A non-reflexive Grothendieck space that does not containl ∞

A compact spaceS is constructed such that, in the dual Banach spaceC(S)^*, every weak^* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol _∞. The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact spaceT is constructed such thatC(T) does not containl _∞ but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.     

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 convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Dunford-pettis operators onL 1 and the Radon-Nikodym property

Using the duality between Dunford-Pettis operators onL ¹ and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL ¹ intoL ¹ form a lattice. We show also that a Banach spaceX has the Radon-Nikodym property provided the Dunford-Pettis members of ℒ(L ¹,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.     

A counterexample to the Bishop-Phelps Theorem in complex spaces

The Bishop-Phelps Theorem asserts that the set of functionals which attain the maximum value on a closed bounded convex subsetS of a real Banach spaceX is norm dense inX ^*. We show that this statement cannot be extended to general complex Banach spaces by constructing a closed bounded convex set with no support points.     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

Spaces of compact operators and their dual spaces

Theω′-topology on the spaceL(X, Y) of bounded linear operators from the Banach spaceX into the Banach spaceY is discussed in [10]. Let ℒ^w' (X, Y) denote the space of allT∈L(X, Y) for which there exists a sequence of compact linear operators (T _n)⊂K(X, Y) such thatT=ω′−lim_nT_n and let . We show that is a Banach ideal of operators and that the continuous dual spaceK(X, Y)* is complemented in . This results in necessary and sufficient conditions forK(X, Y) to be reflexive, whereby the spacesX andY need not satisfy the approximation property. Similar results follow whenX andY are locally convex spaces. Financial support from the Potchefstroom University and Maseno University is greatly acknowledged. Financial support from the NRF and Potchefstroom University is greatly acknowledged.     

On normal and compact operators on Banach spaces

For some normal operators (T=H+iK) on a Banach spaceX we study the dual space of the Banach algebraA (H, K) assuming thatX* is weakly complete and we study the decompositionX=Ker (T) ⊕ (TX)⁻ for spacesX ⊅c ₀.     

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.     

Amenability of Banach algebras of compact operators

In this paper we study conditions on a Banach spaceX that ensure that the Banach algebraК(X) of compact operators is amenable. We give a symmetrized approximation property ofX which is proved to be such a condition. This property is satisfied by a wide range of Banach spaces including all the classical spaces. We then investigate which constructions of new Banach spaces from old ones preserve the property of carrying amenable algebras of compact operators. Roughly speaking, dual spaces, predual spaces and certain tensor products do inherit this property and direct sums do not. For direct sums this question is closely related to factorization of linear operators. In the final section we discuss some open questions, in particular, the converse problem of what properties ofX are implied by the amenability ofК(X). BEJ supported by MSRVP at Australian National University; GAW supported by SERC grant GR-F-74332.     

On the bicanonical map of a surface section of a threefold

Let (ℳ, ℒ) be a 3-fold of log-general type polarized by a very ample line bundle ℒ. We study the pairs (ℳ, ℒ) in the case when there exists at least one smooth surface Ŝ ∈ |ℒ| such that the bicanonical map associated to |2K_Ŝ| is not birational. As one consequence of our classification we obtain the result:if a smooth projective threefold has non- negative Kodaira dimension, then given any smooth very ample divisor Ŝon the threefold, the bicanonical map associated to |2K_Ŝ|is birational.     

Sharp determinants

We introduce a sharp trace Tr ^# ℳ and a sharp determinant Det ^# (1−z ℳ) for an algebra of operators ℳ acting on functions of bounded variation on the real line. We show that the zeroes of the sharp determinant describe the discrete spectrum of ℳ. The relationship with weighted zeta functions of interval maps and Milnor–Thurston kneading determinants is explained. This yields a result on convergence of the discrete spectrum of approximated operators. Oblatum 8-V-1995 & IX-1995     

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.     

An example of a universal Banach space

We construct a separable reflexive Banach spaceX which is complementably universal for all finite dimensional Banach spaces. By this we mean: for every finite dimensional Banach spaceE there is isometric embeddingi:E→X such that there exists a projectionP: → _→ ^onto with ‖P‖=1.     

Weak compactness inL 1(λ) and injective banach spaces

The isomorphic embedding of the Banach spacel ⁱ(Γ) into injective Banach spaces is investigated.     

Properties (u) and (V*) of Pelczynski in symmetric spaces of <Emphasis Type="Italic">τ</Emphasis>-measurable operators

It is shown that order continuity of the norm and weak sequential completeness in non-commutative strongly symmetric spaces of τ-measurable operators are respectively equivalent to properties (u) and (V ^*) of Pelczynski. In addition, it is shown that each strongly symmetric space with separable (Banach) bidual is necessarily reflexive. These results are non-commutative analogues of well-known characterisations in the setting of Banach lattices.     

Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99 Y.A. Abramovich: 1945–2003 The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846     

Banach ideals of operators with applications to the finite dimensional structure of Banach spaces

We present a few applications of the theory of Banach ideals of operators. In particular, we give operator characterizations of the ℒ _p spaces, compute the relative projection constant of isometric embeddings of Hilbert spaces inL _p -spaces, and show that Π₁ (E, F), the space of absolutely summing operators, is reflexive ifE andF are reflexive andE has the approximation property. Research supported by NSF-GP-34193 Research supported by NSF-Science Development Grant     

Certaines formes linéaires pathologiques sur un espace de Banach dual

We construct a Banach spaceE such thatE′ isw ^*-separable, andf∈E″/E, which isw ^*-continuous on every set ofE′ which is thew ^*-closure of a countablebounded set ofE′.      

On amenable groups of automorphisms on Von Neumann algebras

LetG ⊂ Aut ℳ be a countable group, ℳ a Von Neumann algebra. LetE be a set of pure states on ℳ such thatG*E ⊂E, S ^G be the set ofG invariant states on ℳ andS _E ^G =S ^G ∩w* cl coE. We investigate in this paper some geometric properties for the setS _E ^G which turn out to be equivalent to amenability for the groupG. For example, we show thatS _E ^G ⊂ ℳ_* (S _E ^G has the WRNP) implies that ℳ contains minimal projections (ê containsfinite G invariant orbits) hold true, for all ℳ iffG is amenable. Furthermore we show that ifG is amenable thenS ^G ∩M _* ^⊥ contains a big set, thus improving results obtained by Ching Chou in [2]. These results imply that no action of an amenable countable groupG on an arbitraryW* algebra ℳ iss — strongly ergodic. Moreover cardS ^G ∩M _* ^⊥ ≧2 ^c (see M. Choda [4], K. Schmidt [21] and compare with A. Connes and B. Weiss [5]). The author gratefully acknowledges the support of an Izaak Walton Killam Memorial Senior Fellowship.