Transitivity of the norm on Banach spaces¶ having a Jordan structure

We study transitivity conditions on the norm of JB ^*-triples, C ^*-algebras, JB-algebras, and their preduals. We show that, for the predual X of a JBW ^*-triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The closed unit ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the closed unit ball of X is non rare in the unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB ^*-triple with transitive norm is a Hilbert space. We extend to arbitrary C ^*-algebras previously known characterizations of transitivity [20] and convex transitivity [36] of the norm on commutative C ^*-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB-algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB-algebra with almost transitive norm is isomorphic to the field of real numbers. Received: 9 June 1999 / Revised version: 20 February 2000     

A converse to the Oka principle over certain complex Banach manifolds

First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert-Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

Rearrangement Invariant Continuous Linear Functionals on Weak <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We show that in the dual of Weak L¹ the subspace of all rearrangement invariant continuous linear functionals is lattice isometric to a space L¹(μ) and is the linear hull of the maximal elements of the dual unit ball. We also show that the dual of Weak L¹ contains a norm closed weak* dense ideal which is lattice isometric to an ℓ¹-sum of spaces of type C(K). Helmut H. Schaefer in memoriam     

Little Grothendieck`s theorem for real JB*-triples

We prove that given a real JB*-triple E, and a real Hilbert space H, then the set of those bounded linear operators T from E to H, such that there exists a norm one functional and corresponding pre-Hilbertian semi-norm on E such that for all , is norm dense in the set of all bounded linear operators from E to H. As a tool for the above result, we show that if A is a JB-algebra and is a bounded linear operator then there exists a state such that for all . Received June 28, 1999; in final form January 28, 2000 / Published online March 12, 2001     

Representation of a class of locally convex (M)-lattices

We prove a representation theorem for Hausdorff locally convex (M)-lattices which are Dedekind σ-complete, and whose topologies are order σ-continuous and monotonically complete. These turn out to be the weighted spaces c₀(T, H), defined in the paper for T ≠ and H ^T₊. We also characterize the dual of c₀(T, H), as the space l¹ (T, H) defined in the last section. The known representation (on c₀(T)) of Banach (M)-lattices with order continuous norm follows as a particular case. We obtain these results by first proving a new general isomorphism theorem, which seems to be of independent interest. Our notion of “monotonic topological completeness” is weaker than the usual completeness and seems to be very convenient in the framework of topological ordered vector spaces.     

Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞

We study hypercyclicity and supercyclicity of weighted shifts on ℓ^∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c₀ or ℓ^p (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ^∞ is norm nowhere dense. Finally, we show that ℓ^∞ supports an isometry that is weak * sequentially supercyclic.     

Densely two-valued metric projections in uniformly convex Banach spaces

For every uniformly convex Banach spaceX with dimX2 there is a residual setU in the Hausdorff metric spaceB(X) of bounded and closed sets inX such that the metric projection generated by a set fromU is two-valued and upper semicontinuous on a dense and everywhere continual subset ofX. For any two closed and separated subsetsM ₁ andM ₂ ofX the points on the equidistant hypersurface which have best approximations both inM ₁ andM ₂ form a dense G set in the induced topology.The author is partially supported by the National Fund for Scientific Research at the Bulgarian Ministry of Science and Education under contract MM 408/94.     

Jordan operator algebras: basic theory

Jordan operator algebras are norm‐closed spaces of operators on a Hilbert space which are closed under the Jordan product. The discovery of the present paper is that there exists a huge and tractable theory of possibly nonselfadjoint Jordan operator algebras; they are far more similar to associative operator algebras than was suspected. We initiate the theory of such algebras.     

Extension of positive functionals on certain ordered spaces

An ordered linear spaceL is said to satisfy extension property (E1) if for every directed subspaceM ofL and positive linear functional ϕ onM, ϕ can be extended toL. A Riesz spaceL is said to satisfy extension property (E2) if for every sub-Riesz spaceM ofL and every real valued Riesz homomorphism ϕ onM, ϕ can be extended toL as a Riesz homomorphism. These properties were introduced by Schmidt in [5]. In this paper, it is shown that an ordered linear space has extension property (E1) if and only if it is order isomorphic to a function spaceL′ defined on a setX′ such that iff andg belong toL′ there exists a finite disjoint subsetM of the set of functions onX′ such that each off andg is a linear combination of the points ofM. An analogous theorem is derived for Riesz spaces with extension property (E2).     

Exotic Indecomposable Systems of Four Subspaces in a Hilbert Space

We study the relative position of four (closed) subspaces in a Hilbert space. For any positive integer n, we give an example of exotic indecomposable system of four subspaces in a Hilbert space whose defect is . By an exotic system, we mean a system which is not isomorphic to any closed operator system under any permutation of subspaces. We construct the examples using certain nice sequences construced by Jiang and Wang in their study of strongly irreducible operators. Dedicated to Professor Masahiro Nakamura on his 88th birthday     

Stable Norms of Surfaces: Local Structure of the Unit Ball at Rational Directions

Let (M, g) be a closed orientable surface, equipped with a smooth Finsler metric. The metric induces a norm on the real homology of M, called a stable norm. We show this norm is neither strictly convex, nor smooth. Submitted: December 1996, final version: June 1997     

Structure of subspaces of the compact operators having the Dunford-Pettis property

The structure of the subspaces having the Dunford-Pettis property (DPP) is studied, where is the space of all compact operators on and . The following conditions are shown to be equivalent: (i) M has the DPP, (ii) M is isomorphic to a subspace of (iii) the sets and are relatively compact for all and . The equivalence between (i) and (iii) was recently proven in the case of arbitrary Hilbert spaces by Brown and ülger. It is also shown that (i) and (ii) are equivalent for subspaces . This result is optimal in the sense that for there is a DPP-subspace that fails to be isomorphic to a subspace of . Received January 9, 1998; in final form October 1, 1998     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].     

Essentially normal Hilbert modules and <Emphasis Type="Italic">K</Emphasis>-homology

This paper mainly concerns the essential normality of graded submodules. Essentially all of the basic Hilbert modules that have received attention over the years are p-essentially normal—including the d-shift Hilbert module, the Hardy and Bergman modules of the unit ball. Arveson conjectured graded submodules over the unit ball inherit this property and provided motivations to seek an affirmative answer. Some positive results have been obtained by Arveson and Douglas. However, the problem has been resistant. In dimensions d = 2, 3, this paper shows that the Arveson’s conjecture is true. In any dimension, the paper also gives an affirmative answer in the case of the graded principal submodule. Finally, the paper is associated with K-homology invariants arising from graded quotient modules, by which geometry of the quotient modules and geometry of algebraic varieties are connected. In dimensions d = 2, 3, it is shown that K-homology invariants determined by graded quotients are nontrivial. The paper also establishes results on p-smoothness of K-homology elements, and gives an explicit expression for K-homology invariant in dimension d = 2.     

The density condition in subspaces and quotients of Frechet spaces

The aim of this note is to investigate the topological structure (in particular the density condition) of subspaces and separated quotients of Fréchet spaces. Our main result is the following one: LetE be a Fréchet space which is neither Montel nor isomorphic to a closed subspace ofX × , withX a Banach space, also assume thatE can be written asFG withF andG infinite dimensional closed subspaces ofE not isomorphic to , thenE contains a closed subspace with basis and not satisfying the density condition. We also prove that every Köthe echelon space of orderp, 1<p<, which is not quasinormable has a separated quotient with basis which does not satisfy the density condition.     

Some Results on Best Coapproximation in L 1 (S, X)

As a counterpart to best approximation in normed linear spaces, best coapproximation was introduced by Franchetti and Furi. In this paper, we shall show that if M is a separable, coproximinal subspace of X satisfying some additional conditions, then L ¹ (S, M) is coproximinal in L ¹(S, X).      

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.