Interpolation problem for l1 and a uniform algebra

Let K and X be compact plane sets such that K X. Let P (K) be the uniform closure of polynomials on K. Let R(K) be the closure of rational functions K with poles o K. Dene P (X;K) and R(X;K) to be the uniform algebras of functions in C(X) whose restriction to K belongs to P (K) and R(K), respectively. Let CZ(X;K) be the Banach algebra of functions f in C(X) such that fjK = 0. In this paper, we show that every nonzero complex homomorphism ’ on CZ(X;K) is an evaluation homomorphism ez for some z in XnK. By considering this fact, we characterize the maximal ideal space of the uniform algebra P (X;K). Moreover, we show that the uniform algebra R(X;K) is natural.

Consider a uniform algebra 𝔄 on a compact space X having maximal ideal space M. Fix f ∈ 𝔄. Consider an open set W contained in C\f(X). For each λ in W the fiber over λ shall mean {p ∈M|f(p) = λ{ In Section 11 we saw that if f -1 (W) is non-empty and if for sufficiently many λ in W the fiber over λ is a finite set, then f -1 (W) possesses analytic structure.KeywordsCompact SpacePlane DomainUniform AlgebraAnalytic DiskMaximal Ideal SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K , K ∞ ) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L p (T),L∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established. In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J is not closed. Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.

We consider and study Blaschke inductive limit algebrasA(b), defined as inductive limits of disc algebras A(D) linked by a sequence of finite Blaschke products. It is well known that big G‐disc algebras AG over compact abelian groups G with ordered duals can be expressed as Blaschke inductive limit algebras. Any Blaschke inductive limit algebra A(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 of ℚ. It is shown that a big G‐disc algebra AG over a group G with ordered dual is a Blaschke inductive limit algebra if and only if . 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 big G‐disc algebra. These differences are utilized to construct examples of Blaschke inductive limit algebras that are not big G‐disc algebras. A necessary and sufficient condition for a Blaschke inductive limit algebra to be isometrically isomorphic to a big G‐disc algebra is found. We consider also inductive limits H∞(I) of algebras H∞, linked by a sequence of inner functions, and prove a version of the corona theorem with estimates for it. The algebra H∞(I) generalizes the algebra of bounded hyper‐analytic functions on an open big G‐disc, introduced previously by Tonev.

SynopsisA classical theorem of Hartogs gives conditions on the singularity set of an analytic function of several complex variables in order for such a set to be an analytic variety. A result of E. Bishop from 1963 gives an analogous condition of the maximal ideal space of a uniform algebra in order for this space to have analytic structure. We show that algebras of functions satisfying a maximum principle serve to explain both of these results.

In this chapter we introduce shift-invariant algebras, the main objects of this book. These are uniform algebras on a compact connected group G, consisting of continuous functions on G, whose spectrum is contained in a semigroup S of the dual group Ĝ. If Ĝ is a subgroup of ℝ, and S ⊂ ℝ+, then the maximal ideal space of the corresponding shift-invariant algebra is the G-disc, or, big disc over G. In this chapter we describe two important models of shift-invariant algebras, namely, by the means of almost periodic functions on ℝ, and by the means of H∞-functions on the unit circle. The set of automorphisms, and the peak groups of shift-invariant algebras are also characterized. Extensions on G-discs and groups of several classical theorems of Complex Analysis, such as Radó’s theorem for null-sets and the Riemann theorem for removable singularities of analytic functions, are stated and proved. It is shown that these extensions hold for some semigroups, while in general they fail. In principle we state all results for general shift-invariant algebras AS, though they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras. Asymptotically almost periodic functions combine the properties of classical almost periodic functions on ℝ, and of continuous functions on ℝ that vanish at infinity.

We study the cluster value problem for certain Banach algebras ofholomorphic functions defined on the unit ball of a complex Banachspace X. The main results are for spaces of the form X= C(K). 1 Preliminaries. A cluster value problem for a complex Banach space Xis a weak version ofthe corona problem for the open unit ball Bof X, which is a long-standingopen problem in complex analysis when Xhas dimension at least 2. Insteadof studying when B is dense in the spectrum of a uniform algebra H ofbounded analytic functions on Bin the weak topology induced by H(coronaproblem), the cluster value problem investigates the following situation:Let B¯ ∗∗ be the closed unit ball of the bidual X ∗∗ , and let M H be the spec-trum (i.e. maximal ideal space) of a uniform algebra Hof norm continuousfunctions on Bwith H⊃ X ∗ .Then π: M H → B¯ ∗∗ ,given by π(τ) = τ| X ∗ for τ∈ M H ,is surjective (as a consequence of the results in Chapter 2 and5 of [10]). For each x ∗∗ ∈ B¯ ∗∗ ,M x ∗∗ (B) = π −1 (x

We give some conditions under which on certain subsets of the maximal ideal space of a uniform algebra there exist structures of generalized-analytic coverings in the sense of R. Gunning and H. Rossi.

This article examines hereditary f-rings, after first characterizingthe hereditary von Neumann regular commutative rings as those for which the space of maximal ideals is hereditarily paracompact. It is shown that if C ( X , Z ) , the ring of integer-valued continuous functions on a zero-dimensional space X, is hereditary, then X is finite. This is shown two ways; once as a consequence of the following: if A is any singular archimedean f-ring, then A/2A is a boolean ring, and gd(A) >= gd(A/2A)+ 1 , where gd(A) stands for the global dimension of A. As a consequence of this it is also shown that if A is a singular archimedean f-ring and gd(A) <= 2, then Min(A), the space of minimal prime ideals is hereditarily paracompact. The paper concludes with a calculation of the global dimension of a mihereditary singular archimedean f-ring A, in which the cellularityof Min(A) is “much less” than |A| if finite, it is k+2, where Nk = |A|.

Isomorphic sheaf representations of normal lattices

Let H ∞ ( Δ ) {H^\infty }(\Delta ) be the algebra of all bounded analytic functions on the open unit disc Δ \Delta , and let M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) be the maximal ideal space of H ∞ ( Δ ) {H^\infty }(\Delta ) . Using a flow, we represent a reasonable portion of a fiber in M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) . This indicates a relation between the corona theorem and the individual ergodic theorem. As an application, we answer a question of Forelli [3] by showing that there exists a minimal flow on which the induced uniform algebra is not a Dirichlet algebra. The proof rests on the fact that the closure of a nonhomeomorphic part in M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) may contain a homeomorphic copy of M ( H ∞ ( Δ ) ) \mathfrak {M}({H^\infty }(\Delta )) . Taking suitable factors, we may construct a lot of minimal flows which are not strictly ergodic.

Let A be a function algebra, X its Silov boundary, M its maximal ideal space. One of the outstanding problems in function theory is to find a structure of complex analytic space on M such that the functions of A are analytic for this analytic structure. In some cases (algebras with few representative measures, or polynomial hulls of regular curves in ~n) , such structures can be described. (See also the recent work of R. Basener [I] and of Gamelin, Sibony and Wermer, this conference.) The first counterexample to the existence of analytic structure was found by Stolzenberg [14] who exhibited a compact K c ~2 such that the polynomial hull h(K) of K is distinct from K , and there exis~ no analytic structure on h(K) K . More explicit examples were found by Wermer and Basener for rationally comvex hulls (see [2] and also [6] for related class of examples). Moreover, if an analytic structure exists, it would have to live on the Gleason parts of the maximal ideal space. As a consequence,if f is a non constant function in a Gleason part P , then f(P) would have to be interior point to f(M), by the fact that analytic functions are open mappings.~nd that assertion is completely false in general. The interest for the research of analytic structure is that it provides regularity theorems on the functions of A , and also theorems about representative measures, bounded point derivation, approximation theory, etc. But such theorems, in particular differentiability theorems about the functions of A , can be proved if one can find in the maximal ideal space, some kind of dynamical system along the trajectories of which one can study the properties of functions. The aim of this paper is to show how such generalized dynamical systems (brownian paths or even sheets) lives in spectrum of certain function algebras even if there is no analytic structure in these spectrumnand to obtain from that regularity theorems, theorems about Jensen measures, approximation and derivations, just as if there was analytic structure.

It is shown that the space of all regular maximal ideals in the Banach algebra H ∞ ( D ) H^{\infty }(\mathbb {D} ) with respect to the Hadamard product is isomorphic to N 0 . \mathbb {N} _{0}. The multiplicative functionals are exactly the evaluations at the n n -th Taylor coefficient. It is a consequence that for a given function f ( z ) = ∑ n = 0 ∞ a n z n f(z) =\sum _{n=0}^{\infty }a_{n} z^{n} in H ∞ ( D ) H^{\infty }(\mathbb {D} ) and for a function F ( z ) F(z) holomorphic in a neighborhood U U of 0 0 with F ( 0 ) = 0 F(0) =0 and a n ∈ U a_{n} \in U for all n ∈ N 0 n \in \mathbb {N}_{0} the function g ( z ) = ∑ n = 0 ∞ F ( a n ) z n g(z) =\sum _{n=0}^{\infty }F(a_{n} ) z^{n} is in H ∞ ( D ) . H^{\infty }(\mathbb {D} ) .

Let G denote a locally compact abelian topological group (an l.c.a.group) with dual group C\ We will denote by A7(G) the Banach algebra of bounded regular Borel measures on G under convolution multiplication and by L(G) the algebra of bounded measures absolutely continuous with respect to Haar measure on G (for discussions of these Banach algebras cf.[1], [2], and [5]).In this paper we shall be concerned with closed subalgebras 501 of M(G) with the following two properties:(1) if p e 50c and v is absolutely continuous with respect to p, then vs;(2) the maximal ideal space of 501 is G~, where the Gelfand transform p~ of P e 501 is given by p~(x)=J x dp for ^Ga; i.e., the Gelfand transform coincides with the Fourier-Stieltjes transform on 50.Any closed subalgebra of M(G) satisfying (1) will be called an L-subalgebra of M(G).It is well known thatL(G) satisfies ( 1) and (2) (cf.[5, Chapter 1]).In Theorem 1 we show that any L-subalgebra 50c of M(G), with L(G)<=50cc(L(G))1,2) also satisfies (2), where (L(G))112 is the intersection of all maximal ideals of M(G) containing L(G).We conjecture that the converse is also true; i.e., if 501 satisfies (1) and (2) then L(G)c 50c c(L(G))1/2.In Theorem 2 we prove that this is true provided G contains no copy of 7?" for > 1.The problem arises in the following way : in [6] we define the concept of abstract convolution measure algebra and prove that any such algebra 501, provided it is commutative and semisimple, may be represented as an L-subalgebra of M(S), where S is a compact topological semigroup called the structure semigroup of 5D.M(G) and L(G) are convolution measure algebras as is any L-subalgebra of the measure algebra on a semigroup.The map p-> ps which embeds 50c in M(S) is an isometry as well as an algebraic isomorphism and it preserves the order theoretic properties of 50 as a space of measures.The maximal ideal space of 50c may be identified as S~, the set of all semicharacters of S, where the Gelfand transform of p e 50c is given by p~(f) = $fdp.sfor fe S~ (a semicharacter of S is a continuous homomorphism of S into the unit disc which is not identically zero).Under pointwise multiplication S~ is a semigroup provided 50c has an approximate identity.

Point derivations and analytic structure in the spectrum of a Banach algebra