Shift-invariant algebras on compact groups

We consider various algebras of functions on a semitopological left-group S = X × G S = X \times G , the direct product of a left-zero semigroup X and a group G. In §1 we examine various analogues to the theorem of Eberlein that a weakly almost periodic function on a locally compact abelian group is uniformly continuous. Several appealing conjectures are shown by example to be false. In the second section we look at compactifications of products S × T S \times T of semitopological semigroups with right identity and left identity, respectively. We show that the almost periodic compactification of the product is the product of the almost periodic compactifications, thus generalizing a result of deLeeuw and Glicksberg. The weakly almost periodic compactification of the product is not the product of the weakly almost periodic compactifications except in restrictive circumstances; for instance, when T is a compact group. Finally, as an application, we define and study analytic weakly almost periodic functions and derive the theorem, analogous to a classical theorem about almost periodic functions, that an analytic function which is weakly almost periodic on a single line is analytic weakly almost periodic on a whole strip.

Let Hp(m), 0 < p ^ oo, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of Hp{m), p Φ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of Hp(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly. 1* Introduction* Many results have been obtained concerning isometries of Hardy spaces in the theory of uniform algebras. The most fundamental result in this direction is due to de Leeuw, Rudin, and Wermer [2], which states that an automorphism of the classical Hardy space H°°{T) is induced via composition with the unit circle T of a fractional linear transformation of the unit disc onto itself. Their work was carried on independent of Nagasawa [13], who also described the isometries of ίί oo(Tr) onto itself. On the other hand, Arens [1] completely determined the automorphisms of the uniform algebra of analytic functions on a compact abelian group K whose dual group Γ is archimedean ordered (cf. [11]). This result was extended by Muhly [11] to the uniform algebra of analytic functions induced by a flow which has no periodic orbits. Moreover Muhly [12] has recently given, among other things, the following interesting generalization of this result of Arens to the case of isometries of Hardy spaces Hp(m), p Φ 2, on K: Under the assumption that Γ is countable, every isometry of Hp(m), p Φ 2, is induced via composition with an affine map of K such that the adjoint of the additive factor of this map preserves the order of Γ. The purpose of this paper is to remove the assumption on Γ. This result provide a partial positive answer to the following question posed by Muhly in [12; §5]: Is it possible to describe the isometries of ergodic Hardy spaces

The theory of weakly almost periodic functional on the Fourier algebra is herein developed. It is the extension of the theory of weakly almost periodic functions on locally compact abelian groups to the duals of compact groups. The complete direct product of a countable collection of nontrivial compact groups furnishes an important example for some of the constructions.

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.

In this note we point out two properties (Theorems 1 and 2) of the group algebra of the direct product of a compact group and a locally compact abelian group and indicate briefly how these properties enable one to extend to these group algebras theorems which are known in the compact and abelian cases. Our interest in these properties is two-fold; they establish the applicability of known results in [4]1 to the group algebras in question, and they provide an interesting example of a type of B-algebra, a sort of Kroliecker product algebra, which may very well deserve independenlt systematic study. The reader is referred to results of Kaplansky [1 ] and Segal [2 ] of which this note is essentially a continuation. Let C be a compact group, A be a locally compact abelian group, and G = CXA be their direct product group. We are interested in the group algebras L1(C), L1(A), and L1(G). In particular, we ask how the latter is related to the first two. We will show (1) how the B-algebra L1(G) is formed out of the B-algebras L1(C) and L1(A) and (2) how the structure space of L1(G) is formed from the structure spaces of L1(C) and L1(A). DEFINITION 1. The complex algebra R is the Kronecker product of the complex algebras R1 and R2 (written R = R1 XR2) if there exists a map (xl, X2)>X1 X X2 of (R1, R2) into R such that

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

In this paper we discuss the possible generalizations of a lifting theorem of a 2 × 2 2 \times 2 matrix to uniform algebras. These have applications to Hankel operators, weighted norm inequalities for conjugation operators and Toeplitz operators on uniform algebras. For example, the Helson-Szegö theorems for general uniform algebras follow.

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.

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.

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.

Let A be a uniform algebra and M(A) the maximal ideal space of A. A sequence { an }n in M(A) is called .e1-interpolating if for every sequence (an) in f1 there exists a function f in A such that f (an ) = an for all n. In this paper, an f1-interpolating sequence is studied for an arbitrary uniform algebra. For some special uniform algebras, an f1-interpolating sequence is equivalent to an £&lt;)()-interpolating sequence which is familiar for us. However, in general these two interpolating sequences may be different from each other.

Our main focus is on a general class of active rotators with mean field interactions, that is, globally coupled large families of dynamical systems on the unit circle with nontrivial stochastic dynamics. The dynamics of each isolated system is ${\mathrm d} \psi_t =-\delta V'(\psi_t){\mathrm d} t + {\mathrm d} w_t$, where $V'$ is a periodic function, $w$ is a Brownian motion, and $\delta$ is an intensity parameter. It is well known that the interacting dynamics is accurately described, in the limit of infinitely many interacting components, by a Fokker--Planck PDE and the model reduces for $\delta=0$ to a particular case of the Kuramoto synchronization model, for which one can show the existence of a stable normally hyperbolic manifold of stationary solutions for the corresponding Fokker--Planck equation. (We are interested in the case in which this manifold is nontrivial, which happens when the interaction is sufficiently strong, that is, in the synchronized regime of the Kuramoto model.) We use the robustness of normally hyperbolic structures to infer qualitative and quantitative results on the $\vert\delta\vert \le \delta_0$ cases with $\delta_0$ a suitable threshold; in fact, we obtain an accurate description of the dynamics on the invariant manifold for $\delta \neq0$ and we link it explicitly to the potential $V$. This approach allows us to have a complete description of the phase diagram of the active rotator model, at least for $\vert \delta\vert \le \delta_0$, thus identifying for which values of the parameters (notably, noise intensity and/or coupling strength) the system exhibits periodic pulse waves or stabilizes at a quiescent resting state. Moreover, some of our results are very explicit and this brings a new insight into the combined effect of active rotator dynamics, noise, and interaction. The links with the literature on specific systems, notably neuronal models, are discussed in detail.

In the usual construction of a neural network, the individual nodes store and transmit real numbers that lie in an interval on the real line; the values are often envisioned as amplitudes. In this article we present a design for a circular node, which is capable of storing and transmitting angular information. We develop the forward and backward propagation formulas for a network containing circular nodes. We show how the use of circular nodes may facilitate the characterization and parameterization of periodic phenomena in general. We describe applications to constructing circular self-maps, periodic compression, and one-dimensional manifold decomposition. We show that a circular node may be used to construct a homeomorphism between a trefoil knot in R3 and a unit circle. We give an application with a network that encodes the dynamic system on the limit cycle of the Kuramoto-Sivashinsky equation. This is achieved by incorporating a circular node in the bottleneck layer of a three-hidden-layer bottleneck network architecture. Exploiting circular nodes systematically offers a neural network alternative to Fourier series decomposition in approximating periodic or almost periodic functions.

Tingley's problems on uniform algebras

In deriving cohomology algebras of arbitrary compact groups from the cohomology algebras of Lie groups where they are reasonably well understood one has to pass through limit operations. However, these limit operations need to be controlled to such an extent that one can see how morphisms are lifted to the limits In other words one has to pass to the limit functorially. In order to do this efficiently, we develop a theory which connects our situation with the more general concept of Kan extensions of functors. Indeed we derive a quite general existence and uniqueness theorem for Kan extensions—one which covers much more ground than we actually need for our current applications. Among other things we will be able to show as an application that the Cech cohomology of a compact group space is the Kan extension of any of the standard cohomologies on manifolds applied to Lie group spaces, and that the cohomology of a classifying space of a compact totally disconnected group is the Kan extension of the algebraic cohomology of a finite group.