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We prove that irreducible symmetric domains are uniquely determined by the homotopy equivalence classes of their Shilov boundaries. [ABSTRACT FROM AUTHOR]

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let G be a semisimple algebraic R -group such that G = G (R) ∘ is of Hermitian type. If Γ ≤ L is a torsion-free lattice of a finite connected covering of PU (1 , 1) , given a standard Borel probability Γ -space (Ω , μ Ω) , we introduce the notion of Toledo invariant for a measurable cocycle σ : Γ × Ω → G . The Toledo invariant remains unchanged along G-cohomology classes and its absolute value is bounded by the rank of G. This allows to define maximal measurable cocycles. We show that the algebraic hull H of a maximal cocycle σ is reductive and the centralizer of H = H (R) ∘ is compact. If additionally σ admits a boundary map, then H is of tube type and σ is cohomologous to a cocycle stabilizing a unique maximal tube type subdomain. This result is analogous to the one obtained for representations. In the particular case G = PU (n , 1) maximality is sufficient to prove that σ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles. [ABSTRACT FROM AUTHOR]

We prove that Sp(2d, ℝ)-cocycles, HSp(2d)-cocycles and pseudo-unitary cocycles with at least one non-zero Lyapunov exponent are dense in all usual regularity classes for non-periodic dynamical systems. For Schrödinger operators on the strip, we prove a similar result about the density of positive Lyapunov exponents. This generalizes a result of A. Avila [2] to higher dimensions. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit formula for the zonal spherical function on the Lie ball in ℂn. [ABSTRACT FROM AUTHOR]

In this paper, we give the explicit formula for the zonal spherical function on the Lie ball in ℂn. [ABSTRACT FROM AUTHOR]

Let A and B be commutative Banach algebras. Then a multi- plicative linear functional θ on B induces a multiplication on the Cartesian product space A × B given by (a,b)(c,d) = (ac + θ(d)a + θ(b)c,bd) for all (a,b),(c,d) ∊ A × B. We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn{Banach property, and the unique semisimple norm property under certain conditions on θ. [ABSTRACT FROM AUTHOR]

Let D be the Lie ball in C2 and let A′(S) be the space of all hyperfunctions over the Shilov boundary S of D. The aim of this paper is to give a necessary and sufficient condition on the Poisson transform Pλf of an element f in the space A′(S) for f to be in L2(S). More precisely, we establish for any λ … R\{0} that: (i) Let F = Pλf, f … L2(S). Then we have ||F||2* = sup t>0 1/t2 ∫0t (∫r10 ∫S(O(2)×O(2)) |F(kaR.0)|2 sinh (r1 - r2) sinh (r1 + r2) dkdr2)dr1 < ∞. (ii) Let f be a hyperfunction on S such that its image F = Pλf satisfies the growth condition ||F||* < ∞, then necessarily such f is in L2(S). [ABSTRACT FROM AUTHOR]

We continue to discuss the example presented in [4]. In particular, we clarify some gaps and complete the description of the Shilov boundary. [ABSTRACT FROM AUTHOR]

Let Γ denote a lattice in SU(1, p), with p greater than 1. We show that there exists no Zariski dense maximal representation with target SU( m, n) if n > m > 1. The proof is geometric and is based on the study of the rigidity properties of the geometry whose points are isotropic m-subspaces of a complex vector space V endowed with a Hermitian metric h of signature ( m, n) and whose lines correspond to the 2 m dimensional subspaces of V on which the restriction of h has signature ( m, m). [ABSTRACT FROM AUTHOR]

We discuss some generalizations of the classical Fejér and Mittag-Leffler theorems to the case of several complex variables with applications to the Shilov and Bergman boundaries. [ABSTRACT FROM AUTHOR]

The article informs about a study based on a method to develop the multiparameter Hardy space (HS) theory on product spaces of homogeneous type. It includes that the theory comprises of the product HS, its dual, the product bounded mean oscillation space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. It adds that findings of the study prove the boundedness of singular integrals on the classical product HS.

We show that there is a unique functorial way to extend the classical four-point cross ratio on the unit circle to a family of four-point invariants on Shilov boundaries of bounded symmetric domains of tube type. These generalized cross ratios can be used to estimate translation lengths of a large class of isometries of the underlying bounded symmetric domains. They can also be used to associate with every maximal representation of a surface group with Hermitian target a strict weak cross ratio on the circle in the sense of Labourie. In this context our translation length estimates then imply that maximal representations with Hermitian target are well-displacing, and consequently that the action of the mapping class group on the moduli space of maximal representations into a Hermitian Lie group is proper. [ABSTRACT FROM AUTHOR]

If $${\mathcal{A}}$$ is a family of continuous functions on a locally compact Hausdorff space X, a boundary for $${\mathcal{A}}$$ is a subset $${B \subset X}$$ such that every $${f \in \mathcal{A}}$$ attains its maximum modulus on B. In this work we generalize the concept of strong boundary points for families of functions and show that the collection of these generalized strong boundary points is always a boundary for $${\mathcal{A}}$$ . We give conditions under which all boundaries for $${\mathcal{A}}$$ consist of generalized strong boundary points and under which these points coincide with classical strong boundary points. When $${\mathcal{A}}$$ has sufficient algebraic structure it is proven that this construction provides a unique boundary for $${\mathcal{A}}$$ consisting of boundary points, and we conclude by demonstrating how this approach provides an alternate technique for proving the existence of the Choquet and Shilov boundaries in certain function algebras. [ABSTRACT FROM AUTHOR]

Abstract: For bounded symmetric domains of tube type and general domains of type I, we consider the action of G on sections of a homogeneous line bundle over Ω and the corresponding eigenspaces of G-invariant differential operators. The Poisson transform maps hyperfunctions on the Shilov boundary to the eigenspaces. We characterize the image in terms of twisted Hua operators. For some special parameters the Poisson transform is of Szegö type whose image is in a relative discrete series; we compute the corresponding elements in the discrete series. [Copyright &y& Elsevier]

It has been shown that any Banach algebra satisfying | f| = | f| has a representation as an algebra of quaternion-valued continuous functions. Whereas some of the classical theory of algebras of continuous complex-valued functions extends immediately to algebras of quaternion-valued functions, similar work has not been done to analyze how the theory of algebras of complex-valued Lipschitz functions extends to algebras of quaternion-valued Lipschitz functions. Denote by Lip( X, $\mathbb{F}$ ) the algebra over R of F-valued Lipschitz functions on a compact metric space ( X, d), where $\mathbb{F}$ = ℝ, ℂ, or ℍ, the non-commutative division ring of quaternions. In this work, we analyze a class of subalgebras of Lip( X, $\mathbb{F}$ ) in which the closure of the weak peak points is the Shilov boundary, and we show that algebras of functions taking values in the quaternions are the most general objects to which the theory of weak peak points extends naturally. This is done by generalizing a classical result for uniform algebras, due to Bishop, which ensures the existence of the Shilov boundary. While the result of Bishop need not hold in general algebras of quaternion-valued Lipschitz functions, we give sufficient conditions on such an algebra for it to hold and to guarantee the existence of the Shilov boundary. [ABSTRACT FROM AUTHOR]

Let D be the Lie ball in ℂ n , when n is odd. In this paper, we give an explicit expression of the generalized spherical function Φλ,m for D. More precisely, Φλ,m is given explicitly as sum of determinants of the classical Gauss hypergeometric functions. Note that our computation technic is done by integration by parts and induction. Namely, it is based on the key Lemma 2.2. The purpose of this article is to present an explicit expression of the generalized spherical function on odd-dimensional bounded symmetric domains of type IV (Lie balls). [ABSTRACT FROM AUTHOR]

Let K be an algebraically closed field complete with respect to a dense ultrametric absolute value |.|. Let D be an infraconnected affinoid subset of K and let H( D) be the Banach algebra of analytic elements on D. Let f ∈ H( D) be injective in D and let f be the mapping defined on the multiplicative spectrum of H( D) that identifies with the set of circular filters on D. We show that f is injective and maps bijectively the Shilov boundary of H( D) onto this of H( f( D)). Thanks to this property we give a new proof of the equality $\left| {f(x) - f(y)} \right| = \left| {x - y} \right|\sqrt {\left| {f'(x)f'(y)} \right|} $. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to presenting an explicit expression for the Harish-Chandra c-function for the Lie ball. Our computation technique is based on the expression for the generalized spherical function and its asymptotic behavior. [Copyright &y& Elsevier]

Let S be a discrete abelian cancellation semigroup with identity. We consider an algebra of generalized analytic functions defined on the semigroup % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgaru % WqVvNCPvMCamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegqvATv2CG4uz % 3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj-hEe % eu0xXdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-J % b9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaabau % aaaOqaaiqbdofatzaajaaaaa!3ABF! $$ \hat S $$ of semicharacters of S that is wider than the Arens-Singer algebra. We show that the strong boundary and the Shilov boundary of this algebra are unions of maximal subgroups of % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXgaru % WqVvNCPvMCamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegqvATv2CG4uz % 3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0dXdh9vqqj-hEe % eu0xXdbba9frFj0-OqFfea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-J % b9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaacaqabeaadaabau % aaaOqaaiqbdofatzaajaaaaa!3ABF! $$ \hat S $$. If S contains no nontrivial simple ideals, then both boundaries coincide with the character group of S. In this case we calculate the Gelfand spectrum of the algebra under consideration. [ABSTRACT FROM AUTHOR]

Let A and B be unital, semisimple commutative Banach algebras with the maximal ideal spaces M A and M B, respectively, and let r( a) be the spectral radius of a. We show that if T: A → B is a surjective mapping, not assumed to be linear, satisfying r( T( a) + T( b)) = r( a + b) for all a; b ∈ A, then there exist a homeomorphism φ: M B → M A and a closed and open subset K of M B such that for all a ∈ A, where e is unit element of A. If, in addition, $$ \widehat{T\left( e \right)} = 1 $$ and $$ \widehat{T\left( {ie} \right)} = i $$ on M B, then T is an algebra isomorphism. [ABSTRACT FROM AUTHOR]

Abstract: Let and be uniform algebras with Choquet boundaries δA and δB. A map is called norm-linear if ; norm-additive, if , and norm-additive in modulus, if for each and all algebra elements f and g. We show that for any norm-linear surjection there exists a homeomorphism such that for every and . Sufficient conditions for norm-additive and norm-linear surjections, not assumed a priori to be linear, or continuous, to be unital isometric algebra isomorphisms are given. We prove that any unital norm-linear surjection T for which , or which preserves the peripheral spectra of -peaking functions of A, is a unital isometric algebra isomorphism. In particular, we show that if a linear operator between two uniform algebras, which is surjective and norm-preserving, is unital, or preserves the peripheral spectra of -peaking functions, then it is automatically multiplicative and, in fact, an algebra isomorphism. [Copyright &y& Elsevier]

For any n, [image omitted], such that n ≥ m, let { [image omitted]; Im - z * z positive definite} be the bounded symmetric domain of type I of rank m and let A'(S) be the space of all hyperfunctions over the Shilov boundary S of [image omitted]. The aim of this article is to give a necessary and sufficient condition on the generalized Poisson transform Pl,λ f of an element f in the space A'(S) for that f∈Lp(S), 1 < p < ∞. More precisely, we establish for any [image omitted] such that [image omitted] that: (i) Let F = Pl,λ f, f∈Lp(S). Then we have [image omitted] (ii) Let f be a hyperfunction on S such that its image F = Pl,λ f satisfies the growth condition [image omitted], then necessarily such f is in Lp(S). [ABSTRACT FROM AUTHOR]

In this paper we raise a question about the boundary of the crown domain of a Riemannian symmetric space X. In case X is of Hermitian type we give an affirmative answer. [ABSTRACT FROM AUTHOR]

Let D={z∈n/1-2z¯zt+|zzt|2>0 and |zzt|<1} be the Lie ball in n and let Hq be the associated Hua operator. Then, for a complex number λ such that Re(iλ)>n/2-1, we establish that every function F satisfying the following Hua system of differential equations: [image omitted] is the Poisson transform of an Lp-function (1

Abstract: We characterize the -range, , of the Poisson transform on the Shilov boundary for non-tube bounded symmetric domains Ω. We prove that this range is a Hardy type space for solutions of a Hua system, which are eigenfunctions of all invariant differential operators on Ω. [Copyright &y& Elsevier]

Abstract: Let be the Banach algebra of all complex-valued bounded continuous functions on the closed unit ball of a complex Banach space E and holomorphic in the interior of and let be the closed subalgebra of those functions which are uniformly continuous on . For the case whose bidual is a Marcinkiewicz sequence space , we describe some sufficient conditions for a set to be a boundary of either or . Moreover, we consider some analogous problems on to those which were studied on the Gowers space of characteristic p by Grados and Moraes [L.R. Grados, L.A. Moraes, Boundaries for algebras of holomorphic functions, J. Math. Anal. Appl. 281 (2003) 575–586; L.R. Grados, L.A. Moraes, Boundaries for an algebra of bounded holomorphic functions, J. Korean Math. Soc. 41 (1) (2004) 231–242]. [Copyright &y& Elsevier]

Abstract: Let Ω be a bounded symmetric domain of non-tube type in with rank r and S its Shilov boundary. We consider the Poisson transform for a hyperfunction f on S defined by the Poisson kernel , , . For all s satisfying certain non-integral condition we find a necessary and sufficient condition for the functions in the image of the Poisson transform in terms of Hua operators. When Ω is the type I matrix domain in (), we prove that an eigenvalue equation for the second order -valued Hua operator characterizes the image. [Copyright &y& Elsevier]

Let and | zz t | 0, we establish that every &complexes;-valued function F on &Dscr; 2 satisfying the following Hua system of differential equations is the Poisson transform of an L 2 -function on the Shilov boundary S of &Dscr; 2 if and only if it satisfies the following growth condition of Hardy type: [ABSTRACT FROM AUTHOR]

Abstract: Let L be a field complete for a non-trivial ultrametric absolute value and let be a commutative normed L-algebra with unity whose spectral semi-norm is . Let be the set of continuous multiplicative semi-norms of A, let be the Shilov boundary for and let . Then ψ belongs to if and only if for every neighborhood of ψ in , there exists and satisfying and . Suppose A is uniform, let and let . Then f is a topological divisor of zero if and only if there exists such that . Suppose now A is complete. If f is not a divisor of zero, then it is a topological divisor of zero if and only if the ideal fA is not closed in A. Suppose A is ultrametric, complete and Noetherian. All topological divisors of zero are divisors of zero. This applies to affinoid algebras. Let A be a Krasner algebra without non-trivial idempotents: an element is a topological divisor of zero if and only if is not a closed ideal; moreover, is a principal ideal ring if and only if it has no topological divisors of zero but 0 (this new condition adds to the well-known set of equivalent conditions found in 1969). [Copyright &y& Elsevier]

In this paper a *-algebra of regular functions on the Shilov boundary S(픻) of bounded symmetric domain 픻 is constructed. The algebras of regular functions on S(픻) are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harish–Chandra modules related to S(픻) = Un is investigated. [ABSTRACT FROM AUTHOR]

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 $H_{p,l}(B_R^n)$ and $A_{p,l}(B_R^n)$ in Lq(Sn) and Lq(Bn), where R>1, l∈ℤ+, 2≤p≤q≤∞, $B^n=\{z\in {\mathbb C}^n{:}\ |z|=\big(\sum_{j=1}^n|z_j|^2\big)^{1/2} < 1\}$, $S^n=\{z\in {\mathbb C}^n{:}\ |z|=1\}$ are the unit ball and unit sphere in ${\mathbb C}^n$, respectively, and $B_R^n=RB^n$. 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≤∞. [ABSTRACT FROM AUTHOR]

Let D be an irreducible Hermitian symmetric space of tube-type, S its Shilov boundary, G its group of holomorphic diffeomorphisms. For a generic triple of points (σ1,σ2,σ3)∈S×S×S, a characteristic G-invariant ι(σ1,σ2,σ3), called the Maslov index was introduced in [Transform. Groups 6 (2001) 303]. For D of classical type (i.e. for all cases except for the exceptional domain associated to Albert’s algebra), the definition of the Maslov index is extended to all triples, by using a holomorphic embedding of D into a Siegel disc, which corresponds to an embedding of S into a Lagrangian manifold. When D is the Lie ball, the extension of the definition is obtained through a realization of S in the Lagrangian manifold of a spinor space. [Copyright &y& Elsevier]

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<ρ<1 we obtain exact values of the Gel''fand and linear N-widths of A2,λ,l(D) in C(ρΣ). We also obtain the Bernstein N-widths of the Hardy–Sobolev space H∞,l(D) in Aλ2(ρD). [Copyright &y& Elsevier]

We give a counterexample to the following theorem of Bremermann on Shilov boundaries: if D is a bounded domain in Ćn having a univalent envelope of holomorphy, say D, then the Shilov boundary of D with respect to the algebra A(D), call it ∂sD, coincides with the corresponding one for D, called ∂sD. [ABSTRACT FROM AUTHOR]