arXiv 2109.106902021-09math.FAmath.OA

正代数同态的谱定理

Spectral theorems for positive algebra homomorphisms

Marcel de Jeu, Xingni Jiang

备注：: 61 pages. Extended version, with section on JBW-algebras added

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AI总结：

本文证明了局部紧 Hausdorff 空间上正代数同态可由唯一正则谱测度表示，并将其扩展为 $L^1$-空间上的向量格同态，揭示了多种谱定理的共同序理论根源。

摘要：

设 $X$ 为局部紧 Hausdorff 空间，$A$ 为偏序代数，$π\colon \mathrm{C}_{\mathrm c}(X)\to A$ 为正代数同态。在 $A$ 满足诸多实际感兴趣情形的条件时，证明 $π$ 可由 $X$ 的 Borel $σ$-代数上唯一的、取值于 $A$ 中正幂等元的正则谱测度 $μ$ 表示。测度 $μ$ 在序意义下是 $σ$-可加的，通过可追溯到 J.D.M. Wright 并早先由作者们研究过的序积分（Lebesgue 积分的推广）来表示 $π$。正代数同态 $π$ 可从 $\mathrm{C}_{\mathrm c}(X)$ 扩展为从 $μ$ 伴随的 $L^1$-空间到 $A$ 的正线性映射。证明通常该 $L^1$-空间在乘法下封闭，从而为向量格代数；且从 $L^1$ 到 $A$ 的扩展映射不仅是代数同态，即使 $A$ 非向量格，也是文中解释意义上的向量格同态。当 $A$ 具可数上确界性质时，$L^1$（或其正锥）的像可用 $\mathrm{C}_{\mathrm c}(X)$（或其正锥）像的连续上下波动来描述。一般结果应用于三种不同情境，展示各种谱定理的共同序理论根源：Banach 格上的表示、Hilbert 空间上的表示，以及 JBW-代数的谱理论（代数无需由算子构成）。

英文摘要：

Let $X$ be a locally compact Hausdorff space, let $A$ be a partially ordered algebra, and let $π\colon \mathrm{C}_{\mathrm c}(X)\to A$ be a positive algebra homomorphism. Under conditions on $A$ that are satisfied in a good number of cases of practical interest, it is shown that $π$ is represented by a unique regular spectral measure $μ$ on the Borel $σ$-algebra of $X$, taking its values in the positive idempotents in $A$. The measure $μ$, which is $σ$-additive in an ordered sense, represents $π$ via the order integral (a generalisation of the Lebesgue integral) that goes back to J.D.M. Wright and which was investigated earlier by the authors. The positive algebra homomorphism $π$ can be extended from $\mathrm{C}_{\mathrm c}(X)$ to a positive linear map from the accompanying $L^1$-space of $μ$ into $A$. It is shown that, quite often, this $L^1$-space is closed under multiplication, so that it is a vector lattice algebra, and that the extended map from $L^1$ into $A$ is not only an algebra homomorphism but, even when $A$ is not a vector lattice, also a vector lattice homomorphism in a sense that is explained in the paper. When $ A$ has the countable sup property, the image of $L^1$ (or of its positive cone) is described in terms of consecutive ups and downs of the image of ${\mathrm C}_{\mathrm c}(X)$ (or of its positive cone). The general results are applied in three different contexts, showing how various spectral theorems have a common order-theoretical root: representations on Banach lattices, on Hilbert spaces, and (the algebra need not consist of operators) spectral theory for JBW-algebras.

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