Thomas Kriete, Barbara MacCluer, Jennifer Moorhouse
Let $ϕ$ be a linear-fractional self map of the open unit disk, not an automorphism, such that $ϕ(ζ)=η$ for distinct points $ζ,η$ in the unit circle. We consider the question of which composition operators lie in the unital C*-algebra generated by the composition operator $C_ϕ$ and the ideal of compact operators, acting on the Hardy space $H^2$.
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We construct a singly generated subalgebra of ${\mathcal K}({\mathcal H})$ which is non-amenable, yet is boundedly approximately contractible. The example embeds into a homogeneous von Neumann algebra. We also observe that there are singly generated, biflat subalgebras of finite Type I von Neumann algebras, which are not amenable (and hence are not isomorphic to C*-algebras). Such an example can be used to show that a certain extension property for commutative operator algebras, which is shown in arXiv:1012.4259 to follow from amenability, does not necessarily imply amenability.
Hyperrigidity I: singly generated commutative $C^*$-algebras
本文研究了单生成交换C*-代数中单项式集的超rigidity性质，探讨其与泛函分析和数学物理的关联，并通过拓扑方法和Choquet边界技术验证了结果的最优性。
We demonstrate how virtually all common cardinal invariants associated to a von Neumann algebra M can be computed from the decomposability number, dec(M), and the minimal cardinality of a generating set, gen(M). Applications include the equivalence of the well-known generator problem, "Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, "Is every countably-generated von Neumann algebra singly-generated?" and "Is the gen invariant monotone?" Modulo the generator problem, we determine the range of the invariant (gen(M), dec(M)), which is mostly…
We determine when there is a unique conditional expectation from a semifinite von Neumann algebra onto a singly-generated maximal abelian *-subalgebra. Our work extends the results of Kadison and Singer via new methods, notably the observation that a unique conditional expectation onto a singly-generated maximal abelian *-subalgebra must be normal.
We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity, such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0.
Singly generated Selfadjoint-Ideal operator semigroups: spectral density of the generator and simplicity
本文研究单生成 SI 算子半群，揭示其生成元谱密度与半群单性的代数和解析性质，并刻画次正规、本质正规及加权移位等情形。
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.
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…
Let $R$ be a rational function of degree at least two, let $J_R$ be the Julia set of $R$ and let $μ^L$ be the Lyubich measure of $R$. We study the C$^*$-algebra $\mathcal{MC}_R$ generated by all multiplication operators by continuous functions in $C(J_R)$ and the composition operator $C_R$ induced by $R$ on $L^2(J_R, μ^L)$. We show that the C$^*$-algebra $\mathcal{MC}_R$ is isomorphic to the C$^*$-algebra $\mathcal{O}_R (J_R)$ associated with the complex dynamical system $\{R^{\circ n} \}_{n=1} ^\infty$.
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 …
The C$^*$-algebra of a composition reflection
本文研究单位圆盘Hardy空间上由复合反射算子$C_a$生成的C$^*$代数，并考察相关算子，探讨其结构性质。
An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathcal B}({\mathcal H})$ must be similar to an amenable C*-algebra; the question remains open even for singly-generated algebras. In this article we show that any closed, commutative, operator amenable subalgebra of a finite von Neumann algebra ${\mathcal M}$ is similar to a commutative C*-subalgebra of ${\mathcal M}$, with the similarity implemented by an element of ${\mathcal M}$. Our proof makes use of the algebra of measurable operators affiliated to ${\mathcal M}$.
Let $K$ be a compact metric space and let $γ= (γ_1, \dots, γ_n)$ be a system of proper contractions on $K$. We study a C*-algebra $\mathcal{MC}_{γ_1, \dots, γ_n}$ generated by all multiplication operators by continuous functions on $K$ and composition operators $C_{γ_i}$ induced by $γ_i$ for $i=1, \dots, n$ on a certain $L^2$ space. Suppose that $K$ is self-similar. We consider the Hutchinson measure $μ^H$ of $γ$ and the $L^2$ space $L^2(K, μ^H)$. Then we show that the C*-algebra $\mathcal{MC}_{γ_1, \dots, γ_n}$ is isomorphic to the Cuntz algebra $\mathcal{O}_n$ under some conditions.
Let $K$ be a compact metric space and let $φ: K \to K$ be continuous. We study C*-algebra $\mathcal{MC}_φ$ generated by all multiplication operators by continuous functions on $K$ and a composition operator $C_φ$ induced by $φ$ on a certain $L^2$ space. Let $γ= (γ_1, \dots, γ_n)$ be a system of proper contractions on $K$. Suppose that $γ_1, \dots, γ_n$ are inverse branches of $φ$ and $K$ is self-similar. We consider the Hutchinson measure $μ^H$ of $γ$ and the $L^2$ space $L^2(K, μ^H)$. Then we show that the C*-algebra $\mathcal{MC}_φ$ is isomorphic to the C*-algebra $\mathcal{O}_γ(K)$ associat…
We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.
We show that the generalised composition of generalised integral operators is well defined on the space Colombeau algebras of tempered generalised functions.
In this paper, we study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. We study the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
We consider this and related questions: When is a finite linear combination of composition operators, acting on the Hardy space or the standard weighted Bergman spaces on the unit disk, a compact operator?
We study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. The central issue is to understand the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
In this paper we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application we prove that for "quasi-parabolic" composition operators the spectra and the essential spectra are equal.
For 0 < s < 1, let phi_s(z)=sz+(1-s). We investigate the unital C*-algebra generated by the semigroup {C_{phi_s} : 0 < s < 1} of composition operators acting on the Hardy space of the unit disk. We determine the joint approximate point spectrum of a related collection of operators and show that the quotient of the C*-algebra by its commutator ideal is isomorphic to the direct sum of the complex numbers and the algebra of almost periodic functions on the real line. In addition, we show that the C*-algebra is irreducible.
In this paper we consider the C*-algebra $C^{*}(\{C_φ\}\cup\mathcal{T}(PQC(\mathbb{T})))/K(H^{2})$ generated by Toeplitz operators with piece-wise quasi-continuous symbols and a composition operator induced by a parabolic linear fractional non-automorphism symbol modulo compact operators on the Hilbert-Hardy space $H^{2}$. This C*-algebra is commutative. We characterize its maximal ideal space. We apply our results to the question of determining the essential spectra of linear combinations of a class of composition operators and Toeplitz operators.
We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator $D$ of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as $\mbox{cos}(D)$, $De^D$, or $e^D-aI$, where $0<a\le 1$. In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. Non-trivial translations on the space of complex-valued, smooth functions on the real line do su…
Spectrum of Weighted Composition Operators. Part X. The spectrum and essential spectra of weighted automorphisms of the polydisc algebra
本文研究了多圆盘代数加权自同构的谱与本质谱，并在二维情形下给出了这些谱的详细且多数情况下完整的描述。
Minimal invariant subspaces for an affine composition operator
本文刻画了Hardy-Hilbert空间上仿射复合算子的极小不变子空间，给出了函数在1处极限非零或导数有界时的完整特征，并揭示了零点集的作用及Rota普适性与循环性的联系。
A C*-algebra $A$ is said to be stable if it is isomorphic to $A \otimes K(\ell_2)$. Hjelmborg and Rørdam have shown that countable inductive limits of separable stable C*-algebras are stable. We show that this is no longer true in the nonseparable context even for the most natural case of an uncountable inductive limit of an increasing chain of separable stable and AF ideals: we construct a GCR, AF (in fact, scattered) subalgebra $A$ of $B(\ell_2)$, which is the inductive limit of length $ω_1$ of its separable stable ideals $I_α$ ($α<ω_1$) satisfying $I_{α+1}/I_α\cong K(\ell_2)$ for each $α<ω_…
We provide a description of the spectrum and essential spectra of invertible weighted composition operators acting in some algebras of smooth functions on the interval.
Range decreasing group homomorphisms and weighted composition operators
本文提出基于值域递减群同态的方法，给出了李群丛光滑截面空间间群同态成为加权复合算子的充要条件，并推广了 Shanks-Pursell 定理。
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