Inductive limits of normed algebras and 𝑚-convex structures

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.

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.

It will be shown in this paper that certain real rank zero C*-algebras which are inductive limits of C*-algebras of the form ⊕ i M k i (C([inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /])) can be expressed as inductive limits of C*-algebras of the form ⊕ i M k i (C( S 1 )). In particular, if both A and B are of real rank zero and are inductive limits of C*-algebras of the form ⊕ i M k i (C( S 1 )), then also A ⊗ B is an inductive limit of C*-algebras of the form ⊕ i M k i (C( S 1 )). (Hence, A ⊗ B can be classified by its K-theory.) This is a key step in the general classification theory of inductive limit C*-algebras.

We prove that for the inductive limit of sequentially complete spaces regularity or local completeness imply the Banach Disk Closure Property (BDCP ) (an inductive limite enjoys the BDCP if all Banach disks in the steps of the inductive limit are such that their closures, with respect to the inductive limit topology, are Banach discs as well). In particular we obtain that for an inductive limit of sequentially complete spaces, regularity is equivalent to the BDCP plus an “almost regularity” condition. Mathematics Subject Classification: 46A13, 46A30

Let A \mathfrak {A} be a UHF algebra and A ( D ) \mathcal {A}({\mathbf {D}}) the disk algebra. If A = [ ∪ n ≥ 1 A n ] − \mathfrak {A} = {\left [ {{ \cup _{n \geq 1}}{\mathfrak {A}_n}} \right ]^ - } and α \alpha is a product-type automorphism of A \mathfrak {A} which leaves each A n {\mathfrak {A}_n} invariant, then α \alpha defines an embedding \[ A n ⊗ A ( D ) ↪ ı n A n + 1 ⊗ A ( D ) \mathfrak {A}_n \otimes \mathcal {A}({\mathbf {D}}) \stackrel {\imath _n}{\hookrightarrow } {\mathfrak {A}_{n + 1}} \otimes \mathcal {A}({\mathbf {D}}) \] . The inductive limit of this system is a Banach algebra whose maximal ideal space is closely related to that of the disk algebra if the Connes spectrum Γ ( α ) \Gamma (\alpha ) is finite.

Compact group actions on UHF algebras obtained by folding the interval

The principal aim of the present paper is to give a dynamical presentation of the Jiang–Su algebra. Originally constructed as an inductive limit of prime dimension drop algebras, the Jiang–Su algebra has gone from being a poorly understood oddity to having a prominent positive role in George Elliott’s classification programme for separable, nuclear C * {\mathrm{C}^{*}} -algebras. Here, we exhibit an étale equivalence relation whose groupoid C * {\mathrm{C}^{*}} -algebra is isomorphic to the Jiang–Su algebra. The main ingredient is the construction of minimal homeomorphisms on infinite, compact metric spaces, each having the same cohomology as a point. This construction is also of interest in dynamical systems. Any self-map of an infinite, compact space with the same cohomology as a point has Lefschetz number one. Thus, if such a space were also to satisfy some regularity hypothesis (which our examples do not), then the Lefschetz–Hopf Theorem would imply that it does not admit a minimal homeomorphism.

We investigate a natural connection between inductive limits of $C^*$-algebras, ergodic theory and the theory of subfactors.

The notion of Reiter's Segal algebra in commutative group algebras is generalized to a notion of Segal algebra in more general classes of commutative Banach algebras. Then we introduce a family of Segal algebras in commutative Banach algebras under considerations and study some properties of them.

Weierstrass products for inverse limits of Banach algebras

Let X be a complex manifold, let A be a topological discrete valuation ring, and write for the sheaf of functions on X with values in A. We prove Cartan theorems A and B for coherent -modules, when X is a Stein manifold and A satisfies some requirements like being a nuclear direct limit of Banach algebras. The result is motivated by questions in the work of the second author with Kashiwara in the proof of the codimension-three conjecture for holonomic microdifferential systems.

Conditions for the stabilizability of time-delay systems with incommensurable point delays by dynamic state feedback are known in the literature. In this paper it is shown that these conditions are satisfied generically. Although an algebraic approach is used to describe the class of all time-delay systems with point delays, the concept of genericity is formulated in a topological framework. For this, two different topologies are introduced. One turns the parameter-space of all time-delay systems into a metric space. The other is a stronger, inductive limit topology. In both cases, stabilizability is called a generic property, it the subset of all stabilizable delay systems in an open and dense subset. First it is shown that this genericity result holds in the metric space of all time-delay systems. Using the inductive limit topology, it is possible to extend this result to a far more general class of stabilizability problems.

For a compactly generated LCA group G, it is shown that the setH(G) of all generalized characters on G equipped with the compact-open topology is a LCA group andH(G) = Ĝ (the dual group ofG) if and only ifG is compact. Both results fail for arbitrary LCA groups. Further, ifG is second countable, then the Gel’fand space of the commutative convolution algebraC c (G) equipped with the inductive limit topology is topologically homeomorphic toH(G).

A new class of locally convex algebras, called BP*-algebras, is introduced. It is shown that this class properly includes MQ*-algebras which were introduced and studied by the first author andR. Rigelhof [10]. Among other results, it is proved that each positive functional on a BP*-algebraA is admissible but not necessarily continuous as shown by an example. However, ifA, in addition, is either (i) a Q-algebra, or (ii) has an identity and is barrelled, or (iii)A is endowed with the inductive limit topology, then each positive functional onA is continuous.

The topic of this paper is the analysis of abstract linear systems on a locally compact group G, that is, of continuous linear operators $N:\mathfrak{D}(G) \to \mathfrak{D}'(G)$. Here, $\mathfrak{D}(G)$ denotes the space of test functions on G with its inductive limit topology as introduced by Bruhat, Maurin and Kac and $\mathfrak{D}'(G)$ the space of distributions on G with the weak topology. We call such a linear system N periodically invariant with respect to a given closed subgroup $\Gamma $, if N commutes with translations from $\Gamma $. Periodically invariant systems are of interest, e.g., in the theory of electrical networks with periodically varying parameters or in process control theory. Under the assumption that the quotient group ${G / \Gamma }$ is compact, a Fourier series representation for $\Gamma $-periodic distributions on G is derived. From this we conclude via the Schwartz kernel theorem that the classical convolution representation for a translation invariant system $(\Gamma = G)$ generalizes to a Fourier superposition $\Sigma _\chi N_\chi $ of translation invariant systems $N_\chi $, where summation runs over all characters $\chi \in \hat G$ vanishing on $\Gamma $. Finally, it is proved that N is causal with respect to a semigroup $P \subset G$ if and only if all individual systems $N_\chi $ are causal.