卡普兰斯基-希尔伯特模块上单元群表示的分解定理和弗斯滕贝格-齐美尔结构定理

IF 0.6 3区数学 Q3 MATHEMATICS

Analysis Mathematica Pub Date : 2024-05-23 DOI:10.1007/s10476-024-00020-1

N. Edeko, M. Haase, H. Kreidler

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引用次数: 0

摘要

本文建立了斯通代数上卡普兰斯基-希尔伯特模块上（协变）单元群表示的分解定理，该定理推广了著名的希尔伯特空间情况（与雅各布斯、德利乌和格里克斯伯格的分解不谋而合）。该证明在很大程度上依赖于关于卡普兰斯基-希尔伯特模块的算子理论，特别是关于这类模块上希尔伯特-施密特同态的谱定理。作为应用，著名的弗斯滕伯格-齐美尔结构定理被推广到任意群在任意概率空间上的度量保全作用的情况。

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A decomposition theorem for unitary group representations on Kaplansky–Hilbert modules and the Furstenberg–Zimmer structure theorem

In this paper, a decomposition theorem for (covariant) unitary group representations on Kaplansky–Hilbert modules over Stone algebras is established, which generalizes the well-known Hilbert space case (where it coincides with the decomposition of Jacobs, deLeeuw and Glicksberg).

The proof rests heavily on the operator theory on Kaplansky–Hilbert modules, in particular the spectral theorem for Hilbert–Schmidt homomorphisms on such modules.

As an application, a generalization of the celebrated Furstenberg–Zimmer structure theorem to the case of measure-preserving actions of arbitrary groups on arbitrary probability spaces is established.

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来源期刊

Analysis Mathematica MATHEMATICS-

CiteScore

1.00

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Traditionally the emphasis of Analysis Mathematica is classical analysis, including real functions (MSC 2010: 26xx), measure and integration (28xx), functions of a complex variable (30xx), special functions (33xx), sequences, series, summability (40xx), approximations and expansions (41xx). The scope also includes potential theory (31xx), several complex variables and analytic spaces (32xx), harmonic analysis on Euclidean spaces (42xx), abstract harmonic analysis (43xx). The journal willingly considers papers in difference and functional equations (39xx), functional analysis (46xx), operator theory (47xx), analysis on topological groups and metric spaces, matrix analysis, discrete versions of topics in analysis, convex and geometric analysis and the interplay between geometry and analysis.