Uniform embeddings of metric spaces and of banach spaces into hilbert spaces

It is proved using positive definite functions that a normed spaceX is unifomly homeomorphic to a subset of a Hilbert space, if and only ifX is (linearly) isomorphic to a subspace of aL ₀(μ) space (=the space of the measurable functions on a probability space with convergence in probability). As a result we get thatl _p (respectivelyL _p (0, 1)), 2<p<∞, is not uniformly embedded in a bounded subset of itself. This answers negatively the question whether every infinite dimensional Banach space is uniformly homeomorphic to a bounded subset of itself. Positive definite functions are also used to characterize geometrical properties of Banach spaces. Partially supported by the National Science Foundation, Grant MCS-79-03322. Partially supported by the National Science Foundation, Grant MCS-80-06073.     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

Weakly stable Banach spaces

The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL ¹. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in the study of Banach spaces (for whichl ¹ does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l ^p for some p≧1or c_o embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl ² embeds isomorphically in every infinite dimensional subspace ofX (§3).     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Universal non-compact operators between super-reflexive Banach spaces and the existence of a complemented copy of Hilbert space

Suppose that 1<p≦2, 2≦q<∞. The formal identity operatorI:l _p→l _qfactorizes through any given non-compact operator from ap-smooth Banach space into aq-convex Banach space. It follows that ifX is a 2-convex space andY is an infinite dimensional subspace ofX which is isomorphic to a Hilbert space, thenY contains an isomorphic copy ofl ₂ which is complemented inX.     

On smooth, nonlinear surjections of banach spaces

It is shown that (1) every infinite-dimensional Banach space admits aC ¹ Lipschitz map onto any separable Banach space, and (2) if the dual of a separable Banach spaceX contains a normalized, weakly null Banach-Saks sequence, thenX admits aC ^∞ map onto any separable Banach space. Subsequently, we generalize these results to mappings onto larger target spaces. Supported by an NSF Postdoctoral Fellowship in Mathematics.     

Pitt's theorem for the Lorentz and Orlicz sequence spaces

LetL(X, Y) be the Banach space of all continuous linear operators fromX toY, and letK(X, Y) be the subspace of compact operators. Some versions of the classical Pitt theorem (ifp>q, thenK(l _p, l_q)=L(l_p, l_q)) for subspaces of Lorentz and Orlicz sequence spaces are established. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 18–25, January, 1997. Translated by V. N. Dubrovsky     

On the classes of hereditarily ℓp Banach spaces

Let X denote a specific space of the class of X _α,p Banach sequence spaces which were constructed by Hagler and the first named author as classes of hereditarily ℓ_p Banach spaces. We show that for p > 1 the Banach space X contains asymptotically isometric copies of ℓ_p. It is known that any member of the class is a dual space. We show that the predual of X contains isometric copies of ℓ_p where 1/p + 1/q = 1. For p = 1 it is known that the predual of the Banach space X contains asymptotically isometric copies of c ₀. Here we give a direct proof of the known result that X contains asymptotically isometric copies of ℓ₁.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

Optimal properties of contraction semigroups in banach spaces

Suppose that(T _t )_t>0 is aC ₀ semi-group of contractions on a Banach spaceX, such that there exists a vectorx∈X, ‖x‖=1 verifyingJ ⁻¹(Jx)={x}, whereJ is the duality mapping fromX toP(X ^*). If |<T _t x,f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1 thenx is an eigenvector of the generatorA, associated with a purcly imaginary eigenvalue. Because of Lin's example [L], the hypothesis onx∈X is the best possible. If the hypothesisJ ⁻¹(Jx)={x} is not verified, we can prove that ifJx is a singleton and ifJ ⁻¹(Jx) is weakly compact, then if |<T _t x, f>|→1, whent→+∞ for somef∈X ^*, ‖f‖≤1, there existsy∈J ⁻¹(Jx) such thaty is an eigenvector of the generatorA, associated with a purely imaginary eigenvalue. We give also a counter-example in the case whereX is one of the spaces ℓ₁ orL ₁.     

Geometrical implications of the existence of very smooth bump functions in Banach spaces

We show that ifX is a Banach space and if there is a non-zero real-valuedC ^∞-smooth function onX with bounded support, then eitherX contains an isomorphic copy ofc ₀(N), or there is an integerk greater than or equal to 1 such thatX is of exact cotype 2k and, in this case,X contains an isomorphic copy ofl ^2k(N). We also show that ifX is a Banach space such that there is onX a non-zero real-valuedC ⁴-smooth function with bounded support and ifX is of cotypeq forq<4, thenX is isomorphic to a Hilbert space.     

On symmetric basic sequences in lorentz sequence spaces II

It is shown that if {y _n} is a block of type I of a symmetric basis {x _n} in a Banach spaceX, then {y _n} is equivalent to {x _n} if and only if the closed linear span [y _n] of {y _n} is complemented inX. The result is used to study the symmetric basic sequences of the dual space of a Lorentz sequence spaced(a, p). Let {x _n,f _n} be the unit vector basis ofd(a, p), for 1≤p<+∞. It is shown that every infinite-dimensional subspace ofd(a, p) (respectively, [f _n] has a complemented subspace isomorphic tol _p (respectively,l _q, 1/p+1/q=1 when 1<p<+∞ andc ₀ whenp=1) and numerous other results on complemented subspaces ofd(a, p) and [f _n] are obtained. We also obtain necessary and sufficient conditions such that [f _n] have exactly two non-equivalent symmetric basic sequences. Finally, we exhibit a Banach spaceX with symmetric basis {x _n} such that every symmetric block basic sequence of {x _n} spans a complemented subspace inX butX is not isomorphic to eitherc ₀ orl _p, 1≤p<+∞.     

Representation,interpolation, and reiteration theorems for generalized approximation spaces

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l ^q (ℬ))={f∈X:{E _n (f)}∈l ^q (ℬ)} in which the weighted l ^q -space l ^q (ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces. Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003     

Polyhedral banach spaces and extensions of compact operators

LetX be a polyhedral Banach space whose dual is anL ₁(μ) space for some measureμ. Then for each Banach spacesY ⊆Z and each compact operatorT: Y →X there exists a norm preserving compact extension Z →X.     

The problem of envelopes for Banach spaces

LetX be a Banach space. A Banach spaceY is an envelope ofX if (1)Y is finitely representable inX; (2) any Banach spaceZ finitely representable inX and of density character not exceeding that ofY is isometric to a subspace ofY. Lindenstrauss and Pelczynski have asked whether any separable Banach space has a separable envelope. We give a negative answer to this question by showing the existence of a Banach space isomorphic tol ₂, which has no separable envelope. A weaker positive result holds: any separable Banach space has an envelope of density character ≦ℵ₁ (assuming the continuum hypothesis).     

Sequence spaces and inverse of an infinite matrix

We give here some properties of the sets _α(uΔ) generalizing the space of generalized difference sequencesl ^∞(uΔ). Then we study spaces related to the sets of sequences that are strongly convergent or strongly bounded. Next we define from the sets of spaces that are (N,q) summable or bounded the sets of spaces that are (N,q)α-bounded orr-bounded. Then we give some properties of these spaces using Banach space of the forms _α.     

Polynomial algebras on classical banach spaces

For the classical Banach spacesX = ℓ _p ,C(K) we identify alln such that every polynomial of degreen + 1 onX is uniformly approximable on the unit ball by elements of the algebra generated by all polynomials of degree up ton onX.     

Further properties of Azimi-Hagler banach spaces

For the Azimi-Hagler spaces more geometric and topological properties are investigated. Any constructed space is denoted by X _α,p . We show

The ℒ p spaces

The ℒ _p spaces which were introduced by A. Pełczyński and the first named author are studied. It is proved, e.g., that (i)X is an ℒ _p space if and only ifX* is and ℒ _q space (p ⁻¹+q ⁻¹=1). (ii) A complemented subspace of an ℒ _p space is either an ℒ _p or an ℒ₂ space. (iii) The ℒ _p spaces have sufficiently many Boolean algebras of projections. These results are applied to show thatX is an ℒ_∞ (resp. ℒ₁) space if and only ifX admits extensions (resp. liftings) of compact operators havingX as a domain or range space. We also prove a theorem on the “local reflexivity” of an arbitrary Banach space. This research was partially supported by NSF Grant# 8964.     

On dualL 1-spaces and injective bidual Banach spaces

In a previous paper (Israel J. Math.28 (1977), 313–324), it was shown that for a certain class of cardinals τ,l ¹(τ) embeds in a Banach spaceX if and only ifL ¹([0, 1]^τ) embeds inX ^*. An extension (to a rather wider class of cardinals) of the basic lemma of that paper is here applied so as to yield an affirmative answer to a question posed by Rosenthal concerning dual ℒ₁-spaces. It is shown that ifZ ^* is a dual Banach space, isomorphic to a complemented subspace of anL ¹-space, and κ is the density character ofZ ^*, thenl ¹(κ) embeds inZ ^*. A corollary of this result is that every injective bidual Banach space is isomorphic tol ^∞(κ) for some κ. The second part of this article is devoted to an example, constructed using the continuum hypothesis, of a compact spaceS which carries a homogeneous measure of type ω₁, but which is such thatl ¹(ω₁) does not embed in ℰ(S). This shows that the main theorem of the already mentioned paper is not valid in the case τ = ω₁. The dual space ℰ(S)^* is isometric to , and is a member of a new isomorphism class of dualL ¹-spaces.