In an earlier paper, the first two authors have shown that the convolution of a function $f$ continuous on the closure of a Cartan domain and a $K$-invariant finite measure $\mu$ on that domain is again continuous on the closure, and, moreover, its restriction to any boundary face $F$ depends only on the restriction of $f$ to $F$ and is equal to the convolution, in $F$, of the latter restriction with some measure ${\mu }_F$ on $F$ uniquely determined by $\mu$. In this article, we give an explicit formula for ${\mu }_F$ in terms of $F$,...

A previous paper was devoted to the construction of non-trivial holomorphic families of holomorphic isometries for the Carathéodory metric of a bounded domain in a complex Banach space, fixing a point in the domain. The present article shows that such a family cannot exist if it contains a strongly continuous one parameter semigroup.

We explore a condition under which the ideal of polynomials generated by an ideal of multilinear mappings between Banach spaces is a global holomorphy type. After some examples and applications, this condition is studied in its own right. A final section provides applications to the ideals formed by multilinear mappings and polynomials which are absolutely (p;q)-summing at every point.

In this paper spaces of entire functions of $\Theta$-holomorphy type of bounded type are introduced and results involving these spaces are proved. In particular, we “construct an algorithm” to obtain a duality result via the Borel transform and to prove existence and approximation results for convolution equations. The results we prove generalize previous results of this type due to B. Malgrange: Existence et approximation des équations aux dérivées partielles et des équations des convolutions. Annales...

The aim of this paper is to show that the integral and derivative operators defined by local regularities are homeomorphisms for generalized Besov and Triebel-Lizorkin spaces with local regularities. The underlying geometry is that of homogeneous type spaces and the functions defining local regularities belong to a larger class of growth functions than the potentials tα, related to classical fractional integral and derivative operators and Besov and Triebel-Lizorkin spaces.

The classical Erdös spaces are obtained as the subspaces of real separable Hilbert space consisting of the points with all coordinates rational or all coordinates irrational, respectively. One can create variations by specifying in which set each coordinate is allowed to vary. We investigate the homogeneity of the resulting subspaces. Our two main results are: in case all coordinates are allowed to vary in the same set the subspace need not be homogeneous, and by specifying different sets for different...

Let $𝒜$ be a homogeneous algebra on the circle and ${𝒜}^{+}$ the closed subalgebra of $𝒜$ of functions having analytic extensions into the unit disk $D$. This paper considers the structure of closed ideals of ${𝒜}^{+}$ under suitable restrictions on the synthesis properties of $𝒜$. In particular, completely characterized are the closed ideals in ${𝒜}^{+}$ whose zero sets meet the circle in a countable set of points. These results contain some previous results of Kahane and Taylor-Williams obtained independently.

This paper considers the Lipschitz subalgebras $\Lambda (\alpha ,p,𝒜)$ of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in $\Lambda (\alpha ,p;𝒜)$, $\alpha \ne \left[\alpha \right]$. From these estimates the Ditkin and Analytic Ditkin conditions for $\Lambda (\alpha ,p;𝒜)$ follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to $\Lambda (\alpha ;,p;𝒜)$ as does the theory developed in part I of this series which requires the Analytic Ditkin condition.Examples are discussed...

Let $𝔐$ be a Jordan-Banach algebra with identity 1, whose norm satisfies:(i) $\parallel ab\parallel \le \parallel a\parallel \parallel b\parallel$,   $a,b\in 𝔐$(ii) $\parallel a^2\parallel =\parallel a{\parallel }^2$(iii) $\parallel a^2\parallel \le \parallel a^2+b^2\parallel$.$𝔐$ is called a JB algebra (E.M. Alfsen, F.W. Shultz and E. Stormer, Oslo preprint (1976)). The set ${𝔐}^{+}$ of squares in $𝔐$ is a closed convex cone. $(𝔐,{𝔐}^{+},\mathbf{1})$ is a complete ordered vector space with $\mathbf{1}$ as a order unit. In addition, we assume $𝔐$ to be monotone complete (i.e. $𝔐$ coincides with the bidual ${𝔐}^{**}$), and that there exists a finite normal faithful trace $\phi$ on $𝔐$.Then the completion $\{{𝔐}^{+}{\}}_{\phi }$ of ${𝔐}^{+}$ with respect to the Hilbert structure...