论文标题
$ c^*$ - 子产品和产品系统
$C^*$-subproduct and product systems
论文作者
论文摘要
我们介绍和研究$ c^*$ - 代数的两参数子产品和产品系统，作为运算符的代数类似物，以及Tsirelson的Hilbert Space的两参数产品系统。使用几种归纳限制技术，我们表明（i）任何$ C^*$ - 子产品系统都可以扩张到$ C^*$ - 产品系统； （ii）任何$ c^*$ - 副产品系统，即一个单位，即共同培训的投影家族，都可以组装成$ c^*$ - 代数，该代数配备了一个单参数类似的同性恋类似型的代数。我们还介绍和讨论$ C^*$ - 子产品系统的共同单元，由国家的共同家庭组成，并表明它们与相关的$ c^*$ - 代数的同性状态相对应。然后，我们使用GNS构造从共同组中获得希尔伯特空间的Tsirelson次级产品系统，并描述了$ C^*$ - 超级系统的扩张与Hilbert Space的Tsirelson subproduct System的扩张之间的关系。所有这些结果在$ c^*$ - 可交换$ C^*$ - 代数的$ C^*$ - 子产品系统的水平上进行说明。
We introduce and study two-parameter subproduct and product systems of $C^*$-algebras as the operator-algebraic analogues of, and in relation to, Tsirelson's two-parameter product systems of Hilbert spaces. Using several inductive limit techniques, we show that (i) any $C^*$-subproduct system can be dilated to a $C^*$-product system; and (ii) any $C^*$-subproduct system that admis a unit, i.e., a co-multiplicative family of projections, can be assembled into a $C^*$-algebra, which comes equipped with a one-parameter family of comultiplication-like homomorphisms. We also introduce and discuss co-units of $C^*$-subproduct systems, consisting of co-multiplicative families of states, and show that they correspond to idempotent states of the associated $C^*$-algebras. We then use the GNS construction to obtain Tsirelson subproduct systems of Hilbert spaces from co-units, and describe the relationship between the dilation of a $C^*$-suproduct system and the dilation of the Tsirelson subproduct system of Hilbert spaces associated with a co-unit. All these results are illustrated concretely at the level of $C^*$-subproduct systems of commutative $C^*$-algebras.
