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| Format: | Preprint |
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2024
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| Online-Zugang: | https://arxiv.org/abs/2411.12122 |
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| _version_ | 1866909394786058240 |
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| author | Galindo, Jorge Jordá, Enrique Rodríguez-Arenas, Alberto |
| author_facet | Galindo, Jorge Jordá, Enrique Rodríguez-Arenas, Alberto |
| contents | Let G be a locally compact group and let $ϕ$ be a positive definite function on G with $ϕ(e)=1$. This function defines a multiplication operator $M_ϕ$ on the Fourier algebra $A(G)$ of $G$. The aim of this paper is to classify the ergodic properties of the operators $M_ϕ$, focusing on several key factors, including the subgroup $H_ϕ=\{x\in G\colon ϕ(x)=1\}$, the spectrum of $M_ϕ$, or how ``spread-out'' a power of $M_ϕ$ can be. We show that the multiplication operator $M_ϕ$ is uniformly mean ergodic if and only if $H_ϕ$ is open and 1 is not an accumulation point of the spectrum of $M_ϕ$. Equivalently, this happens when some power of $ϕ$ is not far, in the multiplier norm, from a function supported on finitely many cosets of $H_ϕ$. Additionally, we show that the powers of $M_ϕ$ converge in norm if, and only if, the operator is uniformly mean ergodic and $H_ϕ=\{x\in G\colon |ϕ(x)|=1\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12122 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive definite functions as uniformly ergodic multipliers of the Fourier algebra Galindo, Jorge Jordá, Enrique Rodríguez-Arenas, Alberto Functional Analysis 22D25, 22D40, 43A30, 46H99, 47A35 Let G be a locally compact group and let $ϕ$ be a positive definite function on G with $ϕ(e)=1$. This function defines a multiplication operator $M_ϕ$ on the Fourier algebra $A(G)$ of $G$. The aim of this paper is to classify the ergodic properties of the operators $M_ϕ$, focusing on several key factors, including the subgroup $H_ϕ=\{x\in G\colon ϕ(x)=1\}$, the spectrum of $M_ϕ$, or how ``spread-out'' a power of $M_ϕ$ can be. We show that the multiplication operator $M_ϕ$ is uniformly mean ergodic if and only if $H_ϕ$ is open and 1 is not an accumulation point of the spectrum of $M_ϕ$. Equivalently, this happens when some power of $ϕ$ is not far, in the multiplier norm, from a function supported on finitely many cosets of $H_ϕ$. Additionally, we show that the powers of $M_ϕ$ converge in norm if, and only if, the operator is uniformly mean ergodic and $H_ϕ=\{x\in G\colon |ϕ(x)|=1\}$. |
| title | Positive definite functions as uniformly ergodic multipliers of the Fourier algebra |
| topic | Functional Analysis 22D25, 22D40, 43A30, 46H99, 47A35 |
| url | https://arxiv.org/abs/2411.12122 |