$C^{\ast }$-subproduct and product systems

Abstract

We introduce and study two-parameter subproduct and product systems of $C^{\ast }$-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 $C^{\ast }$-subproduct system can be dilated to a $C^{\ast }$-product system; and (ii) any $C^{\ast }$-subproduct system that admis a unit, i.e., a co-multiplicative family of projections, can be assembled into a $C^{\ast }$-algebra, which comes equipped with a one-parameter family of comultiplication-like homomorphisms. We also introduce and discuss co-units of $C^{\ast }$-subproduct systems, consisting of co-multiplicative families of states, and show that they correspond to idempotent states of the associated $C^{\ast }$-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 $C^{\ast }$-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 $C^{\ast }$-subproduct systems of commutative $C^{\ast }$-algebras.

Several changes and corrections have been made