Universal WCG banach spaces and universal Eberlein Compacts

We prove that for cardinalsτ satisfying τ^ω=τ and forτ=ω ₁, there do not exist universal Eberlein Compacts of weightτ, or universal WCG spaces of density characterτ. Ifτ is a strong limit cardinal of countable cofinality such universal spaces do exist. Thus under GCH universal spaces exist forτ iff cof(τ)=ω. The research of the second author was supported by a grant from the United States-Israel Binational Science Foundation and the Fund for the Promotion of Research at the Technion.     

An operator onL pwithout best compact approximation

We construct, for 1<p<∞,p ≠ 2, an operator onL ^pwhose distance to the space of compact operators onL ^pis not attained. We also show that the identity operator onL ^p,p ≠ 1,2, ∞ has a unique best compact approximation. Research partially supported by NSF grant DMS-8201635.     

A predual ofl 1 which is not isomorphic to aC(K) space

We give an example of a Banach spaceX such that (i)X ^* is isometric tol ₁, (ii)X is isometric to a subspace ofC(ω^θ) and (iii)X is not isomorphic to a complemented subspace of anyC(K) space. This is a part of the first author’s Ph. D. Thesis prepared in the Hebrew University of erusalem under the supervision of the second author.     

A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

Wiener's tauberian theorem for spherical functions on the automorphism group of the unit disk

Our main result gives necessary and sufficient conditions, in terms of Fourier transforms, for an ideal in the convolution algebra of spherical integrable functions on the (conformal) automorphism group of the unit disk to be dense, or to have as closure the closed ideal of functions with integral zero. This is then used to prove a generalization of Furstenberg's theorem, which characterizes harmonic functions on the unit disk by a mean value property, and a “two circles” Morera type theorem (earlier announced by Agranovskii). The second author's work was partially supported by the fund for the promotion of research at the Technion-Israel Institute of Technology. The third author's work was partially supported by the Swedish Natural Science Research Council, and by the 1992 Wallenberg Prize from the Swedish Mathematical Society.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

Nonexpansive selections of metric projections in spaces of continuous functions

A subset A of a metric space X is said to be a nonexpansive proximinal retract (NPR) of X if the metric projection from X to A admits a nonexpansive selection. We study the structure of NPR's in the space C(K) of continuous functions on a compact Hausdorff space K. The main results are a characterization of finite-codimensional and of finite-dimensional NPR subspaces of C(K) and a complete characterization of all NPR subsets of .     

AnM-space which is not isomorphic to aC(K) space

We give a new example of anL ¹(μ) predual which is not isomorphic to aC(K) space. The space constructed here is a non-separableM-space. By a previous theorem of the author, the separableM-spaces are isomorphic toC(K) spaces. This research was partially supported by NSF Grant MPS 74-2449.     

Near isometries in the class ofL 1-preduals

We construct preduals ofl ₁ which are nearly isometric without being isometric. We also show that ifX is nearly isometric to aC(K) space withK first countable, then they are in fact isometric.