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Let E be a field provided with an absolute value for which it is complete and let A be a commutative E-algebra with unity provided with a semimultiplicative (or power multiplicative) E-algebra semi-norm ‖.‖. Let Mult(A, ‖. ‖) be the set of multiplicative E-algebra semi-norms continuous with respect to ‖. ‖. We show the existence of a Shilov boundary for ‖. ‖, i.e. a closed subset F of Mult(A, ‖. ‖), minimal for inclusion, such that for every x∈A, there exists ϕ∈F such that ϕ(x)=‖x‖. In particular, if the field E is ultrametric, it applies to the spectral semi-norm of an ultrametric E-algebra.

Let ${\cal D}$ be a bounded symmetric domain of tube-type, S its Shilov boundary, and G the neutral component of its group of biholomorphic transforms. We classify the orbits of G in the set $S\times S\times S.$

Let L be a field complete for a non-trivial ultrametric absolute value and let (A,‖⋅‖) be a commutative normed L-algebra with unity whose spectral semi-norm is ‖⋅‖si. Let Mult(A,‖⋅‖) be the set of continuous multiplicative semi-norms of A, let S be the Shilov boundary for (A,‖⋅‖si) and let ψ∈Mult(A,‖⋅‖si). Then ψ belongs to S if and only if for every neighborhood U of ψ in Mult(A,‖⋅‖), there exists θ∈U and g∈A satisfying ‖g‖si=θ(g) and γ(g)

In this paper a quantum analog of the $*$-algebra of regular functions on the Shilov boundary $S(\mathbb D)$ of bounded symmetric domain $\mathbb D$ is constructed. The algebras of regular functions on $S(\mathbb D)$ are described in terms of generators and relations for two particular series of bounded symmetric domains. Also, the degenerate principal series of quantum Harich-Chandra modules related to $S(\mathbb D)=U_n$ is investigated., Comment: 17 pages

In this paper we study a quantum analogue of a degenerate principal series of $U_q \mathfrak{su}_{n,n}$-modules ($0, Comment: 33 pages, 4 figures

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 $(\sigma_1, \sigma_2, \sigma_3) \in S \times S \times S$, a characteristic $G$-invariant $\iota (\sigma_1,\sigma_2,\sigma_3)$, 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.

Let G/H be a semisimple symmetric space. The main tool to embed a principal series representation of G into L^2(G/H) are the H-invariant distribution vectors. If G/H is a non-compactly causal symmetric space, then G/H can be realized as a boundary component of the complex crown $\Xi$. In this article we construct a minimal G-invariant subdomain $\Xi_H$ of $\Xi$ with G/H as Shilov boundary. Let $\pi$ be a spherical principal series representation of G. We show that the space of H-invariant distribution vectors of $\pi$, which admit a holomorphic extension to $\Xi_H$, is one dimensional. Furthermore we give a spectral definition of a Hardy space corresponding to those distribution vectors. In particular we achieve a geometric realization of a multiplicity free subspace of L^2(G/H)_mc in a space of holomorphic functions., Comment: Due to a very careful and professional referee's work we could strongly improve on both exposition and detail. To appear in Invent. math., 35 pages, 1 figure

Let D be a Hermitian symmetric space of the non-compact type, ! its Kaehler form. Fora geodesic triangle in D, we compute explicitly the integral R � !, generalizing previous results (see (D-T)). As a consequence, if X is a manifold which admits D as universal cover, we calculate the Gromov norm of (!) 2 H2(X, R). The formula for R � ! is extended to ideal triangles. Precise estimates are given and triangles for which the bound is achieved are studied. For tube-type domains we show the connection of these integrals with the Maslov index we introduced in a previous paper (see (C-O)). 0. Introduction. Let M be a Hermitian symmetric space of the non-compact type, which for simplicity, we assume to be irreducible. Let G be the neutral com- ponent of the group of biholomorphic automorphisms of M. The space M admits a natural (G-invariant) Kaehler form !. This real differential form of degree 2 is closed and hence can be integrated along any 2-cycle, in particular geodesic triangles (to mean triangles the sides of which are geodesic segments). When M is of type I,II or III (in E. Cartan's classification), the integral R � ! (the symplectic area of the geodesic triangle �) was computed in (D-T). By using their techniques, we give the result in the general case. It turns out that these quantities have an upper bound, and with the appropriate normalization, the bound depends only on the rank r of M. We extend these computations to ideal triangles, and we prove (new) sharp estimates for the areas. In particular, we determine precisely the triangles for which the upper bound is achieved. This turns out be of great geometric significance, as the summits of such an extremal triangle are contained in the image of a tight holomorphic totally geodesic imbedding of the complex unit disc into M (Theorem 4.7). Generally speaking, our study of the integrals R � ! requires the fine structure of Hermitian symmetric spaces : special role played by the tube-type case, behaviour of geodesics at infinity and structure of G-orbits in the boundary, use of partial Cayley transforms. This study is also related to a previous work (see (C-O)) where we extended the notion of Maslov index to the Shilov boundary S of a Hermitian symmetric space of tube-type. The Maslov index is (up to a factor �) nothing but the symplectic area of ideal triangles with summits in S, and in some sense the present work can be understood as a continuation of (C-O). The computation of the integrals was used in (D-T) to calculate the Gromov norm of the Kaehler class of a compact Hermitian locally symmetric manifold X = \M, where M is of type I and a discrete, torsion-free, co-compact subgroup of the group G. They observed that it has a nice topological corollary. Let S be a Riemann surface of genus g > 1 and f : S → X a continuous map. Then

We generalize some facts about function algebras to operator algebras, using the `noncommutative Shilov boundary' or $C^*$-envelope first considered by Arveson. In the first part we study and characterize complete isometries between operator algebras. In the second part we introduce and study a notion of logmodularity for operator algebras. We also give a result on conditional expectations. Many miscellaneous applications are given., 24 pages

The Future Tube $T^+_n$ of n-dimensional Minkowski spacetime may be identified with the reduced phase space or "space of motions" of a particle moving in (n+1)-dimensional Anti-de-Sitter spacetime. Both are isomorphic to a bounded homogeneous domain in ${\bf C}^{n}$ whose Shilov boundary may be identified with $n$-dimensional conformally compactified Minkowski spacetime., Comment: 13 pp Latex: written version of a talk given at Strings99 to appear in the proceedings

We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator exists. Using such 'trace theorems' it is possible to construct discrete increments to spectra of some unitary representation and to catch singular unitary representations in the spectra. We also discuss spectral problems related to Berezin type kernels on riemann symmetric spaces.

where ||/||oo is the L°° = L°°(dD,dθ) norm of the non-tangential limit /. Thus we can view H°° as a uniformly closed subalgebra of L°°. An inner function is a function / G H°° with \I(e)\ = 1, a.e. dθ. For example S(z) = e ^ is an inner function, though it has radial limit 0 at ζ = 1. Inner functions are of central importance in the study of analytic functions on D. For example, Beurling used inner functions (and coined the name) to characterize the translation invariant subspaces of the Hardy space H. Rudin and Carleson used inner functions to characterize the ideals of the disk algebra, and Newman used them to characterize the Shilov boundary of if. If {zn} C D with Σ n ( l — \zn\) < oo then a Blaschke product with zero set {zn} is an inner function of the form

Max Born’s reciprocity principle is revisited and complex four dimensional Kahler manifold D4 ≈ SU (2, 2)/S (U (2) × U(2)) is proposed as a replacement for spacetime on the micro scale. It is suggested that the geodesic distance in D 4 plays a role of a quark binding super-Hamiltonian.

Let D be a classical domain in ℂ n , and S its distinguished boundary (Shilov boundary). We define the Hardy classes H P (D) as follows: a function ƒ ∈ A(D) belongs to the class HP(D), 0 < p < ∞, if $$ \mathop {\sup }\limits_{0 \leqslant {\text{r < 1}}} \int\limits_s {|f(r\varsigma )|} ^p d\mu < \infty , $$ (24.1)

Let A be a uniform dual algebra with separable predual, denoted be Q A , and such that the product is separately weak -continuous. We show that there exists a positive measure μ on the Shilov boundary Γ of A such that A is isometrically embedded in L∞(Γ,μ) = L∞(μ) and a contractive weak*-continuous homomorphism Φ from H∞(μ), the weak*-closure of A in L∞(μ), into A(= Q* A ) such that Φ(f) = f, f e A. Consequently, for any isometric representation of a uniform algebra A on a separable complex Hilbert space of infinite dimension H has a weak*-continuous homomorphic contractive extension to H∞ (μ) for some positive measure μ on the Shilov boundary of A.

We give a short proof of Basener Slodkowski theorem for multituple Shilov boundaries of tensor products of two uniform algebraas.

Bounded symmetric domains (Cartan domains and exceptional domains) are higher-dimensional generalizations of the open unit disc. In this note we give a structure theory for the C*-algebra T generated by all Toeplitz operators Tf(h) := P{fh) with continuous symbol function ƒ G C(S) on the Shilov boundary 5 of a bounded symmetric domain D of arbitrary rank r. Here h belongs to the Hardy space H(S), and P : L(S) —• H(S) is the Szego projection. For domains of rank 1 and tube domains of rank 2, the structure of T has been determined in [1, 2]. In these cases Toeplitz operators are closely related to pseudodifferential operators. For the open unit disc, T is the C*-algebra generated by the unilateral shift. The structure theory for the general case [12] is based on the fact that D can be realized as the open unit ball of a unique Jordan triple system Z [7, Theorem 4.1]. Denoting the Jordan triple product by {uv*w}, a tripotent e G Z satisfies {ee*e} = e. Tripotents generalize the partial isometries of matrix algebras and determine the boundary structure of D C Z (cf. [7, Theorem 6.3]). Our principal result ([12]; cf. also [3, 4, 8]) is the following

Let be a bounded domain and let be a modulus of continuity. This article is devoted to the following problem: which closed sets with possess the property that, for an arbitrary function belonging to the algebra of all functions analytic in and continuous in , the relation for all 0$ SRC=http://ej.iop.org/images/0025-5734/50/2/A12/tex_sm_2841_img10.gif/> implies for all 0$ SRC=http://ej.iop.org/images/0025-5734/50/2/A12/tex_sm_2841_img10.gif/>, where the constant depends only on and .The main result is a theorem which asserts that if is a regular Weil domain then can be taken to be the Shilov boundary.Bibliography: 20 titles.

The concepts of Choquet boundary and Shilov boundary are well-established in the context of a complex function algebra (see [2] for example). There have been a few attempts to develop the concept of a Shilov boundary for real algebras, [4], [6] and [7]. But there seems to be none to develop the concept of Choquet boundary for real algebras.The aim of this paper is to develop the theory of Choquet boundary of a real function algebra (see Definition (1.8)) along the lines of the corresponding theory for a complex function algebra.In the first section we define a real-part representing measure for a continuous linear functional ϕ on a real function algebra A with the property ║ϕ║ = 1 = ϕ(1). The elements of A are functions on a compact, Hausdorff space X. The Choquet boundary is then defined as the set of those points x ∊ X such that the real part of the evaluation functional, Re(ex), has a unique real part representing measure.

A generalization of the concept of the Shilov boundary of a uniform algebra is introduced. This makes it possible to formulate and prove several-dimensional analogues of certain well-known results which guarantee the existence of one-dimensional analytic structure when a function in the algebra is finite-to-one over a suitable part of its spectrum. A major problem in the study of uniform algebras is to find interesting sufficient conditions for the existence of analytic structure in their maximal ideal spaces; see, e.g., [6, Chapter 3 and ?30]. With one notable exception (a result due to Gleason 13], or see [6, Theorem 15.2]), most of these results are one dimensional. In this paper we indicate how one kind of condition ([7, Theorems 10.7 and 11.2], which were derived from Bishop's paper [2]) can be generalized to yield several-dimensional analytic structure. First we must define a generalization of the Shilov boundary of a uniform algebra. Notation. A will always denote a uniform algebra defined on the compact Hausdorff space X with maximal ideal space M. OOA is the usual Shilov boundary of A. Let An = (fl, *0 fn)111 ... In fn e A} so that each F = ... , * /E fn) E An maps M to cn, and F(M) is the joint spectrum of / fn. If K is a compact subset of M, let AK = I/E C(K) I f is a uniform limit on K of functions from A}. If F E An, let V(F)= x E M I F(x) = 0 be the A-variety corresponding to F and note that V(F) is A-convex, i.e., the maximal ideal space of AV(F) is V(F). Received by the editors August 15, 1973. AMS (MOS) subject classifications (1970). Primary 46J10, 46J15.

In this paper Jordan algebraic methods are applied to study Toeplitz operators on the Hardy space H 2 ( S ) {H^2}(S) associated with the Shilov boundary S S of a bounded symmetric domain D D in C n {{\mathbf {C}}^n} of arbitrary rank. The Jordan triple system Z ≈ C n Z \approx {{\mathbf {C}}^n} associated with D D is used to determine the relationship between Toeplitz operators and differential operators. Further, it is shown that each Jordan triple idempotent e ∈ Z e \in Z induces an irreducible representation (" e e -symbol") of the C ∗ {C^{\ast } } -algebra T \mathcal {T} generated by all Toeplitz operators T f {T_f} with continuous symbol function f f .

Let H∞ be the Banach algebra of all bounded analytic functions in the unit disc. We present a complete description of the closed primary (respectively prime) ideals contained in a maximal ideal of the Shilov boundary of H∞. The paper is also concerned with chains of prime ideals in H∞.

For any uniform algebraA and any pointq of the maximal ideal space ofA there exists a Jensen measureλ forq carried on the Shilov boundary forA such thatλ admits the generalized Brownian maximal function to each nonnegativeA-subharmonic function inC R(X). The maximal function and its original function satisfy Doob’s inequality, Burkholder-Gundy-Silverstein inequalities and Fefferman-Stein inequality.

O. Introduction Let A be a uniform algebra on a compact subset K of a complex space X such that A contains a subset F of function holomorphic in an open neighborhood Y of K such that F separates points in Y. It is shown that the Shilov boundary of A is not almost thin in K. As an example, let G ~ 0 be an open and bounded subset of ~n and consider the uniform algebra of functions which are uniform limits on of polynomials. Then there exists a unique minimal closed subset S(A) of G such that for every function f in A, the maximum of Ifl is taken on S(A). The set S(A) is contained in the boundary of G and called the Shilov boundary. Let U be the unit disc and let T be the unit circle. In the case G = U n is the polydisc, the Shilov boundary is the torus S(A) = T n. Here S(A) is small compared with G or its boundary dG = G-G. The topological dimension of S(A) is half that of G in this example. On the other hand, if G is a ball or any open, bounded subset of C" having a smooth, strongly pseudo-convex boundary OG, then $(A)= OG. These two examples make it seem that the size of S(A) varies from extremely small to as large as possible. The purpose of this paper is to demonstrate that, from a holomorphic point of view, S(A) and G have exactly the same size. Essentially the same fact is also true when G is taken to be any compact subset K of a complex space X and if A is any uniform algebra on K which contains sufficiently many holomorphic functions. These results simplify some proofs in [6].

It is shown that if T λ {T_\lambda } is an analytic operator valued function (or if f f , g g belong to a uniform algebra A A ) then n n th diameters and logarithmic capacity of σ ( T λ ) \sigma ({T_\lambda }) (or of the set g ( f − 1 ( λ ) ) g({f^{ - 1}}(\lambda )) ) are subharmonic functions of λ \lambda (on a suitable domain).

The transformation group of the universal covering of the real projective line, obtained by lifting ordinary projective transformations, is given explicitly in terms of canonical coordinates. A similar formulation is given of the action of the universal covering of SU(2, 2) upon the universal covering of the Shilov boundary of its associated bounded Hermitian symmetric domain, structured as R1 × SU(2). The former group G , isomorphic to S U(1, 1) , has a unique and continuous bi-invariant global partial ordering ⩽ (similar to that expressing space-time causality relations) corresponding to its bivariant Lorentzian metric; the partial ordering is the same as that induced by the ordering of the real line which the transformation group preserves. As an application, the compactness of the intervals [g 1 , g 2 ] = {g ϵ G : g 1 ⩽ g ⩽ g 2 } for g 1 , g 2 ϵ G , necessary for global hyperbolicity of the metric, is studied. It is shown that [g1, g2] is compact if and only if g ⩽ ζg1 for all g in a neighborhood of g2, where ζ is the generator of the center of G satisfying ζ ⩾ e. In particular, the interior of [e, ζ] is a maximal open global hyperbolic submanifold; S U(1, 1) is not globally hyperbolic.

Let n ⩾ 2 n \geqslant 2 and D ⊂⊂ C n D \subset \subset {{\mathbf {C}}^n} be a strictly pseudoconvex domain with C k {C^k} boundary for k > 2 k > 2 . There is a closed nowhere dense subset of the maximal ideal space of L ∞ ( b D ) {L^\infty }({\text {b}}D) which defines a closed boundary for H ∞ ( D ) {H^\infty }(D) .

We have proved earlier that a certain class of functions, which is useful in the study of analytic structure associated with a uniform algebra, is plurisubharmonic, if the algebra satisfies Basener’s conjecture. In this paper we prove the plurisubharmonicity of these functions without the hypothesis of Basener’s conjecture.

We show that a finitely generated projective extension B of a commuta- tive complex unitary Banach algebra A induces an open mapping it between the carrier spaces. We next prove that if tr is a local homeomorphism then B contains an inertial subalgebra. Finally we present a necessary and sufficient condition for B to be uniform if A is. 1. Introduction. Throughout this paper Banach algebra means a commutative Banach algebra over the complex field C with an identity element 1. If A is a Banach algebra, then its carrier space, endowed with the Gelfand topology, is denoted by M(A) and its Shilov boundary by o(A). For each a G Av/e denote by â the Gelfand transform of a. The Jacobson radical of A is written R(A). From now on B will denote a finitely generated projective extension of a fixed Banach algebra A. We suppose B to be endowed with Magid's norm under which B is also a Banach algebra (10, Theorem 4, p. 138). We call it the projection of M(B) onto M(A) induced by the inclusion of A into B. The above situation is illustrated by the following example: take B = A(x)/(a(x)) where a(x) is a monic polynomial in A(x) (the so-called Arens-Hoffman extensions of A). For such extensions, the structure of the projection it as well as the study of the properties that B can inherit from A have been the main subject in a series of papers ((4), (7), (8)). The purpose of this paper is to establish some results along these lines for the more general situation. To state the main results it is convenient to introduce a definition. If \p G M(B), = w(i|/), and m^ = Ker

Let H ∞ ( Δ ) denote the Banach algebra of bounded analytic functions on the open unit disc, let M denote its maximal ideal space, and let ∂ denote its Shilov boundary. D. J. Newman has shown that a homomorphism ϑ in M will be in ∂ if and only if ϑ is unimodular on all Blaschke products. We answer a question of K. Hoffman by showing that ϑ will be in ∂ if and only if ϑ is unimodular on every Blaschke product whose zero set is an interpolating sequence. Our method is based on a construction due to L. Carleson, originally developed for the proof of the Corona theorem.

Let Un be the unit polydisc in Cn and let Tn be its distinguished boundary. It is shown that a function f ∈ H∞(Un) is inner if and only if ∣f(Φ)∣ = 1 for all Φ in the maximal ideal space of L∞(Tn). This generalizes a result of Csordas.

A differential geometric method is introduced to study the modulus of boundary values of holomorphic functions on smoothly bounded pseudoconvex domains D in C n , n ⩾ 2 {{\mathbf {C}}^n},n \geqslant 2 . It is shown that functions in A ( D ) A(D) are determined up to a constant factor by their modulus on an open subset of the Shilov boundary. For the case of H ∞ ( D ) {H^\infty }(D) , it is shown that inner functions which satisfy a certain local condition are constant.

In this note we give an index theory for Toeplitz operators on the Hardy space of the Shilov boundary of an arbitrary bounded symmetric domain. Our results generalize earlier work of Gohberg-Krein and Venugopalkrishna [12] for domains of rank 1 and of Berger-Coburn-Koranyi [1] for domains of rank 2. Bounded symmetric domains (Cartan domains, classical or exceptional) are the natural higher-dimensional analogues of the open unit disk. Each such domain is homogeneous under a semisimple Lie group of biholomorphic transformations and has an (essentially unique) realization as a convex circular domain which is conveniently described in Jordan algebraic terms: The underlying vector space Z « C n carries a Jordan triple product, denoted by (uyV^w) •-• {uv*w}, and the bounded symmetric domain D is the open unit ball of Z with respect to the "spectral norm" [7]. The basic example is the space Z = C of rectangular matrices, with Jordan triple product {uv*w} := (uv*w + wv*u)/2) giving rise to the "hyperbolic matrix ball" D = {z G Z: spectrum (z*z) < 1}. The symmetric domains of rank 1 are the (smooth) Hilbert balls in C n . For domains D of higher rank r, the boundary dD forms a nonsmooth "stratified space". In Jordan algebraic terms, the structure of dD can be described as follows: An element e € Z satisfying {ee*e} = e is called a tripotent. For matrices, the tripotents are just the partial isometries. Every tripotent e induces a splitting of Z into the "Peirce spaces" Z\(e) := {z G Z: {ee*z} = Xz} for A = 0 , ^ , 1 . The set

If G is a locally compact abelian topological group and M(G) denotes the algebra of bounded regular Borel measures on G under convolution multiplication, then M(G) is a convolution measure algebra in the sense of [1]. In [1 ] we showed that the maximal ideal space of any such algebra 9Y can be represented as the semigroup 3 of all semicharacters on some compact abelian topological semigroup S. S is called the structure semigroup of the algebra P. If H= { hGC: I h(s) I =0 or 1 for sES}, then the Gelfand transform ,u of each element ,u of 9 attains its maximum modulus on H (cf. [1, Theorem 3.3]). Hence the closure H7 of H in the Gelfand topology contains the Shilov boundary of P. In [1] we show that when 9 = M(G) for some nondiscrete locally compact topological group G, then H is a proper subset of S. In this paper we show that there is at least one group G for which H1 is a proper subset of 3. Hence, for this group G, the Shilov boundary of M(G) is a proper subset of the maximal ideal space of M(G). For each positive integer n let Tn be the multiplicative two point group {i, -1 } and set G = 1J, i T,n. G is a compact abelian topological group. For each n we let Xn be the function which projects G onto its nth coordinate. Each Xn is a character in the dual group 0 of G and each kEd is either the identity or a finite product of distinct xn'S. S will denote the structure semigroup of M(G) and 3 the semigroup of all continuous semicharacters on S. it-+Is is the natural imbedding of M(G) into M(S) (cf. [1, Theorem 2.3]). The Gelfand transform A of ,AG M(G) is described by the equation fi(f) =ff d,us for fG3. We are interested in a particular class of measures ,u in M(G). Let r,}Z.? 1 be a sequence of numbers in [0, 1) and for each n let j.& be the measure on Tn defined by y,(1) = 2-1(1 + rn) and u,(-1) = 2-1(1 -rn). Each , is a strictly positive measure of norm one on Tn. Let ,u be the measure in M(G) which is the infinite product of the j.. That is, if U is any neighborhood in G of the form U= { gGG: xn(g)

1. Introduction. The first result presented here is that an element c of a commutative normed algebra A has an inverse in some norm-preserving extension B ("superalgebra") of A if and only if can—>0 implies an—*0 lor every sequence in A, i.e., c is not a topological zero-divisor. The method may be described as consisting in constructing the algebra A (x) of polynomials with coefficients in A, and then dividing out the ideal generated by the (first degree!) polynomial 1—cx. The resulting algebra B is always of infinite degree over A, except when c already has an inverse in A. In the classical applications of this method (usually, to fields) the coefficient of the highest power may be taken as 1, and that sort of polynomial extension goes over in manageable fashion to normed algebras [see bibliography: Hoffman and Arens]. However, the present construction is nontrivial only because c has no inverse (and effective only because it is not a topological zero-divisor). Most of our space is taken up with quantitative refinements, with systems of elements, and finally with a result on semi-simplicity, interesting perhaps mainly in view of the complexity of our proof. The question (unsolved at present) is whether, when A is semi-simple, it can be arranged that B is semi-simple (if it can be arranged, then it can be done by our method (see 3.71)). We are able to answer affirmatively only in the case where c generates A over the scalars (or some other similarly-behaving subalgebra). We remark that all problems treated here are trivial when A has the sup norm, i.e., ||a||=||a||M [see Loomis]: the algebra of all bounded (complex valued) functions on the Shilov boundary provides an extension solving all reasonable problems. C. E. Rickart has exhibited a technique for simultaneously advancing to regularity a certain class of elements. This class is more or less the obvious one for his technique—reminiscent of the making a field out of an integral domain. Our individual- (or finitely many) element-method takes care of some elements not in Rickart's class. Some unsolved problems already familiar to people in this field are men

Let A ⊂ C ( X ) and B ⊂ C ( Y ) be uniform algebras with Choquet boundaries δA and δB. A map T : A → B is called norm-linear if ‖ λ T f + μ T g ‖ = ‖ λ f + μ g ‖ ; norm-additive, if ‖ T f + T g ‖ = ‖ f + g ‖ , and norm-additive in modulus, if ‖ | T f | + | T g | ‖ = ‖ | f | + | g | ‖ for each λ , μ ∈ C and all algebra elements f and g. We show that for any norm-linear surjection T : A → B there exists a homeomorphism ψ : δ A → δ B such that | ( T f ) ( y ) | = | f ( ψ ( y ) ) | for every f ∈ A and y ∈ δ B . 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 T ( i ) = i , or which preserves the peripheral spectra of C -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 C -peaking functions, then it is automatically multiplicative and, in fact, an algebra isomorphism.

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.

It is shown that the inner functions on the polydisc, unlike the classical case of the unit disc, fail to separate the points of the maximal ideal space of H ∞ ( U n ) {H^\infty }({U^n}) . From this we deduce that the inner functions generate a proper closed subalgebra of H ∞ ( U n ) {H^\infty }({U^n}) .

The existence of a carrier in the sense of potential theory is proved for rather general cones of functions. The existence of the Shilov boundary is an immediate consequence of this result. 1. The main results. Let X be a compact space and E C X such that E = X. Let F be a set of lower bounded numerical functions on E. We assume that there exists a convex cone 5 C g which possesses the following properties: (1) u + s C ' for any u C I,s C S. ( 2) For any s E 5, there exists a continuous extension sto X (which is obviously unique). ( 3) There exists a function s C 5 such that sis strictly positive on X. (54) For any x, y X, x #y, there exists s, t E 5 such that s(x)-t(y) # s(y) * t(x). A closed set K C X is called a carrier set of a numerical function f on E iff, for any open V C X, K C V and any u E C, f u on vn E implies f? u throughout E. Obviously { f > 0) is a carrier set of f if F is a cone of nonnegative functions. For any numerical function f on E, we denote by carr( f) the smallest carrier set of f if it exists. carr( f ) is called the carrier of f (with respect to f ). 1.1. THEOREM. For every upper bounded numerical function f on E, carr( f ) exists.

H. G. Dales has proved in [1] that if A A is a Banach function algebra on a compact metrizable space X X , then S ¯ 0 ( A , X ) = Γ ( A , X ) {\bar S_0}(A,X) = \Gamma (A,X) , where S 0 ( A , X ) {S_0}(A,X) is the set of peak points of A A (w.r.t. X X ) and Γ ( A , X ) \Gamma (A,X) is the Shilov boundary of A A (w.r.t. X X ). Here, by considering the relation between peak sets and peak points of a Banach function algebra A A and its uniform closure A ¯ \bar A , we present an elementary and constructive proof of the density of peak points in the Shilov boundary.

A notion of size for subsets of the spectrum of a function algebra is described, relative to which each open subset of the Shilov boundary has the same size as the spectrum. Let A be a function algebra; denote its spectrum (space of nonzero, continuous, complex-valued hotnomorphisms, with the weak*-topology) by T,A, and its Shilov boundary by FA. In a topological sense, TA may be much smaller than ~^A; for instance it is easy to construct function algebras for which YA is zero-dimensional, while the dimension of ~LA is as large as we please (see Stout (8, p. 372)). In this note we describe another natural notion of size, an analytic notion derived from the theory of several complex variables, relative to which YA and £? have precisely the same size. This generalizes and simplifies some of the recent results of Huckleberry and Stoll (3). The proof uses techniques similar to the function algebra techniques in (3). We will make use of the /14iolomorphic functions introduced by Rickart (4), (5), (6), which may be defined by transfinite induction as follows: Let A0=A, and assume that the class Av has been defined for all ordinal numbers v

For the case of the polydisc, Range has shown that the Shilov boundary ∂ n {\partial _n} of H ∞ ( U n ) {H^\infty }({U^n}) is a proper subset of τ X n \tau {X_n} , the set of all restrictions of complex homomorphisms of L ∞ ( T n ) {L^\infty }({T^n}) to H ∞ ( U n ) {H^\infty }({U^n}) . In this paper, we show that τ X n \tau {X_n} is a proper subset of those complex homomorphisms of H ∞ ( U n ) {H^\infty }({U^n}) which are unimodular on the class of all inner functions.