卷积代数的模型论

中文摘要 AI 辅助

本文研究局部紧群 G 的卷积代数 (L¹(G),*) 的模型论，将其视为配备卷积乘积 * 的巴拿赫格。我们首先证明当基础群 G 离散时，初等等价和初等嵌入的转移原理：即 (ℓ¹(G),*) ≡ (ℓ¹(H),*) 蕴含 G ≡ H，而当 G 和 H 为 ω-饱和时逆命题成立（初等子结构同理）；若无 ω-饱和性，逆命题不成立。尽管纯巴拿赫格在模型论意义上是温顺的，但我们的结果表明，加入卷积会产生剧烈行为。例如，我们证明若 G 是任意局部紧非离散群，则公式 d(x,x*y) 关于 Th(L¹(G),*) 是不稳定的。此外，我们证明若 G 是离散群且包含特定的可和子群构型，则公式 d(x*y,z) ˙− 1/2 见证 Th(ℓ¹(G),*) 的 TP₂ 性质；由此，若 G 包含无限阿贝尔子群，则 Th(ℓ¹(G),*) 具有 TP₂ 性质。我们利用近似单位的概念在局部紧非离散情形下证明了类似结果。最后，我们证明一个“连续-离散结合”逼近定理：连通阿贝尔李群的卷积代数可度量嵌入到有限阿贝尔群卷积代数的超幂中。

英文摘要

This paper deals with the model theory of convolution algebras $(L^1(G),*)$ for locally compact groups $G$, seen as Banach lattices equipped with the convolution product $*$. We first prove transfer principles for elementary equivalence and elementary embeddings when the underlying group $G$ is discrete, namely, $(\ell^1(G),*) \equiv (\ell^1(H),*)$ implies $G \equiv H$, while the converse holds when $G$ and $H$ are $ω$-saturated (likewise for elementary substructures). Without $ω$-saturation, the converse fails. Although pure Banach lattices are model-theoretically tame, our results imply that adding convolution yields wild behavior. For example, we prove that if $G$ is any locally compact, non-discrete group, then the formula $d(x,x*y)$ is unstable with respect to $\mathrm{Th}(L^1(G),*)$. Moreover, we show that if $G$ is discrete and contains a particular configuration of amenable subgroups, then the formula $d(x*y,z)\mathbin{\dot{-}}\frac{1}{2}$ witnesses $\mathrm{TP}_2$ with respect to $\mathrm{Th}(\ell^1(G),*)$. As a consequence, if $G$ contains an infinite abelian subgroup, then $\mathrm{Th}(\ell^{1}(G),*)$ has $\mathrm{TP}_{2}$. We prove similar results in the locally compact non-discrete setting using the notion of an approximate identity. Finally, we prove a `continuous-by-discrete' approximation theorem. Namely, convolution algebras of connected abelian Lie groups admit metric embeddings into ultraproducts of convolution algebras over finite abelian groups.

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34 pages