C⁎-subproduct and product systems

Abstract

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 admits 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⁎-subproduct 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.