Algebra Of Continuous Functions Research Articles

Endomorphisms of the Toeplitz algebra

This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry V in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries {V P}, where P is a projection of finite codimension. The notions of T -equivalence of endomorphisms and T -equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the discalgebra and generate unbranched coverings over the unit circle, were singled out.

Universal continuous calculus for Su*‐algebras

AbstractUniversal continuous calculi are defined and it is shown that for every finite tuple of pairwise commuting Hermitian elements of a Su*‐algebra (an ordered *‐algebra that is symmetric, i.e., “strictly” positive elements are invertible and uniformly complete), such a universal continuous calculus exists. This generalizes the continuous calculus for ‐algebras to a class of generally unbounded ordered *‐algebras. On the way, some results about *‐algebras of continuous functions on locally compact spaces are obtained. The approach used throughout is rather elementary and especially avoids any representation theory.

Function algebras on the n-dimensional quantum complex space

This paper introduces a (universal) C*-algebra of continuous functions vanishing at infinity on the [Formula: see text]-dimensional quantum complex space. To this end, the well-behaved Hilbert space representations of the defining relations are classified. Then these representations are realized by multiplication operators on an [Formula: see text]-space. The C*-algebra of continuous functions vanishing at infinity is defined by considering an *-algebra such that its classical counterpart separates the points of the [Formula: see text]-dimensional complex space and by taking the operator norm closure of a universal representation of this algebra.

On Harmonic Hilbert Spaces on Compact Abelian Groups

Harmonic Hilbert spaces on locally compact abelian groups are reproducing kernel Hilbert spaces (RKHSs) of continuous functions constructed by Fourier transform of weighted L^2 spaces on the dual group. It is known that for suitably chosen subadditive weights, every such space is a Banach algebra with respect to pointwise multiplication of functions. In this paper, we study RKHSs associated with subconvolutive functions on the dual group. Sufficient conditions are established for these spaces to be symmetric Banach ^*-algebras with respect to pointwise multiplication and complex conjugation of functions (here referred to as RKHAs). In addition, we study aspects of the spectra and state spaces of RKHAs. Sufficient conditions are established for an RKHA on a compact abelian group G to have the same spectrum as the C^*-algebra of continuous functions on G. We also consider one-parameter families of RKHSs associated with semigroups of self-adjoint Markov operators on L^2(G), and show that in this setting subconvolutivity is a necessary and sufficient condition for these spaces to have RKHA structure. Finally, we establish embedding relationships between RKHAs and a class of Fourier–Wermer algebras that includes spaces of dominating mixed smoothness used in high-dimensional function approximation.

Cauchy transform and uniform approximation by polynomial modules

For a compact subset K of the complex plane C, let C(K) denote the algebra of continuous functions on K. For an open subset U⊂K, let A(K,U)⊂C(K) be the algebra of functions that are analytic in U. We show that there exists ϕ∈A(K,U) so that each f∈A(K,U) can uniformly be approximated by {pn+qnϕ} on K, where pn and qn are analytic polynomials in z. In particular, ϕ can be chosen as a Cauchy transform of a finite positive measure η compactly supported in C∖U. Recent developments of analytic capacity and Cauchy transform provide us useful tools in our proofs.

On generalization of Petryshyn's fixed point theorem and its application to the product of $ n $-nonlinear integral equations

<abstract><p>Regarding the Hausdorff measure of noncompactness, we provide and demonstrate a generalization of Petryshyn's fixed point theorem in Banach algebras. Comparing this theorem to Schauder and Darbo's fixed point theorems, we can skip demonstrating closed, convex and compactness properties of the investigated operators. We employ our fixed point theorem to provide the existence findings for the product of $ n $-nonlinear integral equations in the Banach algebra of continuous functions $ C(I_a) $, which is a generalization of various types of integral equations in the literature. Lastly, a few specific instances and informative examples are provided. Our findings can successfully be extended to several Banach algebras, including $ AC, C^1 $ or $ BV $-spaces.</p></abstract>

Tolerance relations and quantization

It is well known that "bad" quotient spaces (typically: non-Hausdorff) can be studied by associating to them the groupoid C*-algebra of an equivalence relation, that in the "nice" cases is Morita equivalent to the C*-algebra of continuous functions vanishing at infinity on the quotient space. It was recently proposed by A. Connes and W.D. van Suijlekom that a similar procedure for relations that are reflexive and symmetric but fail to be transitive (i.e. tolerance relations) leads to an operator system. In this paper we observe that such an operator system carries a natural product that, although in general non-associative, arises in a number of relevant examples. We relate this product to truncations of (C*-algebras of) topological spaces, discuss some geometric aspects and a connection with positive operator valued measures.

Equivariant Callias index theory via coarse geometry

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the $K$-theory of a group $C^*$-algebra.

Z◦−ideals in MV−algebras of continuous functions

In this paper, we study MV?algebra of continuous functions C(X) and maximal ideals of C(X). Furthermore, Z?ideal and Z??ideal of C(X) are introduced and proved that every Z??ideal in C(X) is a Z?ideal but the converse is not true and every finitely generated Z?ideal is a basic Z??ideal. Also, we investigate meet and join of two Z?ideals (Z??ideal) of C(X). Complemented elements of C(X) are examined and their properties have been studied. In particular, the relationship between generated ideal by them and Z?ideals (Z??ideals) is proved. Finally, we investigate some property of Z??ideals in basically disconnected space and extremally disconnected space.

Каноническое представление $C^*$-алгебры эйконалов метрического графа

Алгебра эйконалов $\mathfrak E$ метрического графа $\Omega$ - это операторная $C^*$-алгебра, определяемая динамической системой, которая описывает распространение волн, порожденных источниками в граничных вершинах $\Omega$. В работе описывается каноническая блочная форма алгебры $\mathfrak E$ произвольного компактного связного метрического графа. Переход к этой форме равносилен построению функциональной модели, реализующей $\mathfrak E$ в виде алгебры непрерывных матричнозначных функций на ее спектре $\widehat{\mathfrak{E}}$. Результаты предполагается использовать в обратной задаче, состоящей в реконструкции графа по спектральным и динамическим граничным данным. Библиография: 28 наименований.

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.

Existence and Rigidity of Quantum Isometry Groups for Compact Metric Spaces

We prove the existence of a quantum isometry groups for new classes of metric spaces: (i) geodesic metrics for compact connected Riemannian manifolds (possibly with boundary) and (ii) metric spaces admitting a uniformly distributed probability measure. In the former case it also follows from recent results of the second author that the quantum isometry group is classical, i.e. the commutative $C^*$-algebra of continuous functions on the Riemannian isometry group.

Gelfand theory for real Banach algebras

We are concerned with the development of the more general real case of the classical theorem of Gelfand on representation of a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. To that end, we use only intrinsic methods which do not depend on the complexification of the algebra, and obtain two representation theorems for commutative unital real Banach algebras as algebras of continuous real (respectively, complex) functions on the compact space of real-valued (respectively, complex-valued) $$\mathbb R$$ -algebra homomorphisms.

Translations in quantum groups

\lt p \gt Let $H$ be the Hopf $C^*$-algebra of continuous functions on a (locally) compact quantum group of either reduced or full type. We show that endomorphisms of $H$ that respect its right regular comodule structure are translations by elements of th

Standard homogeneous C*-algebras as compact quantum metric spaces

Given a compact metric space $X$ and a unital C*-algebra $A$, we introduce a family of seminorms on the C*-algebra of continuous functions from $X$ to $A$, denoted by $C(X, A)$, induced by classical Lipschitz seminorms that produce compact quantum metrics

2-Local isometries between Banach algebras of continuous functions with involution

We give a description of 2-local real isometries between $$C(X,\tau )$$ and $$C(Y,\eta )$$ where X and Y are compact Hausdorff spaces, X is also first countable and $$\tau $$ and $$\eta $$ are topological involutions on X and Y, respectively. In particular, we show that every 2-local real isometry T from $$C(X,\tau )$$ to $$ C(Y,\eta )$$ is a surjective real linear isometry whenever X is also separable.

Finite codimensional maximal ideals in subalgebras of ultrametric uniformly continuous functions

Let $\rm I\!E$ be a complete ultrametric space, let $\rm I\!E$ be a perfect complete ultrametric field and let $A$ be a Banach $\rm I\!E$-algebra which is either a full $\rm I\!E$-subalgebra of the algebra of continuous functions from $\rm I\!E$ to $\rm I\!E$ owning all characteristic functions of clopens of $\rm I\!E$, or a full $\rm I\!E$-subalgebra of the algebra of uniformly continuous functions from $\rm I\!E$ to $\rm I\!E$ owning all characteristic functions of uniformly open subsets of $\rm I\!E$. We prove that all maximal ideals of finite codimension of $A$ are of codimension $1$.

Spectral triples for nonarchimedean local fields

Using associated trees, we construct a spectral triple for the $\mathrm {C}^*$-algebra of continuous functions on the ring of integers $R$ of a nonarchimedean local field $F$ of characteristic zero, and investigate its properties. Remarkably, the spectrum of the spectral triple operator is closely related to the roots of a $q$-hypergeometric function. We also study a noncompact version of this construction for the $\mathrm {C}^*$-algebra of continuous functions on $F$, vanishing at infinity.

Equilibrium states on higher-rank Toeplitz non-commutative solenoids

We consider a family of higher-dimensional non-commutative tori, which are twisted analogues of the algebras of continuous functions on ordinary tori and their Toeplitz extensions. Just as solenoids are inverse limits of tori, our Toeplitz non-commutative solenoids are direct limits of the Toeplitz extensions of non-commutative tori. We consider natural dynamics on these Toeplitz algebras, and we compute the equilibrium states for these dynamics. We find a large simplex of equilibrium states at each positive inverse temperature, parametrized by the probability measures on an (ordinary) solenoid.

Inductive limits of semiprojective C⁎-algebras

We prove closure properties for the class of C⁎-algebras that are inductive limits of semiprojective C⁎-algebras. Most importantly, we show that this class is closed under shape domination, and so in particular under shape and homotopy equivalence. It follows that the considered class is quite large. It contains for instance the stable suspension of any nuclear C⁎-algebra satisfying the UCT and with torsion-free K0-group. In particular, the stabilized C⁎-algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives.