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In this paper, we give a necessary and sufficient condition on eigenfunctions of the Hua operator on a Hermitian symmetric space of tube type X = G / K , to have an L p -Poisson integral representations over the Shilov boundary of X . More precisely, let λ ∈ C such that R ( λ ) > η − 1 (2 η being the genus of X ) and let F be a C -valued function on X satisfying the following Hua system of second order differential equations: H q F = ( λ 2 − η 2 ) 32 η 2 F Z . Then F has an L p -Poisson integral representation ( 1 p + ∞ ) over the Shilov boundary of X if and only if it satisfies the following growth condition of Hardy type: sup t > 0 e r ( η − R λ ) t { ∫ K | F ( k a t ) | p d k } 1 / p + ∞ . In particular for λ = η , we obtain that a Hua-harmonic function on X has an L p -Poisson integral representation over the Shilov boundary of X if and only if its Hardy norm is finite.

Let D be a bounded symmetric domain and Σ be the Shilov boundary of D. For R≥1, l∈ℤ+ and 1≤p≤∞, let DR=RD, Hp,l(DR) and Ap,l(DR) be the Hardy–Sobolev and Bergman–Sobolev spaces on DR, respectively. In this paper we show that the Kolmogorov, linear, Gel'fand, and Bernstein N-widths of Hp,l(DR) in Lp(Σ) all coincide, calculate the exact value, and identify optimal subspaces or optimal linear operators. We also do the same for N-widths of Ap,l(DR) in Lp(D). Moreover, we obtain new asymptotic estimates for the linear and Gel'fand N-widths of [Formula: see text] and [Formula: see text] in Lq(Sn) and Lq(Bn), where R>1, l∈ℤ+, 2≤p≤q≤∞, [Formula: see text], [Formula: see text] are the unit ball and unit sphere in [Formula: see text], respectively, and [Formula: see text]. Furthermore, we obtain asymptotic estimates for the linear and Gel'fand N-widths of Hp,l(DR) in Lq(Σ), where R>1, l∈ℤ+ and 2≤p≤q≤∞.

Let Ab(BE) be the Banach algebra of all complex val- ued bounded continuous functions on the closed unit ball BE of a complex Banach space E, and holomorphic in the interior of BE, endowed with the sup norm. We present some sucient conditions for a set to be a boundary for Ab(BE) in case E belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in (6). We also prove the non-existence of the Shilov boundary for Ab(BE) and give some examples of bound- aries.

RésuméSoit D un espace hermitien symétrique de type tube, soit G=G(D) son groupe de difféomorphismes holomorphes, et S sa frontière de Shilov. A tout triplet (σ1,σ2,σ3)∈S×S×S on associe un entier ι(σ1,σ2,σ3), appelé son indice de Maslov. L'indice de Maslov est invariant sous l'action de G, est antisymétrique par permutation des arguments et satisfait une relation de cocycle. On généralise ainsi la théorie classique de l'indice de Maslov, où S est la variété lagrangienne et G le groupe symplectique. La définition de l'indice de Maslov suit des travaux antérieurs [Clerc, Ørsted, Transformation Groups 6 (2001) 303–320 ; Clerc, Ørsted, Asian J. Math. 7 (2003) 269–296], où la définition était restreinte aux triplets mutuellement transverses. La clé de l'extension est l'utilisation de la convergence Γ-radiale en un point de la frontière de Shilov.AbstractLet D be a Hermitian symmetric space of tube type, G=G(D) its group of holomorphic diffeomorphisms, and S its Shilov boundary. To any triple (σ1,σ2,σ3)∈S×S×S is associated an integer ι(σ1,σ2,σ3), called its Maslov index. The Maslov index is invariant under the action of G, is skew-symmetric with respect to the three arguments and satisfies a cocycle relation. It generalizes the classical theory of the Maslov index, where S is the Lagrangian manifold and G the symplectic group. The definition of the Maslov index follows previous work [Clerc, Ørsted, Transformation Groups 6 (2001) 303–320; Clerc, Ørsted, Asian J. Math. 7 (2003) 269–296], where the definition was restricted to mutually transverse triples. The key to the present extension is the use of Γ-radial convergence at a point of the Shilov boundary.

if it is annihilated by every element of DG(X), the algebra of all G-invariant differential operators without constant term. One of the most beautiful results in the harmonic analysis of symmetric spaces is the Helgason conjecture, which states that on a Riemannian symmetric space of noncompact type, a function is harmonic if and only if it is the Poisson integral of a hyperfunction over the Furstenberg boundary G/Po where Po is a minimal parabolic subgroup. (See [14], [17].) One of the more remarkable aspects of this theorem is its generality; one obtains a complete description of all solutions to the system of invariant differential operators on X without imposing any boundary or growth conditions. If X is a Hermitian symmetric space, then one is typically interested in complex function theory, in which case one is interested in functions whose boundary values are supported on the Shilov boundary rather than the Furstenberg boundary. (The Shilov boundary is G/P where P is a certain maximal parabolic containing Po.) In this case, it turns out that the algebra of G invariant differential operators is not necessarily the most appropriate one for

Let A u ( B G ) be the Banach algebra of all complex valued functions defined on the closed unit ball B G of a complex Banach space G which are uniformly continuous on B G and holomorphic in the interior of B G , endowed with the sup norm. A characterization of the boundaries for A u ( B G ) is given in case G belongs to a class of Banach spaces that includes the pre-dual of a Lorentz sequence space studied by Gowers in Israel J. Math. 69 (1990) 129–151. The non-existence of the Shilov boundary for A u ( B G ) is also proved.

In this paper, we present a geometric norm equality involving an admissible linear form ω for the Shilov boundary of a homogeneous Siegel domain D. We prove that the validity of this norm equality is equivalent to the symmetry of D and the reduction of ω essentially to the Koszul form. This, in particular, reveals a geometric reason that the Poisson kernel is annihilated by the Laplace–Beltrami operator if and only if D is symmetric, a theorem due to Hua, Look, Korányi and Xu.

Let D be a bounded symmetric domain and Σ be the Shilov boundary of D. For λ∈W, the Wallach set, and a nonnegative integer l, we study the weighted Bergman space Aλ2(D) and the weighted Bergman–Sobolev space A2,λ,l(D). For 0

Sibony and the author independently defined a higher order generalization of the usual Shilov boundary of a function algebra which yielded extensions of results about analytic structure from one dimension to several dimensions. Tonev later obtained an alternative characterization of this generalized Shilov boundary by looking at closed subsets of the spectrum whose image under the spectral mapping contains the topological boundary of the joint spectrum. In this note we define two related notions of what it means to be a higher order/higher dimensional boundary for a space of functions without requiring that the boundary be a closed set. We look at the relationships between these two boundaries, and in the process we obtain an alternative proof of Tonev's result. We look at some examples, and we show how the same concepts apply to convex sets and linear functions.

The paper is devoted to the conditions of existence of the holomorphic expansion, in a domain D, of a function f ∈C(M) defined on a set M with a positive measure μ of the Shilov boundary S of the complete circular domain D⊂Cn. These conditions are as follows: integrals of the product of f by certain polynomials that are orthogonal holomorphic functions tend to 0.

We describe the Shilov boundary ideal for a q-analog of algebra of holomorphic functions on the unit ball in the space of $2\times 2$ matrices., 14 pages

We introduce the concept of an $E$-valued function algebra, a type of Banach algebra that consist of continuous $E$-valued functions on some compact Hausdorff space, where $E$ is a Banach algebra. We present some basic results about such algebras, having to do with the Shilov boundary and the set of peak points of some commutative $E$-valued function algebras. We give some specific examples., 13 pages. This version corrects the published version by deleting two sentences on page 2

LetD be a Hermitian symmetric space of tube type,S its Shilov boundary andG the neutral component of the group of bi-holomorphic diffeomorphisms ofD. In the model situationD is the Siegel disc,S is the manifold of Lagrangian subspaces andG is the symplectic group. We introduce a notion of transversality for pairs of elements inS, and then study the action ofG on the set of triples of mutually transversal points inS. We show that there is a finite number ofG-orbits, and to each orbit we associate an integer, thus generalizing theMaslov index. Using the scalar automorphy kernel ofD, we construct a ℂ*,G-invariant kernel onD×D×D. Taking a specific determination of its argument and studying its limit when approaching the Shilov boundary, we are able to define a ℤ-valued,G-invariant kernel for triples of mutually transversal points inS. It is shown to coincide with the Maslov index. Symmetry properties and cocycle properties of the Maslov index are then easily obtained.

We consider and studyBlaschke inductive limit algebrasA(b), defined as inductive limits of disc algebrasA(D)linked by a sequenceb={Bk}k=1∞of finite Blaschke products. It is well known that bigG-disc algebrasAGover compact abelian groupsGwith ordered dualsΓ=Gˆ⊂ℚcan be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebraA(b)is a maximal and Dirichlet uniform algebra. Its Shilov boundary∂A(b)is a compact abelian group with dual group that is a subgroup ofℚ. It is shown that a bigG-disc algebraAGover a groupGwith ordered dualGˆ⊂ℝis a Blaschke inductive limit algebra if and only ifGˆ⊂ℚ. The local structure of the maximal ideal space and the set of one-point Gleason parts of a Blaschke inductive limit algebra differ drastically from the ones of a bigG-disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not bigG-disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a bigG-disc algebra is found. We consider also inductive limitsH∞(I)of algebrasH∞, linked by a sequenceI={Ik}k=1∞of inner functions, and prove a version of the corona theorem with estimates for it. The algebraH∞(I)generalizes the algebra of bounded hyper-analytic functions on an open bigG-disc, introduced previously by Tonev.

Let $A$ be a unital Banach function algebra with character space $\Phi_A$. For $x\in \Phi_A$, let $M_x$ and $J_x$ be the ideals of functions vanishing at $x$, and in a neighbourhood of $x$, respectively. It is shown that the hull of $J_x$ is connected, and that if $x$ does not belong to the Shilov boundary of $A$ then the set $\{y\in\Phi_A: M_x\supseteq J_y\}$ has an infinite, connected subset. Various related results are given.

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)0, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)0=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991).

E. Damek, A. Hulanicki, and R. Penney ( J. Funct. Anal. , in press) studied a canonical system of differential equations (the Hua system) denoted HJK which is definable on any Kahlerian manifold M . Functions annihilated by this system are called “Hua-harmonic.” In the case where M is a bounded homogeneous domain in C n with its Bergman metric, it was shown that every bounded Hua-harmonic function has a boundary value on the Bergman–Shilov boundary and that the function is reproducible from the Shilov boundary by integration against the reduction of the Poisson kernel for the Laplace–Beltrami operator to the Shilov boundary. This then provided a partial generalization of the results of Johnson and Koranyi to the stated context. Significantly, however, no characterization of the resulting space of boundary functions so obtained was given. The current work extends these results in several ways. We show that for a tube domain (i.e., a Siegel domain of type I), the Cauchy–Szego Poisson kernel also reproduces the Hua-harmonic functions. Since the two kernels agree only in the symmetric case, it follows that the space of boundary functions is dense in L ∞ if and only if the domain is symmetric. We also show that an L 2 function is the boundary function for a Hua-harmonic function if and only if its Fourier transform is supported in a certain (typically non-convex) cone. This cone is characterized in terms of the Fourier transformation of the Cauchy–Szego Poisson kernel.

Let M = G / H \mathcal {M}=G/H be a irreducible symmetric space of Cayley type. Then M \mathcal {M} is diffeomorphic to an open and dense G G -orbit in the Shilov boundary of G / K × G / K G/K\times G/K . This compactification of M \mathcal {M} is causal and can be used to give answers to questions in harmonic analysis on M \mathcal {M} . In particular we relate the Hardy space of M \mathcal {M} to the classical Hardy space on the bounded symmetric domain G / K × G / K G/K\times G/K . This gives a new formula for the Cauchy-Szegö kernel for M \mathcal {M} .

We consider the operator of restriction of functions holomorphic in a ball or a polydisc to curves lying on the Shilov boundary. It turns out that any function with polynomial growth near the boundary has such a restriction if the position of the curve satisfies a certain condition: if the domain is a ball, then the curve must be transversal to the standard contact distribution on the sphere, and if the domain is a polydisc, then the curve must be monotonic increasing with respect to all coordinates in the standard coordinatization of the torus. We use assertions of this kind to obtain a simple description of discrete inclusions in spectra (of minimal invariant subspaces) for several problems of -harmonic analysis.

(S,α) on a complex open Ol'shanskii semigroup S = G Exp (iW), where 1 ≤p≤∞ and α is an absolute value on the involutive semigroup X. For 1 < p < ∞ we prove the existence of an isometric boundary value map H p (S,α) → L p (G) generalizing the corresponding result of Ol'shanskii for p = 2 and α = 1. In the second part we use the fine structure of the space H 2 (S,1) to prove the existence of a bounded holomorphic function on S whose absolute value has a unique maximum in the boudary point 1Β G and therefore complete the proof of the approximation property of the Poisson kernel and the uniqueness of G as a Shilov boundary of S whenever W does not contain affine line.

We give a characterization of those solutions F of the Hua system of differential equations HDF(Z) = − (λ2 + 4)F(Z) · I, that are Poisson-Shilov integrals of L2-functions on the Shilov boundary of the rank two matrix ball.

Let D = G / K D=G/K be a bounded symmetric domain and K / L K/L the Shilov boundary of D D . Let N \mathcal {N} be the Shilov boundary of the Siegel domain realization of G / K G/K . We consider the case when D D is the exceptional non-tube type domain of the type ( e 6 ( − 14 ) , s o ( 10 ) × s o ( 2 ) ) (\mathfrak {e}_{6(-14)}, \mathfrak {so}(10)\times \mathfrak {so}(2)) . We prove that ( N ⋊ L , L ) (\mathcal {N}\rtimes L, L) is not a Gelfand pair and thus resolve an open question of G. Carcano.

We present a result on the existence of some kind of peak functions for C-convex domains and for the symmetrized polydisc. Then we apply the latter result to show the equivariance of the set of peak points for A(D) under proper holomorphic mappings. Additionally, we present a description of the set of peak points in the class of bounded pseudoconvex Reinhardt domains.

In this paper we consider several conditions for sequences of points in M(H ∞) and establish relations between them. We show that every interpolating sequence for QA of nontrivial points in the corona $$M(H^\infty )\backslash \mathbb{D}$$ of H ∞ is a thin sequence for H ∞, which satisfies an additional topological condition. The discrete sequences in the Shilov boundary of H ∞ necessarily satisfy the same condition.

In this article the images of the Poisson transform on the degenerate series representations attached to the Shilov boundary of a bounded symmetric domain of tube type are considered. We characterize the images by means of second-order differential equations.

Semisimple commutative Banach algebras A admitting exactly one uniform norm (not necessarily complete) are investigated. A has this Unique Uniform Norm Property iff the completion U(A) of A in the spectral radius r(·) has UUNP and, for any non-zero spectral synthesis ideal L of U(A), L ∩ A is non-zero. A is regular iff U (A) is regular and, for any spectral synthesis ideal L of A, A/I has UUNP iff U(A) is regular and for any spectral synthesis ideal I of U(A), L = k(h(A∩I)) (hulls and kernels in U(A)). A has UUNP and the Shilov boundary coincides with the Gelfand space iff A is weakly regular in the sense that, given a proper, closed subset F of the Gelfand space, there exists a non-zero x in A having its Gelfand transform vanishing on F. Several classes of Banach algebras that are weakly regular but not regular, as well as those that are not weakly regular but have UUNP are exhibited. The UUNP is investigated for quotients, tensor products, and multiplier algebras. The property UUNP compares with the unique C*-norm property on (not necessarily commutative) Banach *-algebras. The results are applied to multivariate holomorphic function algebras as well as to the measure algebra of a locally compact abelian group G. For a continuous weight ω on G, the Beurling algebra L1(G,ω) (assumed semisimple) has UUNP iff it is regular.

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over C whenever all the closed ideals in the algebra are (algebraically) finitely generated. In 1974, Sinclair and Tullo obtained a non-commutative version of this result. In 1978, Ferreira and Tomassini improved the result of Grauert and Remmert by showing that the statement is also true if one replaces `closed ideals' by `maximal ideals in the Shilov boundary of A'. We give a shorter proof of this latter result, together with some extensions and related examples. We study the following conjecture. Suppose that all maximal left ideals in a unital Banach algebra A are finitely generated. Then A is finite-dimensional.

This paper is a continuation of the program started by Ruan (Acta Math Sin (Engl Ser) 19(3):485–496, 2003, Illinois J Math 47(4):1047–1062, 2003), of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, \(C^*\)-envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided \(M\)-ideal theory and characterize one-sided \(M\)-ideals in real \(C^*\)-algebras and real operator algebras with contractive approximate identity.

An affirmative answer is given to a question of S. Axler and A. Shields regarding the possibility of the continuous extension to the Shilov boundary M(L∞) of the algebra H∞ of arbitrary continuous (not necessarily bounded) functions in the unit circle, having almost everywhere nontangential limits. At the same time a description is obtained, in terms of the continuous extension to part of the boundary M(H∞), of the maximal class of continuous functions for which Fatou's theorem on nontangential limits holds.

Among all plastic deformations of the gravitational Lorentz vacuum \cite{wr1} a particular role is being played by conformal deformations. These are conveniently described by using the homogeneous space for the conformal group SU(2,2)/S(U(2)x U(2)) and its Shilov boundary - the compactified Minkowski space \tilde{M} [1]. In this paper we review the geometrical structure involved in such a description. In particular we demonstrate that coherent states on the homogeneous Kae}hler domain give rise to Einstein-like plastic conformal deformations when extended to \tilde{M} [2]., 10 pages, 1 figure; four misprints in the original version corrected: one lacking closing parenthesis, two letters, and an overall sign in front of the primitive function on p. 3

In this paper, we will discuss a class of nonlinear boundary value problems for second–order, overdetermined elliptic systems. In particular, we consider the following nonlinear Dirichlet problem: where G: = G 1 × G 2 is a polycylinderG j:={z j :|z j

and the closed Lie ball is defined by B[r] = { z ∈ C; L(z) ≤ r}. The Shilov boundary of the Lie ball, which we call the Lie sphere, is given by Σr = { ex ∈ C, θ ∈ R, x ∈ Sr } , where Sr = {x ∈ R;x = r2} is the real sphere. We denote by O(B(r)) the space of holomorphic functions on B(r) and by C(B[r]) the space of continuous functions on B[r]. For holomorphic functions on B(r) we have the following Cauchy-Hua integral formula. Theorem 1.1. (Cauchy-Hua integral formula, ) For f ∈ O(B(r)) ∩ C(B[r]), we have the following integral representation

Let $I$ be an index set, not necessarily a subset of any Banach algebra. Let $\mathcal{A}$ and $\mathcal{B}$ be unital semisimple commutative Banach algebras with maximal ideal spaces $M_{\mathcal{A}}$ and $M_{\mathcal{B}}$, respectively. If surjective mappings $S_1, S_2 \colon I \to \mathcal{A}$ and $T_1, T_2 \colon I \to \mathcal{B}$ satisfy $\mathrm{r}(T_1(\lambda) - T_2(\mu)) = \mathrm{r}(S_1(\lambda) - S_2(\mu))$ for all $\lambda, \mu \in I$, where $\mathrm{r}(a)$ is the spectral radius of $a$, then there exist $p, w \in \mathcal{B}$, a homeomorphism $\varphi \colon M_\mathcal{B} \to M_\mathcal{A}$ and a closed and open subset $K$ of $M_\mathcal{B}$ such that $|\hat{w}| = 1$ on $M_\mathcal{B}$ and that ¶ $$ \widehat{T_k(\lambda)}(y) - \hat{p}(y) = \begin{cases} \hat{w}(y)\widehat{S_k(\lambda)}(\varphi(y)) & y \in K \\[2pt] \hat{w}(y)\overline{\widehat{S_k(\lambda)}(\varphi(y))} & y \in M_\mathcal{B} \setminus K \end{cases} $$ ¶ for all $\lambda \in I$ $(k = 1, 2)$. In particular, if $\mathcal{A}$ and $\mathcal{B}$ are uniform algebras, and if $S_1, S_2 \colon I \to \mathcal{A}$ and $T_1, T_2 \colon I \to \mathcal{B}$ satisfy ¶ $$ \sigma_\pi ({T_1(\lambda) - T_2(\mu)}) \cap \sigma_pi ({S_1(\lambda) - S_2(\mu)} ) \neq \emptyset \qquad (\forall \lambda, \mu \in I), $$ ¶ where $\sigma_pi (f)$ is the peripheral spectrum of $f$, then $\widehat{T_k(\lambda)}(y) = \hat{p}(y) + \widehat{S_k(\lambda)}(\varphi(y))$ for all $\lambda \in I$ and $y \in M_\mathcal{B}$ $(k = 1, 2)$.

Extreme points of positive functionals and spectral states on real commutative Banach algebras are investigated and characterized as multiplicative functionals extending the well-known results from complex to real Banach algebras. As an application a new and short proof of the existence of the Shilov boundary of a real commutative Banach algebra with nonempty maximal ideal space is given.

We define a family of four-point invariants for Shilov boundaries of bounded symmetric domains of tube type, which generalizes the classical four-point cross ratio on the unit circle. This generalization, which is based on a similar construction of Clerc and {\O}rsted, is functorial and well-behaved under products; these two properties determine our extension uniquely. Our generalized cross ratios can be used to estimate translation lengths of a large class of isometries of the underlying bounded symmetric domain. Our main application concerns maximal representations of surface groups with Hermitian target. For any such representation we can construct a strict cross ratio on the circle in the sense of Labourie via pullback of our generalized cross ratio along a suitable limit curve. In this context our translation length estimates then imply that maximal representations with Hermitian target are well-displacing; this implies in particular that the action of the mapping class group on the moduli space of maximal representations into a Hermitian Lie group is proper., Comment: 50 pages, completely revised and extended version; earlier name "Cross ratios associated with maximal representations" extended to the present one

We prove that the Shilov boundary is invariant under proper holomorphic mappings between some classes of domains (containing among others quasi-balanced domains with the continuous Minkowski functionals). Moreover, we obtain an extension theorem for proper holomorphic mappings between quasi-circular domains. Using these results we show that there are no non-trivial proper holomorphic self-mappings in the tetrablock. Another important result of our work is a description of Shilov boundaries of a large class of domains (containing among other the symmetrized polydisc and the tetrablock). It is also shown that the tetrablock is not $\mathbb C$-convex., 11 pages

International audience; In the first part, the theory of bounded symmetric domains is presented along two main approaches : as special cases of Riemannian symmetric spaces of the noncompact type on one hand, as unit balls in positive Hermitian Jordan triple systems on the other hand. In the second part, an invariant for triples in the Shilov boundary of such a domain is constructed. It generalizes an invariant constructed by E. Cartan for the unit sphere in $\Bbb C^2$ and also the triple Maslov index on the Lagrangian manifold.

Let A be a Banach function algebra on a compact Hausdorff space X. In this paper we consider some relations between the maximal ideal space, the Shilov boundary and the Choquet boundary of A and its uniform closure A. As an application we determine the maximal ideal space, the Shilov boundary and the Choquet boundary of algebras of infinitely differentiable functions which were introduced by Dales and Davie in 1973. For some notations, definitions, elementary and known results, one can refer to [2] and [3]. Let X be a compact Hausdorff space and let C(X) denote the space of all continuous complex valued functions on X. A function algebra on X is a subalgebra of C(X) which contains the constants and separates the points of X. If there is an algebra norm on A so that Ilf gII < IlfH *HgII for all f, g E A, then A is called a normed function algebra. A complete normed function algebra on X is called a Banach function algebra on X. If the norm of a Banach function algebra is the uniform norm on X; i.e. Ilf IX = supxEX If(x)I, it is called a uniform algebra on X. If A is a function algebra on X, then A, the uniform closure of A, is a uniform algebra on X. If (A, 11 11) is a Banach function algebra on X, for every x E X the map ox: A -* C, defined by Ox(f) = f(x), is a nonzero continuous complex homomorphism on A and so Ox E MA, where MA is the maximal ideal space of A. We call ox the evaluation homomorphism at x. Clearly for every || fKx = sup If(x)I = sup IO'(f)I < sup 10(f)l = 1flM ? llfll xEX XEX OEMA where E is the Gelfand transform of f. The Banach function algebra A on X is called natural, if every 0 E MA is given by an evaluation homomorphism ox at some x E X; or, in other Received by the editors February 9, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46J10; Secondary 46J20.

We study the relations between boundaries for algebras of holomorphic functions on Banach spaces and complex convexity of their balls. In addition, we show that the Shilov boundary for algebras of holomorphic functions on an order continuous sequence space $X$ is the unit sphere $S_X$ if $X$ is locally c-convex. In particular, it is shown that the unit sphere of the Orlicz-Lorentz sequence space $\lambda_{\varphi, w}$ is the Shilov boundary for algebras of holomorphic functions on $\lambda_{\varphi, w}$ if $\varphi$ satisfies the $\delta_2$-condition.

This is a companion to recent papers of the authors; here we construct the `noncommutative Shilov boundary' of a (possibly nonunital) selfadjoint ordered space of Hilbert space operators. The morphisms in the universal property of the boundary preserve order. As an application, we consider `maximal' and `minimal' unitizations of such ordered operator spaces., 11 pages. To appear, Positivity

Let $\cal D$ be a bounded domain, $G$ its group of biholomorphic diffeomorphisms, and $S$ its Shilov boundary. We define a function $ \iota : S\times S\times S \longrightarrow \Bbb R$ which is invariant under $G$. This invariant generalizes the Maslov index as defined earlier by B. Osrted and the author and the angular invariant constructed by E. Cartan for the unit sphere in $\Bbb C^2$.

Geometric and Functional Analysis, 19 (3), ISSN:1016-443X, ISSN:1420-8970

We study positivity in C*-modules and operator spaces using open tripotents, and an ordered version of the `noncommutative Shilov boundary'. Because of their independent interest, we also systematically study open tripotents and their properties., To appear