We introduce the notion of additive units (addits) of a pointed inclusion system of Hilbert modules over the \(C^*\)-algebra of all compact operators acting on a Hilbert space G. By a pointed inclusion system, we mean an inclusion system with a fixed normalised reference unit. We prove that if G is a Hilbert space of finite dimension, then there is a bijection between the set of addits of a pointed inclusion system and the set of addits of the generated product system. We also consider addits of spatial product systems of Hilbert modules and, as an example, we find all continuous addits in the product system from Barreto et al. (J Funct Anal 212:121–181, 2004, Example 4.2.4).

中文翻译：

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$$ C ^ * $$ C ∗-紧算子的代数上的希尔伯特模块包含系统的加法单元

我们在作用于希尔伯特空间G的所有紧致算子的\（C ^ * \）-代数上引入希尔伯特模块的有尖包含系统的加法单元（加法）的概念。尖锐的包含系统是指具有固定的标准化参考单位的包含系统。我们证明如果G是有限维的希尔伯特空间，那么在尖锐包含系统的加法集合与生成的乘积系统的加法集合之间存在双射。我们还考虑了希尔伯特模块的空间乘积系统的加法，例如，我们从Barreto等人的乘积系统中找到了所有连续加法。（J Funct Anal 212：121-181，2004，Example 4.2.4）。