Compactification of perception pairs and spaces of group equivariant non-expansive operators

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2026-02-01 Epub Date: 2025-11-24 DOI:10.1016/j.topol.2025.109668

Faraz Ahmad

引用次数: 0

Abstract

We study some metric and topological properties of spaces of equivariant operators. We define the notions of a compact perception pair, compactification of a perception pair, and compactification of a space of group equivariant non-expansive operators. We prove that every perception pair with totally bounded space of measurements, which is also rich enough to endow the common domain with a metric structure, can be isometrically embedded in a compact perception pair. Likewise, we prove that if the images of group equivariant non-expansive operators in a given space form a cover for their common codomain, then the space of such operators can be isometrically embedded in a compact space of group equivariant non-expansive operators, such that the new reference perception pairs are compactifications of the original ones having totally bounded data sets. Meanwhile, we state some compatibility conditions for these embeddings and show that they too are satisfied by our constructions.

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群等变非膨胀算子的感知对与空间的紧化

研究了等变算子空间的一些度量和拓扑性质。定义了紧致感知对、紧致感知对和紧致群等变非膨胀算子空间的概念。我们证明了每一个具有完全有界测量空间的感知对都可以等距嵌入到紧致感知对中，并且这些感知对也足够丰富，足以赋予公共域一个度量结构。同样地，我们证明了如果给定空间中的群等变非膨胀算子的像形成了它们的公共上域的覆盖，那么这些算子的空间可以等距嵌入到群等变非膨胀算子的紧化空间中，使得新的参考感知对是具有完全有界数据集的原始参考感知对的紧化。同时，我们提出了这些嵌入的一些兼容条件，并表明我们的结构也能满足这些条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.