$C^*$-algebras generated by multiplication operators and composition operators with rational symbol

Hamada, Hiroyasu

doi:10.7900/jot.2015mar03.2085

Cited by 4 publications

(3 citation statements)

References 21 publications

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“…Indeed, ∆ pos = J R is uncountable and ∆ pos \ ∆ reg is finite by [Bea91, Corollary 2.7.2, Theorem 4. As a by product we also recover the main result of [Ham16]. Namely, let µ L be the Lyubich measure.…”

Section: Simplicity and Pure Infinitenesssupporting

confidence: 62%

On $C^*$-algebras associated to transfer operators for countable-to-one maps

Bardadyn¹,

Kwaśniewski²,

Lebedev³

2022

Preprint

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Our initial data is a transfer operator L for a continuous, countable-to-one map ϕ : ∆ → X defined on an open subset of a locally compact Hausdorff space X. Then L may be identified with a 'potential', i.e. a map ̺ : ∆ → X that need not be continuous unless ϕ is a local homeomorphism. We define the crossed product C0(X) ⋊ L as a universal C * -algebra with explicit generators and relations, and give an explicit faithful representation of C0(X) ⋊ L under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver C * -algebras of Muhly and Tomforde, C * -algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid C * -algebras associated to Deaconu-Renault groupoids.We describe spectra of core subalgebras of C0(X) ⋊ L and use it to characterise simplicity of C0(X) ⋊ L and prove the uniqueness theorem for C0(X) ⋊ L. We give efficient criteria for C0(X) ⋊ L to be a Kirchberg algebra, and we discuss relationship between KMS states on the core subalgebra of C0(X) ⋊ L and conformal measures for ϕ.

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Section: Simplicity and Pure Infinitenesssupporting

confidence: 62%

On $C^*$-algebras associated to transfer operators for countable-to-one maps

Bardadyn¹,

Kwaśniewski²,

Lebedev³

2022

Preprint

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“…Several authors considered C * -algebras generated by composition operators (and Toeplitz operators) on the Hardy space H 2 (D) on the open unit disk D ( [2,5,7,8,14,15,16,18,20,21,22]). On the other hand, there are some studies on C *algebras generated by composition operators on L 2 spaces, for example [3,4,17]. Matsumoto [17] introduced some C * -algebras associated with cellular automata generated by composition operators and multiplication operators.…”

Section: Introductionmentioning

confidence: 99%

“…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. In [3], we studied the C * -algebra 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 showed that the C *algebra MC R is isomorphic to the C * -algebra O R (J R ) associated with the complex dynamical system {R •n } ∞ n=1 introduced in [10].…”

Section: Introductionmentioning

confidence: 99%

C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches II

Haba¹

2021

Preprint

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Let K be a compact metric space and let ϕ : K → K be continuous. We study a C * -algebra MCϕ generated by all multiplication operators by continuous functions on K and a composition operator Cϕ induced by ϕ on a certain L 2 space. Let γ = (γ 1 , . . . , γn) be a system of proper contractions on K. Suppose that γ 1 , . . . , γn 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 MCϕ is isomorphic to the C * -algebra Oγ(K) associated with γ under the open set condition and the measure separation condition. This is a generalization of our previous work, in which we studied the case where γ satisfied the finite branch condition.

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$$C^*$$-Algebras Associated to Transfer Operators for Countable-to-One Maps

Bardadyn,

Kwaśniewski,

Lebedev

2024

Integr. Equ. Oper. Theory

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Our initial data is a transfer operator L for a continuous, countable-to-one map $$\varphi :\Delta \rightarrow X$$ φ : Δ → X defined on an open subset of a locally compact Hausdorff space X. Then L may be identified with a ‘potential’, i.e. a map $$\varrho :\Delta \rightarrow X$$ ϱ : Δ → X that need not be continuous unless $$\varphi $$ φ is a local homeomorphism. We define the crossed product $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L as a universal $$C^*$$ C ∗ -algebra with explicit generators and relations, and give an explicit faithful representation of $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L under which it is generated by weighted composition operators. We explain its relationship with Exel–Royer’s crossed products, quiver $$C^*$$ C ∗ -algebras of Muhly and Tomforde, $$C^*$$ C ∗ -algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid $$C^*$$ C ∗ -algebras associated to Deaconu–Renault groupoids. We describe spectra of core subalgebras of $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L , prove uniqueness theorems for $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L and characterize simplicity of $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L . We give efficient criteria for $$C_0(X)\rtimes L$$ C 0 ( X ) ⋊ L to be purely infinite simple and in particular a Kirchberg algebra.

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$C^*$-algebras generated by multiplication operators and composition operators with rational symbol

Cited by 4 publications

References 21 publications

On $C^*$-algebras associated to transfer operators for countable-to-one maps

On $C^*$-algebras associated to transfer operators for countable-to-one maps

C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches II

$$C^*$$-Algebras Associated to Transfer Operators for Countable-to-One Maps

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