Algébres topologiquement uniformes et algébres fortement topologiquement uniformes

We characterize locally convex topological algebrasA satisfying: a sequence (x _n) inA converges to 0 if, and only if, (x _n ²) converges to 0. We also show that a real Banach algebra such thatx _n ²+y _n ²→0 if, and only if,x _n → 0 andy _n → 0, for every sequences (x _n) and (y _n) inA, is isomorphic to, whereX is a compact space.      

Optimization of convex functions on w*-compact sets

We give a direct, self-contained, and iterative proof that for any convex, Lipschitz andw ^*-lower semicontinuous function ϕ defined on aw ^*-compact convex setC in a dual Banach spaceX ^* and for any ε>0 there is anx∈X, with ‖x‖≤ε, such that ϕ+x attains its supremum at an extreme point ofC. This result is implicitly contained in the work of Lindenstrauss [9] and the work of Ghoussoub and Maurey on strongw ^*−H _σ sets [8]. In addition, we discuss the applications of this result to the geometry of convex sets. Research supported in part by the NSERC of Canada under grant OGP41983 for the first author and grant OGP7926 for the second author.     

On a new geometric property for Banach spaces

In this paper we study a geometric property for Banach spaces called condition (*), introduced by de Reynaet al in [3], A Banach space has this property if for any weakly null sequencex _n of unit vectors inX, ifx _* ⁿ is any sequence of unit vectors inX ^* that attain their norm at x_n’s, then . We show that a Banach space satisfies condition (*) for all equivalent norms iff the space has the Schur property. We also study two related geometric conditions, one of which is useful in calculating the essential norm of an operator.     

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.     

The minimal cardinality where the Reznichenko property fails

A topological spaceX has the Fréchet-Urysohn property if for each subsetA ofX and each elementx inĀ, there exists a countable sequence of elements ofA which converges tox. Reznichenko introduced a natural generalization of this property, where the converging sequence of elements is replaced by a sequence of disjoint finite sets which eventually intersect each neighborhood ofx. In [5], Kočinac and Scheepers conjecture: The minimal cardinality of a setX of real numbers such thatC _p(X) does not have the weak Fréchet-Urysohn property is equal to b. (b is the minimal cardinality of an unbounded family in the Baire space^Nℕ.) We prove the Kočinac-Scheepers conjecture by showing that ifC _p(X) has the Reznichenko property, then a continuous image ofX cannot be a subbase for a non-feeble filter on ℕ. The author is partially supported by the Golda Meir Fund and the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).     

Multivalued perturbations ofm-accretive differential inclusions

Given anm-accretive operatorA in a Banach spaceX and an upper semicontinuous multivalued mapF: [0,a]×X→2 ^X , we consider the initial value problemu′∈−Au+F(t,u) on [0,a],u(0)=x ₀. We concentrate on the case when the semigroup generated by—A is only equicontinuous and obtain existence of integral solutions if, in particular,X* is uniformly convex andF satisfies β(F(t,B))≤k(t)β(B) for all boundedB⊂X wherek∈L ¹([0,a]) and β denotes the Hausdorff-measure of noncompactness. Moreover, we show that the set of all solutions is a compactR _δ-set in this situation. In general, the extra condition onX* is essential as we show by an example in whichX is not uniformly smooth and the set of all solutions is not compact, but it can be omited ifA is single-valued and continuous or—A generates aC _o-semigroup of bounded linear operators. In the simpler case when—A generates a compact semigroup, we give a short proof of existence of solutions, again ifX* is uniformly (or strictly) convex. In this situation we also provide a counter-example in ℝ⁴ in which no integral solution exists. The author gratefully acknowledges financial support by DAAD within the scope of the French-German project PROCOPE.     

Extremal properties of contraction semigroups onc 0

For any complex Banach spaceX, letJ denote the duality mapping ofX. For any unit vectorx inX and any (C ₀) contraction semigroup (T _t ) _t>0 onX, Baillon and Guerre-Delabriere proved that ifX is a smooth reflexive Banach space and if there isx ^*∈J(x) such that ÷〈(T(t)x, J(x)〈÷→1 ast→∞, then there is a unit vectory∈X which is an eigenvector of the generatorA of (T _t ) _t>0 associated with a purely imaginary eigenvalue. They asked whether this result is still true ifX is replaced byc ₀. In this article, we show the answer is negative Partial results of this paper were obtained when the author attended the International Conference of Convexity at the University of Marne-La-Vallée. He would like to express his gratitude for the kind hospitality offered to him. He would also like to thank Profs. Goldstein and Jamison for their valuable suggestions.     

Stability of <Emphasis Type="Italic">C</Emphasis>-regularized Semigroups

Let T = (T(t))t≥0 be a bounded C-regularized semigroup generated by A on a Banach space X and R(C) be dense in X. We show that if there is a dense subspace Y of X such that for every x ∈ Y, σu(A, Cx), the set of all points λ ∈ iR to which (λ - A)^-1 Cx can not be extended holomorphically, is at most countable and σr(A) N iR = Ф, then T is stable. A stability result for the case of R(C) being non-dense is also given. Our results generalize the work on the stability of strongly continuous senfigroups.     

On Jakovlev spaces

A topological spaceX is called weakly first countable, if for every pointx there is a countable family {C _n ^x |n ∈ω} such thatx ∈C _n ^+1x ⊆C _n ^x and such thatU ⊂X is open iff for eachx ∈U someC _n ^x is contained inU. This weakening of first countability is due to A. V. Arhangelskii from 1966, who asked whether compact weakly first countable spaces are first countable. In 1976, N. N. Jakovlev gave a negative answer under the assumption of continuum hypothesis. His result was strengthened by V. I. Malykhin in 1982, again under CH. In the present paper we construct various Jakovlev type spaces under the weaker assumption b=c, and also by forcing. The second author was supported by the Ben-Gurion University Center for Advanced Studies in Mathematics, Be’er Sheva. The third author was supported by OTKA grant no. 37758 of Hungary.     

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.     

On weak positive supercyclicity

A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT ⁿ x: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σ _p (T ^*) = 0, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT ⁿ x: n ∈ ℂ, r ∈ ℝ₊} we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions. Partially supported by MEC MTM2006-09060 and MTM2006-15546, Junta de Andalucía FQM-257 and P06-FQM-02225. Partially supported by Junta de Andalucía FQM-257, and P06-FQM-02225     

Purely imaginary eigenvalues of operator semigroups

For aC ₀-contraction semigroup (S(t)) _t≥0 of bounded linear operators on a complex Banach spaceX, J. A. Goldstein and B. Nagy [6] have shown that, givenx∈X, S(t)x=e ^iλt x, t≥0, for some λ∈ℝ, provided lim _t→∞ |<S(t)x,x ^* >|=|<x,x ^* >| for allx ^*∈X^*. We present (a) an extension to the case of nonlinear nonexpansive mapsS(t), t≥0, and (b) various generalizations in the linear context.     

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.     

Classification of Homomorphisms from C (X) to Simple C*-Algebras of Real Rank Zero

Abstract Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K ₀₍ A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism. Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon)     

Simple C*-Algebras with Unique Tracial States and Quantized Topological Spaces

Let X be a connected finite CW complex and d _X : K ₀(C(X)) →ℤ be the dimension function. We show that, if A is a unital separable simple nuclear C*-algebra of TR(A) = 0 with the unique tracial state and satisfying the UCT such that K ₀(A) = ℚ⊕ kerd _x and K ₁(A) = K ₁(C(X)), then A is isomorphic to an inductive limit of M _{n !}(C(X)). Received April 19, 2001, Accepted April 27, 2001.     

On the average distance property and certain energy integrals

One of our main results is the following: LetX be a compact connected subset of the Euclidean spaceR ⁿ andr(X, d ₂) the rendezvous number ofX, whered ₂ denotes the Euclidean distance inR ⁿ . (The rendezvous numberr(X, d ₂) is the unique positive real number with the property that for each positive integern and for all (not necessarily distinct)x ₁,x ₂,...,x _n inX, there exists somex inX such that .) Then there exists some regular Borel probability measure μ₀ onX such that the value of ∫ _X d ₂(x, y)dμ₀ (y) is independent of the choicex inX, if and only ifr(X, d ₂) = sup_μ ∫ _X ∫ _X d ₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ onX.     

On<Emphasis Type="Italic">M</Emphasis>-structure,the asymptotic-norming property and locally uniformly rotund renormings

Letr, s ∈ [0, 1], and letX be a Banach space satisfying theM(r, s)-inequality, that is,

Bi-convexity and bi-martingales

A set in a product spaceX×Y isbi-convex if all itsx- andy-sections are convex. Abi-martingale is a martingale with values inX×Y whosex- andy-coordinates change only one at a time. This paper investigates the limiting behavior of bimartingales in terms of thebi-convex hull of a set — the smallest bi-convex set containing it — and of several related concepts generalizing the concept of separation to the bi-convex case. Research partially supported by NSF grants at the Institute for Mathematical Studies in the Social Sciences, Standford University. The second author has also been partially supported by the Deutche Forschungsgemeinschaft. We thank Andreu Mas-Colell, Jean-Francois Mertens, Abraham Neyman and Lloyd S. Shapley for many useful discussions.     

The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (x_n) → x in X and (x_n*) → 0 in X* with ||x_n|| = ||x||= 1 we have (x_n*(x_n)) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S ↦ Sh, S ↦ S^th are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.