Ideals in Toeplitz Algebras

旋转不变 Toeplitz 代数中 Toeplitz 算子的稠密性

Density of Toeplitz operators in rotation-invariant Toeplitz algebras

本文运用 Werner 的量子调和分析技术，证明了仿射酉群子群下不变及准径向 Toeplitz 算子在相应代数中的稠密性，并给出有界算子空间中 SOT 稠密性的构造性证明。

Amalgamated R-transform Theory on Toeplitz-Probability Spaces over Toeplitz Matricial Algebras

In this paper, we will consider a noncommutative probability space, $(T,E),$ over a Toeplitz matricial algebra $B=\QTR{cal}{C}^{N},$ for $N\in \QTR{Bbb}{N}$, induced by a (scalar-valued) noncommutative probability space, $(A,\QTR{cal}ϕ),$ with a suitable $B$-functional, $E:T\to B,$ defined by $ϕ.$ On this framework, we will observe amalgamated R-transform calculus with respect to a $B$-functional, $E.$ The technique and idea are came from those in [12] and Free Probability of type $B$ studied in [31].

Correspondence theory on $p$-Fock spaces with applications to Toeplitz algebras

We prove several results concerning the theory of Toeplitz algebras over $p$-Fock spaces using a correspondence theory of translation invariant symbol and operator spaces. The most notable results are: The full Toeplitz algebra is the norm closure of all Toeplitz operators with bounded uniformly continuous symbols. This generalizes a result obtained by J. Xia (J. Funct. Anal. 269:781-814, 2015) in the case $p = 2$, which was proven by different methods. Further, we prove that every Toeplitz algebra which has a translation invariant $C^\ast$ subalgebra of the bounded uniformly continuous functi…

Commutative Banach Algebras Generated by Toeplitz Operators on the Bergman Space

We present and study commutative Banach algebras generated by Toeplitz operators with generalized quasi-radial pseudo-homogeneous symbols acting on the Bergman space over the unit ball. We develop the Gelfand theory of these algebras and give some structural information about them. In particular, we provide a description of the radical of these algebras. This paper generalizes and completes the results from previous works related to Toeplitz operators with quasi-radial quasi-homogeneous symbols.

Toeplitz algebras over Fock and Bergman spaces

In this paper, we study Toeplitz algebras generated by certain class of Toeplitz operators on the $p$-Fock space and the $p$-Bergman space with $1<p<\infty$. Let BUC($\mathbb C^n$) and BUC($\mathbb B_n$) denote the collections of bounded uniformly continuous functions on $\mathbb C^n$ and $\mathbb B_n$ (the unit ball in $\mathbb C^n$), respectively. On the $p$-Fock space, we show that the Toeplitz algebra which has a translation invariant closed subalgebra of BUC($\mathbb C^n$) as its set of symbols is linearly generated by Toeplitz operators with the same space of symbols. This answers a ques…

Essential Commutants and Characterizations of the Toeplitz Algebra

In this paper we study the Toeplitz algebra, which is generated by Toeplitz operators with bounded symbols on the Fock space $F^p_α$. We show that the Toeplitz algebra coincides with each of the algebras generated by band-dominated, sufficiently localized and weakly localized operators, respectively. Moreover, we determine its essential commutant and its essential bicommutant. For $p = 2$ these results were obtained recently by Xia. However, Xia's ideas are mostly connected to Hilbert space theory and methods which are not applicable for $p \neq 2$. Instead, we use a recent result of Fulsche t…

Berger-Coburn theorem, localized operators, and the Toeplitz algebra

We give a simplified proof of the Berger-Coburn theorem on the boundedness of Toeplitz operators and extend this theorem to the setting of $p$-Fock spaces $(1\leq p \leq \infty)$. We present an overview of recent results by various authors on the compactness characterization via the Berezin transform for certain operators acting on the Fock space. Based on these results we present three new characterizations of the Toeplitz $C^*$ algebra generated by Toeplitz operators with bounded symbols.

算子的拉普拉斯算子与径向 Toeplitz 代数

The Laplacian of an operator and the radial Toeplitz algebra

本文利用量子调和分析证明算子拉普拉斯算子的定义域在 Fock 空间上的 Toeplitz 代数中稠密，并据此简化了径向 Toeplitz 代数的 Gelfand 理论。

有界对称域的 Toeplitz 代数：一种经由局部化的量子调和分析方法

Toeplitz algebra of bounded symmetric domains: A quantum harmonic analysis approach via localization

本文利用表示论、量子调和分析与弱局部化算子，证明了有界对称域加权 Bergman 空间上 Toeplitz 算子在 Toeplitz 代数中范数稠密，并刻画了 α-弱局部化算子自伴代数的范数闭包。

Nica-Toeplitz代数的理想结构

Ideal structure of Nica-Toeplitz algebras

该文研究N^n上乘积系统的Nica-Toeplitz代数的规范不变理想结构，刻画X-不变理想，用A中理想族分类真乘积系统下的规范不变理想，并应用于高阶图。

$\mathbb{T}^m$-不变Toeplitz代数的局部化框架：Gelfand理论的应用

A localization framework for $\mathbb{T}^m$-Invariant Toeplitz Algebras: Applications to Gelfand Theory

本文为$\mathbb{T}^m$-不变Toeplitz算子代数建立局部化框架，通过符号延拓识别局部商代数，并显式描述极大理想空间与Gelfand变换。

The Toeplitz corona problem for algebras of multipliers on a Nevanlinna-Pick space

Suppose $\fA$ is an algebra of operators on a Hilbert space $H$ and $A_1,..., A_n \in \fA$. If the row operator $[A_1,..., A_n] \in B(H^{(n)},H)$ has a right inverse in $B(H, H^{(n)})$, the Toeplitz corona problem for $\fA$ asks if a right inverse can be found with entries in $\fA$. When $H$ is a complete Nevanlinna-Pick space and $\fA$ is a weakly-closed algebra of multiplication operators on $H$, we show that under a stronger hypothesis, the corona problem for $\fA$ has a solution. When $\fA$ is the full multiplier algebra of $H$, the Toeplitz corona theorems of Arveson, Schubert and Ball-Tr…

Dense quasi-free subalgebras of the Toeplitz algebra

We introduce a family of dense subalgebras of the Toeplitz algebra and give conditions under which our algebras are quasi-free. As a corollary, we show that the smooth Toeplitz algebra introduced by Cuntz is quasi-free.

Non-commutative Clark measures for the free and abelian Toeplitz algebras

We construct a non-commutative Aleksandrov-Clark measure for any element in the operator-valued free Schur class, the closed unit ball of the free Toeplitz algebra of vector-valued full Fock space over $\mathbb{C} ^d$. Here, the free (analytic) Toeplitz algebra is the unital weak operator topology (WOT)-closed algebra generated by the component operators of the free shift, the row isometry of left creation operators. This defines a bijection between the free operator-valued Schur class and completely positive maps (non-commutative AC measures) on the operator system of the free disk algebra, t…

Toeplitz-composition C*-algebras for certain finite Blaschke products

Let R be a finite Blaschke product of degree at least two with R(0)=0. Then there exists a relation between the associated composition operator C_R on the Hardy space and the C*-algebra associated with the complex dynamical system on the Julia set of R. We study the C*-algebra generated by both the composition operator C_R and the Toeplitz operator T_z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C*-algebra associated with the complex dynamical system, which is simple and purely infinite.

The Cuntz-Toeplitz algebras have nuclear dimension one

We prove that unital extensions of Kirchberg algebras by separable stable AF algebras have nuclear dimension one. The title follows.

交换 $G$-不变 Toeplitz C$^\ast$ 代数在 Fock 空间上及其通过量子调和分析的 Gelfand 理论

Commutative $G$-invariant Toeplitz C$^\ast$ algebras on the Fock space and their Gelfand theory through Quantum Harmonic Analysis

本文基于量子调和分析中的谱综合概念，研究了具有特定不变符号的 Toeplitz 代数子代数及其交换子，发现了一类新的 Fock 空间上的交换 Toeplitz C$^\ast$ 代数，并探讨了其 Gelfand 理论。

$C^*$-algebras generated by truncated Toeplitz operators

We obtain an analogue of Coburn's description of the Toeplitz algebra in the setting of truncated Toeplitz operators. As a byproduct, we provide several examples of complex symmetric operators which are not unitarily equivalent to truncated Toeplitz operators having continuous symbols.

Spatial isomorphisms of algebras of truncated Toeplitz operators

We examine when two maximal abelian algebras in the truncated Toeplitz operators are spatially isomorphic. This builds upon recent work of N. Sedlock, who obtained a complete description of the maximal algebras of truncated Toeplitz operators.

Quotient algebras of Toeplitz-composition C*-algebras for finite Blaschke products

Let R be a finite Blaschke product. We study the C*-algebra TC_R generated by both the composition operator C_R and the Toeplitz operator T_z on the Hardy space. We show that the simplicity of the quotient algebra OC_R by the ideal of the compact operators can be characterized by the dynamics near the Denjoy-Wolff point of R if the degree of R is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of OC_R such that R is a finite Blaschke product of degree at least two and the Julia set of R is the unit circle, using the …

Radial Toeplitz operators on the Fock space and square-root-slowly oscillating sequences

In this paper we show that the C*-algebra generated by radial Toeplitz operators with $L_{\infty}$-symbols acting on the Fock space is isometrically isomorphic to the C*-algebra of bounded sequences uniformly continuous with respect to the square-root-metric $ρ(j,k)=|\sqrt{\vphantom{jk}j}-\sqrt{\vphantom{jk}k}\,|$. More precisely, we prove that the sequences of eigenvalues of radial Toeplitz operators form a dense subset of the latter C*-algebra of sequences.

Toeplitz algebras associated to Endomorphisms of Ore semigroups

In this paper, we consider the Toeplitz algebra associated to actions of Ore semigroups on $C^{*}$-algebras. In particular, we consider injective and surjective actions of such semigroups. We use the theory of groupoid dynamical systems to represent the Toeplitz algebra as a groupoid crossed product. We also discuss the K-theory of the Toeplitz algebra in some examples. For instance, we show that for the semigroup of positive matrices, the K-theory of the associated Toeplitz algebra vanishes.

The homotopy groups of the automorphism groups of Cuntz-Toeplitz algebras

The Cuntz-Toeplitz algebra $E_{n+1}$ for $n\geq1$ is the universal C*-algebra generated by $n+1$ isometries with mutually orthogonal ranges. In this paper, we investigate the automorphism groups of the Cuntz-Toeplitz algebras and determine their homotopy groups.

Spectral theorem approach to commutative C* -algebras generated by Toeplitz operators on the unit ball: quasi-elliptic related cases

We consider commutative C* -algebras of Toeplitz operators in the weighted Bergman space on the unit ball in $\mathbb{C}^{\mathbf{n}}$. For the algebras of elliptic type we find a new representation, namely as the algebra of operators which are functions of certain collections of commuting unbounded self-adjoint operators in the Bergman space.

Realising the Toeplitz algebra of a higher-rank graph as a Cuntz-Krieger algebra

For a row-finite higher-rank graph Λ, we construct a higher-rank graph TΛ such that the Toeplitz algebra of Λ is isomorphic to the Cuntz-Krieger algebra of TΛ. We then prove that the higher-rank graph TΛ is always aperiodic and use this fact to give another proof of a uniqueness theorem for the Toeplitz algebras of higher-rank graphs.

Some Exact Sequences for Toeplitz Algebras of Spherical Isometries

A family $\{T_j\}_{j\in J}$ of commuting Hilbert space operators is said to be a spherical isometry if $\sum_{j\in J}T^*_jT_j=1$ in the weak operator topology. We show that every commuting family $\Cal F$ of spherical isometries has a commuting normal extension $\hat{\Cal F}$. Moreover, if $\hat{\Cal F}$ is minimal, then there exists a natural short exact sequence $0\to\Cal C\to C^*(\Cal F)\to C^*(\hat{\Cal F})\to 0$ with a completely isometric cross-section, where $\Cal C$ is the commutator ideal in $C^*(\Cal F)$. We also show that the space of Toeplitz operators associated to $\Cal F$ is com…

Equilibrium states on the Toeplitz algebras of small higher-rank graphs

We consider a family of operator-algebraic dynamical systems involving the Toeplitz algebras of higher-rank graphs. We explicitly compute the KMS states (equilibrium states) of these systems built from small graphs with up to four connected components.

Unbounded Derivations in Bunce-Deddens-Toeplitz Algebras

In this paper we study decompositions and classification problems for unbounded derivations in Bunce-Deddens-Toeplitz and Bunce-Deddens algebras. We also look at implementations of these derivations on associated GNS Hilbert spaces.

Fock空间上H-Toeplitz算子的代数与图形分析

Algebraic and Graphical Analysis of H-Toeplitz operators on Fock space

本文全面研究了Fock空间上H-Toeplitz算子的代数与谱性质，推导其矩阵表示，建立交换性条件与正规性分析准则，并引入有向H-Toeplitz图揭示其组合结构。

The Toeplitz algebra has nuclear dimension one

We prove the title by constructing 2-colourable completely positive approximations for the Toeplitz algebra. Besides results about nuclear dimension and completely positive contractive order zero maps, our argument involves projectivity of the cone over a finite dimensional C*-algebra and Lin's theorem on almost normal matrices.

Toeplitz代数与Heisenberg群

Toeplitz algebras and the Heisenberg group

本文证明了由d-移位生成的Toeplitz代数自然构造的代数与(2d+1)维Heisenberg群的C*-代数理想同构，并指出这是分级幂零Lie群更一般结果的特例，且证明仅用基础泛函分析。

The Toeplitz algebra of a Hilbert bimodule

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz algebras O_n, and the Cuntz-Krieger algebras O_B. Here we analyse the representations of the corresponding Toeplitz algebra. One consequence is a uniqueness theorem for the Toeplitz-Cuntz-Krieger algebras of directed graphs, which includes Cuntz's uniqueness theorem for O_\infty.

Classifying Higher Rank Analytic Toeplitz Algebras

To a higher rank directed graph $(Λ, d)$, in the sense of Kumjian and Pask, one can associate natural noncommutative analytic Toeplitz algebras, both weakly closed and norm closed. We introduce methods for the classification of these algebras in the case of single vertex graphs.

Reconstructing directed graphs from generalised gauge actions on their Toeplitz algebras

We show how to reconstruct a finite directed graph E from its Toeplitz algebra, its gauge action, and the canonical finite-dimensional abelian subalgebra generated by the vertex projections. We also show that if E has no sinks, then we can recover E from its Toeplitz algebra and the generalised gauge action that has, for each vertex, an independent copy of the circle acting on the generators corresponding to edges emanating from that vertex. We show by example that it is not possible to recover E from its Toeplitz algebra and gauge action alone.

Inductive Limits for Systems of Toeplitz Algebras

This article deals with inductive systems of Toeplitz algebras over arbitrary directed sets. For such a system the family of its connecting injective $*$-homomorphisms is defined by a set of natural numbers satisfying a factorization property. The motivation for the study of those inductive systems comes from our previous work on the inductive sequences of Toeplitz algebras defined by sequences of numbers and the limit automorphisms for the inductive limits of such sequences. We show that there exists an isomorphism in the category of unital $C^*$-algebras and unital $*$-homomorphisms between …

A Class of Toeplitz Operators in Several Complex Variables

In order to study the Toeplitz algebras related to a Dirac operators in a neighborhood of a closed bounded domain D with smooth boundary in C^n we introduce a singular Cauchy type integral. We compute its principal symbol, thus initiating the index theory.

Fixed Point Composition and Toeplitz-Composition C*-algebras

Let $φ$ be a linear-fractional, non-automorphism self-map of $\mathbb{D}$ that fixes $ζ\in \mathbb{T}$ and satisfies $φ^{\prime}(ζ) \neq 1$ and consider the composition operator $C_φ$ acting on the Hardy space $H^2(\mathbb{D}).$ We determine which linear-fractionally-induced composition operators are contained in the unital C$^*$-algebra generated by $C_φ$ and the ideal $\mathcal{K}$ of compact operators. We apply these results to show that $C^*(C_φ, \mathcal{K})$ and $C^*(\mathcal{F}_ζ)$, the unital C$^*$-algebra generated by all composition operators induced by linear-fractional, non-automor…

Commutative $C^*$-algebras generated by Toeplitz operators on the super unit ball

We extend known results about commutative $C^*$-algebras generated Toeplitz operators over the unit ball to the supermanifold setup. This is obtained by constructing commutative $C^*$-algebras of super Toeplitz operators over the super ball $\mathbb{B}^{p|q}$ and the super Siegel domain $\mathbb{U}^{p|q}$ that naturally generalize the previous results for the unit ball and the Siegel domain. In particular, we obtain one such commutative $C^*$-algebra for each even maximal Abelian subgroup of automorphisms of the super ball.

Boundary quotients of the right Toeplitz algebra of the affine semigroup over the natural numbers

We consider the semigroup crossed product of the additive natural numbers by the multiplicative natural numbers. We study its Toeplitz C*-algebra generated by the right-regular representation, which we call the right Toeplitz algebra. We analyse its structure by studying three distinguished quotients. We show that the multiplicative boundary quotient is isomorphic to a crossed product of the Toeplitz algebra of the additive rationals by an action of the multiplicative rationals, and study its ideal structure. We identify the Crisp-Laca boundary quotient as the C*-algebra of the corresponding g…

On the universal property of Pimsner-Toeplitz C*-algebras and their continuous analogues

We consider C*-algebras generated by a single Hilbert bimodule (Pimsner-Toeplitz algebras) and by a product systems of Hilbert bimodules. We give a new proof of a theorem of Pimsner, which states that any representation of the generating bimodule gives rise to a representation of the Pimsner-Toeplitz algebra. Our proof does not make use of the conditional expectation onto the subalgebra invariant under the dual action of the circle group. We then prove the analogous statement for the case of product systems, generalizing a theorem of Arveson from the case of product systems of Hilbert spaces.

Rigidity results for automorphisms of Hardy-Toeplitz $C^*$-algebras

We prove a number of results on the automorphisms of and isomorphisms between Hardy-Toeplitz algebras $\mathcal{T}(D)$ associated to bounded symmetric domains $D$: that the stable isomorphism class of $\mathcal{T}(D)$ determines $D$ (even when it is reducible), that for reducible domains $D=D_1\times\cdots \times D_s$ the automorphisms of the Shilov boundary $\check{S}(D)$ induced by those of $\mathcal{T}(D)$ permute the Shilov boundaries $\check{S}(D_i)$, and that by contrast to arbitrary solvable algebras, automorphisms of $\mathcal{T}(D)$ that are trivial on their character spaces $\check{S…

On Equivalence for Representations of Toeplitz Algebras

Two new notions of equivalence for representations of a Toeplitz algebra $\mathcal{E}_n$, $n<\infty$, on a common Hilbert space are defined. Our main results apply to $C^*$-dynamics and the conjugacy of certain $*$-endomorphisms. One particular case of the relations is shown to coincide with the multiplicity of a representation. Previously known results due to Laca and Enomoto-Watatani are recovered as special cases.

The C*-algebra generated by irreducible Toeplitz and composition operators

We describe the C*-algebra generated by an irreducible Toeplitz operator $T_ψ$, with continuous symbol $ψ$ on the unit circle $\mathbb{T}$, and finitely many composition operators on the Hardy space $H^2$ induced by certain linear-fractional self-maps of the unit disc, modulo the ideal of compact operators $K(H^2)$. For composition operators with automorphism symbols, we show that this algebra is not isomorphic to the one generated by the shift and composition operators.

Relative graphs and pullbacks of relative Toeplitz graph algebras

In this note we generalize a result from a recent paper of Hajac, Reznikoff and Tobolski (2020). In that paper they give conditions they call admissibility on a pushout diagram in the category of directed graphs implying that the $C^*$-algebras of the graphs form a pullback diagram. We consider a larger category of relative graphs that correspond to relative Toeplitz graph algebras. In this setting we give necessary and sufficient conditions on the pushout to get a pullback of $C^*$-algebras.

Nica-Toeplitz algebras associated with product systems over right LCM semigroups

We prove uniqueness of representations of Nica-Toeplitz algebras associated to product systems of $C^*$-correspondences over right LCM semigroups by applying our previous abstract uniqueness results developed for $C^*$-precategories. Our results provide an interpretation of conditions identified in work of Fowler and Fowler-Raeburn, and apply also to their crossed product twisted by a product system, in the new context of right LCM semigroups, as well as to a new, Doplicher-Roberts type $C^*$-algebra associated to the Nica-Toeplitz algebra. As a derived construction we develop Nica-Toeplitz cr…

正则2步 nilpotent 群主符号的 Toeplitz 群场

Fields of Toeplitz algebras form the principal symbol of regular 2-step nilpotent groups

该研究通过连续 Toeplitz 群场和交叉乘积恢复正则2步 nilpotent 群的C*-代数，并推广到多接触流形及H型群模型的流形。

Isometries of the Toeplitz Matrix Algebra

We study the structure of isometries defined on the algebra $\mathcal{A}$ of upper-triangular Toeplitz matrices. Our first result is that a continuous multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either $A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the complex conjugation and $U$ is a unitary matrix. In our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where $U$ and $V$ are two unitary matrices. This implies, in particular,…

由 Cuntz 代数与 Cuntz-Toeplitz 代数及其交叉积产生的拓扑满群

Topological full groups arising from Cuntz and Cuntz-Toeplitz algebras and their crossed products

论文研究 Cuntz 与 Cuntz-Toeplitz 代数及其 Cartan 交叉积所诱导的拓扑满群，分析其正规子群和交换化，并完全刻画相关约化交叉积在规范作用下的 KMS 态。

Factoring in Non-commutative Analytic Toeplitz Algebras

The non-commutative analytic Toeplitz algebra is the weak operator topology closed algebra generated by the left regular representation of the free semigroup on $n$ generators. The structure theory of contractions in these algebras is examined. Each is shown to have an $H^\infty$ functional calculus. The isometries defined by words are shown to factor only as the words do over the unit ball of the algebra. This turns out to be false over the full algebra. The natural identification of weakly closed left ideals with invariant subspaces of the algebra is shown to hold only for a proper subcollec…

Toeplitz Algebras of Correspondences and Endomorphisms of Sums of Type I Factors

It is a well-known fact that endomorphisms of $B(H)$ are intimately connected with families of mutually orthogonal isometries, i.e. with representations of the so-called Toeplitz $C^*$-algebras. In this paper we consider a natural generalization of this connection between the representation theory of certain $C^*$-algebras associated to graphs and endomorphisms of certain von Neumann subalgebras of $B(H)$. Our primary results give criteria by which it may be determined if two representations give rise to equal or conjugate endomorphisms.

Boundary quotients and ideals of Toeplitz C*-algebras of Artin groups

We study the quotients of the Toeplitz C*-algebra of a quasi-lattice ordered group (G,P), which we view as crossed products by a partial actions of G on closed invariant subsets of a totally disconnected compact Hausdorff space, the Nica spectrum of (G,P). Our original motivation and our main examples are drawn from right-angled Artin groups, but many of our results are valid for more general quasi-lattice ordered groups. We show that the Nica spectrum has a unique minimal closed invariant subset, which we call the boundary spectrum, and we define the boundary quotient to be the crossed produc…

群可嵌入幺半群上积系统的Toeplitz代数

On Toeplitz algebras of product systems

研究群可嵌入幺半群上积系统的Toeplitz C*-代数核性与系数代数核性的关系，在技术假设下证明相关等价性，结果适用于交换幺半群等特定情形。

Algebras of Toeplitz operators on the $n$-dimensional unit ball

We study $C^*$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space $\mathcal{A}_λ^2(\mathbb{B}^n)$ over the unit ball $\mathbb{B}^n$ in $\mathbb{C}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type. By choosing $a$ and $c$ in different function algebras $\mathcal{S}_a$ and $\mathcal{S}_c$ over lower dimensional unit balls $\mathbb{B}^{\ell}$ and $\mathbb{B}^{n-\ell}$, respectively, and by assuming the invariance of $a\in \mathcal{S}_a$ under some torus action we obtain $C^*$-algebras $\boldsymbol{\mathcal{T}}_λ(\math…

Toeplitz-Composition C*-Algebras

Let $ζ$ and $η$ be distinct points on the unit circle and suppose that $ϕ$ is a linear-fractional self-map of the unit disk D, not an automorphism, with $ϕ(ζ)=η$. We describe the C*-algebra generated by the associated composition operator $C_ϕ$ and the shift operator, acting on the Hardy space on D.