Flows in fibers

Focusing on the role that automorphisms and equivalence relations play in the algebraic theory of minimal sets provides an original treatment of some key aspects of abstract topological dynamics. Such an approach is presented in this lucid and self-contained book, leading to simpler proofs of classical results, as well as providing motivation for further study. Minimal flows on compact Hausdorff spaces are studied as icers on the universal minimal flow M. The group of the icer representing a minimal flow is defined as a subgroup of the automorphism group G of M, and icers are constructed explicitly as relative products using subgroups of G. Many classical results are then obtained by examining the structure of the icers on M, including a proof of the Furstenberg structure theorem for distal extensions. This book is designed as both a guide for graduate students, and a source of interesting new ideas for researchers.

The totally minimal flow (X, T) is said to have finite almost periodic rank if there is a positive integer n such that whenever (x1, x2,…, xn+1) is an almost periodic point of the product flow (Xn+1, T×…×T) then, for some i≠j, xi, and xj are in the same orbit. The rank of (X, T) is the smallest such integer. If (Y, S) is a graphic flow, (Y, Sn) has rank |n| and it is shown that every finite rank flow has, modulo a proximal extension, a graphic power factor. Various classes of finite rank flows are defined, and characterized in terms of their Ellis groups. There are four disjoint types which have basic structural differences.

Let A be a uniform algebra on X and P a set of all probability measures on X. For each µ in P, H2 (µ) is the closure of A in L2 (µ) and Tt is a Toeplitz operator on H2 (µ) for a continuous function cf> on X. In this paper we study the invertibility and the spectrum of Tip = L EB Tt. We show that if Tip is invertible then the index of cf> is zero and if the converse is true for an arbitrary continuous function cf> then A is a Dirichlet algebra on X. Moreover we study the spectrum of Tip.

We prove that the multiplier algebra of the Drury-Arveson Hardy space H-n(2) on the unit ball in C-n has no corona in its maximal ideal space, thus generalizing the corona theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space B-p(sigma) has the "baby corona property" for all sigma >= 0 and 1 < p < infinity. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.

Uniform algebras that can be expressed as inductive limits of standard simpler algebras are of particular interest. For instance, some G-disc algebras are inductive limits of sequences of disc algebras, connected with finite Blaschke products, called also Blaschke inductive limit algebras. Here we show, among others, that only G-disc algebras, and the spaces \( H^\infty (\mathbb{D}_G ) \) with G ⊂ ℚ, can be expressed as limits of countable inductive sequences of algebras of type \( A(\mathbb{D}) \) and H∞ correspondingly. We study also inductive limits of sequences of spaces of type H∞ and prove corresponding corona theorems. Further, we establish relationships between Bourgain algebras of coordinate algebras, and the Bourgain algebra of their inductive limit, and also between H∞-spaces on \( \mathbb{D} \), as coordinate algebras in an inductive sequence, and their inductive limit. While we state all results for general shift-invariant algebras AS, they apply automatically to the particular cases of algebras APS of almost periodic functions, and of H S ∞ -algebras.

This paper deals with some operator representations \(\Phi\) of a weak*-Dirichlet algebra \(A\), which can be extended to the Hardy spaces \(H^{p}(m)\), associated to \(A\) and to a representing measure \(m\) of \(A\), for \(1\leq p\leq\infty\). A characterization for the existence of an extension \(\Phi_p\) of \(\Phi\) to \(L^p(m)\) is given in the terms of a semispectral measure \(F_\Phi\) of \(\Phi\). For the case when the closure in \(L^p(m)\) of the kernel in \(A\) of \(m\) is a simply invariant subspace, it is proved that the map \(\Phi_p|H^p(m)\) can be reduced to a functional calculus, which is induced by an operator of class \(C_\rho\) in the Nagy-Foias sense. A description of the Radon-Nikodym derivative of \(F_\Phi\) is obtained, and the log-integrability of this derivative is proved. An application to the scalar case, shows that the homomorphisms of \(A\) which are bounded in \(L^p(m)\) norm, form the range of an embedding of the open unit disc into a Gleason part of \(A\).

In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\mathcal {X}$ of analytic functions on the open unit disc $\mathbb {D}$ . We develop a general framework where a method based on a corona theorem can be used to show that if $f,g\in \mathcal {X}$ satisfy $|g(z)|\leq |f(z)|$ , for every $z\in \mathbb {D}$ , and if g is cyclic, then f is cyclic. We also give sufficient conditions for cyclicity in this context. This enable us to recapture some recent results obtained in de Branges–Rovnayk spaces, in Besov–Dirichlet spaces and in weighted Dirichlet type spaces.

The corona theorem in algebras of bounded analytic functions

We prove an alternate Toeplitz corona theorem for the algebras of pointwise kernel multipliers of Besov-Sobolev spaces on the unit ball in $\mathbb{C}^{n}$, and for the algebra of bounded analytic functions on certain strictly pseudoconvex domains and polydiscs in higher dimensions as well. This alternate Toeplitz corona theorem extends to more general Hilbert function spaces where it does not require the complete Pick property. Instead, the kernel functions $k_{x}\left(y\right)$ of certain Hilbert function spaces $\mathcal{H}$ are assumed to be invertible multipliers on $\mathcal{H}$, and then we continue a research thread begun by Agler and McCarthy in 1999, and continued by Amar in 2003, and most recently by Trent and Wick in 2009. In dimension $n=1$ we prove the corona theorem for the kernel multiplier algebras of Besov-Sobolev Banach spaces in the unit disk, extending the result for Hilbert spaces $H^\infty\cap Q_p$ by A. Nicolau and J. Xiao.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

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.

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|.

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.

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.

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