Large-\( N\) limits of spaces and structures. (English) Zbl 07680182

Alam, Irfan; Sengupta, Ambar N.

Commun. Pure Appl. Anal. 22, No. 4, 1009-1042 (2023).

Summary: For a sequence of spaces \(X_N\), with topological, algebraic or measure-theoretic structures, we show how a large-\(N\) limit \(X_\infty\) with corresponding structures is obtained. For example, when each space is a topological group \(G_N\), such as \(G_N=U(N)\), a limiting group \(G_\infty\) with topology results. Using the Weil-Kodaira construction, for compact topological groups \(G_N\) equipped with normalized Haar measures, we obtain a topological structure on \(G_\infty\) that also makes the group operations continuous. When each \(G_N\) is a Lie group we describe a Lie algebra associated to \(G_\infty\).

MSC:

54J05	Nonstandard topology
28C10	Set functions and measures on topological groups or semigroups, Haar measures, invariant measures
28C20	Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.)

Keywords:

large-\(N\) limits; nonstandard analysis; Loeb measure; Haar measure; ultraproducts

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