We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of $C^{\ast }$‐algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved $C^{\ast }$‐algebras.

中文翻译：

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超乘C *-代数的Cuntz半群

我们证明抽象Cuntz半群的类别是双完全的。因此，该类别承认产品和超产品。我们进一步证明了一个族的（超）乘积的按比例缩放的Cuntz半群$C^{\ast }$-代数与所涉及的按比例缩放的Cuntz半群的（超）乘积一致 $C^{\ast }$-代数