New Moduli of Smoothness on the Unit Ball and Other Domains,Introduction and Main Properties

A new set of moduli of smoothness on a large variety of Banach spaces of functions on the unit ball is introduced. These measures of smoothness utilize uniformly bounded holomorphic semigroups on the Banach space in question. The new moduli are “correct” in the sense that they satisfy direct (Jackson) and weak converse inequalities. The method used also applies to spaces of functions on the simplex and the unit sphere, and while the main goal is the investigation of properties and relations concerning the unit ball, many of the results will be given for other domains and situations. The classic properties, including equivalence with appropriate \(K\) -functionals or realization functionals, will be established. Bernstein- and Kolmogorov-type inequalities are proved.     

Smoothness of Subharmonic Functions and Potentials of the Bergman Metric in the Unit Ball

We consider Lipschitz smoothness of an arbitray invariant potential U on the unit ball B in . We establish some Lipschitz estimates for both U and its gradient vector field U with respect to the Bergman metric. These estimates are taken with respect an invariant distance on B and shown to hold outside on open sets with arbitrarily small Hausdorff conttent. We also prove that for an M-subharmonic function u which satisfies Littelwood's integrability condition, there are such open sets , such that u is Lipschitz smooth on B\.     

Prime JB*‐Triples and Extreme Dual Ball Density

Properties of the extreme points ∂_e(E*₁) of the closed dual ball E*₁ of a JB*‐triple E are studied. It is shown that the canonical mapping from ∂_e(E*₁) onto the structure space, Prim (E), of primitive M‐ideals of E is an open mapping. This property is utilised to show that ∂_e(E*₁) is weak* dense in E*₁ if and only if E is an infinite dimensional Hilbert space, an infinite dimensional spin factor or E is prime with zero socle.     

On the Local Hölder Boundary Smoothness of an Analytic Function in the Unit Ball Compared with the Smoothness of Its Modulus

The Paley–Littlewood square function     

Closed Tripotents and Weak Compactness in the Dual Space of a Jb*-Triple

We revise the concept of compact tripotent in the bidual spaceof a JB*-triple. This concept was introduced by Edwards andRüttimann generalizing the ideas developed by Akemann forcompact projections in the bidual of a C*-algebra. We also obtainsome characterizations of weak compactness in the dual spaceof a JC*-triple, showing that a bounded subset in the dual spaceof a JC*-triple is relatively weakly compact if and only ifits restriction to any abelian maximal subtriple C is relativelyweakly compact in the dual of C. This generalizes a very usefulresult by Pfitzner in the setting of C*-algebras. As a consequencewe obtain a Dieudonné theorem for JC*-triples which generalizesthe one obtained by Brooks, Saitô and Wright for C*-algebras.     

On Completeness Properties of Functors of the Unit Ball of Borel Measures

In this paper, we study the questions of uniform and topological completeness of functors of the unit ball U and U _R of -additive measures and Radon measures respectively.     

Manifolds of tripotents in JB*-triples

Abstract. We describe the affine connections, geodesics and symmetries of various Banach manifolds of tripotents in JB*-triples which include the C*-algebras and Hilbert spaces where the nonzero tripotents are respectively the partial isometries and the extreme points of the closed unit ball. Received July 7, 1998; in final form November 16, 1998     

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

Self-adjointness in unitary isotopes of JB*-algebras

We prove that x is in the norm closure of invertibles whenever x is self-adjoint in some unitary isotope of a JB*-algebra. In the sequel, we investigate certain links between the elements of the unit ball which are self-adjoint in some unitary isotope of a JB*-algebra and the elements which are average of two unitaries. Received: 9 September 2005     

Asymptotic Properties of Harmonic and M-Harmonic Functions near the Boundary of the Unit Ball

It is shown that decompositions of a plurisubharmonic measure on a sphere of ⁿ with respect to the Hausdorff dimension scales related to the Euclidean and Korányi metrics coincide under a linear correspondence of indices. Bibliography: 7 titles.     

On real forms of JB*-triples

We introduce real JB^*-triples as real forms of (complex) JB^*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB^*-triples is an isometry if and only if it commutes with the cube mappinga→a ³={aaa}. This generalizes a result of Dang for complex JB^*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB^*-triple numerical invariants that are respected by surjective linear isometries.     

Equivalence of the Stone-Weierstrass conjectures for C* and JB*-algebras

We prove that the Stone-Weierstrass conjectures for C*-algebras and JB*-algebras are equivalent.     

Separate weak*-continuity of the triple product in dual real JB*-triples

We prove that, if E is a real JB*-triple having a predual then is the unique predual of E and the triple product on E is separately $sigma (E,E_{*_{}})-$continuous. Received February 1, 1999; in final form March 29, 1999 / Published online May 8, 2000     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

Weak continuity of holomorphic automorphisms in JB*-triples

Supported by DAAD grant 313, programm II (1991)     

Representable Functions on the Unit Ball of a Banach Space

In this paper we treat the problem of integral representation of analytic functions over the unit ball of a complex Banach space X using the theory of abstract Wiener spaces. We define the class of representable functions on the unit ball of X and prove that this set of functions is related with the classes of integral k–homogeneous polynomials, integral holomorphic functions and also with the set of L ^p –representable functions on a Banach space.     

Weak limits of the extreme dual ball in JB*-triples

We study weak limits of the extreme points, ∂ _e (E ^* ₁), of the dual ball of a JB*-triple, E. We show that all such weak limits, except possibly the zero functional, are weak sequential limits and we discuss implications for the structure of E. Received: 9 April 2001     

Continuity of generalized derivations on JB*-algebras

We prove that every generalized Jordan derivation D from a JB?-algebra 𝒜 into itself or into its dual space is automatically continuous. In particular, we establish that every generalized Jordan derivation from a C?-algebra to a Jordan Banach module is continuous. As a consequence, every generalized derivation from a C?-algebra to a Banach bimodule is continuous.     

Algebraic Properties of Toeplitz Operators with Radial Symbols on the Bergman Space of the Unit Ball

In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in . We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form is studied, where and φ is a radial function. Ze-Hua Zhou: supported in part by the National Natural Science Foundation of China (Grand Nos.10671141, 10371091).