A function from a closed interval [a, b] to a Banach space X is regulated if all one-sided limits exist at each point of the interval. A function $$\alpha$$ from [a, b] to the space of all bounded linear transformations from X to a Banach space Y is an integrator for the regulated functions if, for each regulated function f, the Riemann-Stieltjes sums of f, with sampling points from the interiors of subintervals, converge to a vector in Y. When X and Y are Banach ($$C^{^{*}}$$-)algebras, we give a complete description of the class of all integrators that induce Banach (resp. $$C^{^{*}}$$-)algebra homomorphisms. Each multiplicative integrator is associated with a nested family of idempotents (resp. selfadjoint projections). The main result of Fernandes and Arbach (Ann Funct Anal 3(2):21–31, 2012) exhibits a very special subclass of such integrators (which have $$\left\{ 0 \preceq \mathbb {1} _{_{\mathcal {B}}}\right\}$$ as the associated family of projections).

中文翻译：

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Banach 代数值调节函数的积分器诱导同态

从闭区间 [a, b] 到 Banach 空间 X 的函数如果在区间的每一点都存在所有单边极限，则该函数被调节。从 [a, b] 到从 X 到 Banach 空间 Y 的所有有界线性变换的空间的函数 $$\alpha$$ 是调节函数的积分器，如果对于每个调节函数 f，Riemann-Stieltjes 和f，用子区间内部的采样点，收敛到Y中的一个向量。当X和Y是Banach($$C^{^{*}}$$-)代数时，我们给出了类的完整描述所有诱导 Banach（分别为 $$C^{^{*}}$$-）代数同态的积分器。每个乘法积分器都与一个嵌套的幂等族（分别是 selfadjoint 投影）相关联。Fernandes 和 Arbach (Ann Funct Anal 3(2):21–31,