$; hence $B^*$ fails to be weakly sequentially complete. (A weak-Cauchy sequence is called {\it non-trivial\/} if it is {\it non-weakly convergent\/}.) Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either an {\rm (s.s.)} subsequence, or a convex block basis equivalent to the summing basis. Remark : The two alternatives of the Theorem are easily seen to be mutually exclusive. Corollary 1. A Banach space $B$ contains no isomorph of $c_0$ if and only if every non-trivial weak-Cauchy sequence in $B$ has an {\rm (s.s.)} subsequence. Combining the $c_0$ and $\ell^1$ Theorems, we obtain Corollary 2. If $B$ is a non-reflexive Banach space such that $X^*$ is weakly sequentially complete for all linear subspaces $X$ of $B$, then $c_0$ embeds in $X$; in fact, $B$ has property~$(u)$

On Differences of Semi-Continuous Functions

Extrinsic and intrinsic characterizations are given for the class DSC(K) of differences of semi-continuous functions on a Polish space K, and also decomposition characterizations of DSC(K) and the class PS(K) of pointwise stabilizing functions on K are obtained in terms of behavior restricted to ambiguous sets. The main, extrinsic characterization is given in terms of behavior restricted to some subsets of second category in any closed subset of K. The concept of a strong continuity point is introduced, using the transfinite oscillations osc$_\alpha f$ of a function $f$ previously defined by the second named author. The main intrinsic characterization yields the following DSC analogue of Baire's characterization of first Baire class functions: a function belongs to DSC(K) iff its restriction to any closed non-empty set L has a strong continuity point. The characterizations yield as a corollary that a locally uniformly converging series $\sum \phi_j$ of DSC functions on K converges to a DSC function provided $\sum{osc}_\alpha \phi_j$ converges locally uniformly for all countable ordinals $\alpha$.Comment: 20 pages, AMSTe

Strictly semi-transitive operator algebras

An algebra A of operators on a Banach space X is called strictly semi-transitive if for all non-zero x,y in X there exists an operator S in A such that Sx=y or Sy=x. We show that if A is norm-closed and strictly semi-transitive, then every A-invariant linear subspace is norm-closed. Moreover, Lat A is totally and well ordered by reverse inclusion. If X is complex and A is transitive and strictly semi-transitive, then A is WOT-dense in L(X). It is also shown that if A is an operator algebra on a complex Banach space with no invariant operator ranges, then A is WOT-dense in L(X). This generalizes a similar result for Hilbert spaces proved by Foias.Comment: To appear in Journal of Operator Theor

On Functions of Finite Baire Index

It is proved that every function of finite Baire index on a separable metric space $K$ is a $D$-function, i.e., a difference of bounded semi-continuous functions on $K$. In fact it is a strong $D$-function, meaning it can be approximated arbitrarily closely in $D$-norm, by simple $D$-functions. It is shown that if the $n^{th}$ derived set of $K$ is non-empty for all finite $n$, there exist $D$-functions on $K$ which are not strong $D$-functions. Further structural results for the classes of finite index functions and strong $D$-functions are also given

On Weakly Null FDD's in Banach Spaces

In this paper we show that every sequence (F_n) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be ``refined'' to yield an F.D.D. (G_n), still having increasing dimensions, so that either every bounded sequence (x_n), with x_n in G_n for n in N, is weakly null, or every normalized sequence (x_n), with x_n in G_n for n in N, is equivalent to the unit vector basis of l_1. Crucial to the proof are two stabilization results concerning Lipschitz functions on finite dimensional normed spaces. These results also lead to other applications. We show, for example, that every infinite dimensional Banach space X contains an F.D.D. (F_n), with lim_{n to infty} dim (F_n)=infty, so that all normalized sequences (x_n), with x_n in F_n, n in N, have the same spreading model over X. This spreading model must necessarily be 1-unconditional over X

On certain classes of Baire-1 functions with applications to Banach space theory

Certain subclasses of $B_1(K)$, the Baire-1 functions on a compact metric space $K$, are defined and characterized. Some applications to Banach spaces are given

Banach embedding properties of non-commutative L^p-spaces

Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach embed in L^p(M) for N infinite, M finite, 1 < or = p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class C_p embeds in L^p(N) for N infinite). Theorem: Let 1 < or = p < 2 and let X be a Banach space with a spanning set (x_{ij}) so that for some C < or = 1: (i) any row or column is C-equivalent to the usual ell^2-basis; (ii) (x_{i_k,j_k}) is C-equivalent to the usual ell^p-basis, for any i_1 < i_2 < ... and j_1 < j_2 < ... . Then X is not isomorphic to a subspace of L^p(M), for M finite. Complements on the Banach space structure of non-commutative L^p-spaces are obtained, such as the p-Banach-Saks property and characterizations of subspaces of L^p(M) containing ell^p isomorphically. The spaces L^p(N) are classified up to Banach isomorphism, for N infinite-dimensional, hyperfinite and semifinite, 1 < or = p< infty, p not= 2. It is proved that there are exactly thirteen isomorphism types; the corresponding embedding properties are determined for p < 2 via an eight level Hasse diagram. It is also proved for all 1 < or = p < infty that L^p(N) is completely isomorphic to L^p(M) if N and M are the algebras associated to free groups, or if N and M are injective factors of type III_lambda and III_{lambda'} for 0 < lambda, lambda' < or = 1.Comment: 54 pp., LaTe