C⁎-subproduct and product systems

We introduce and study two-parameter subproduct product systems of C⁎-algebras as the operator-algebraic analogues of, in relation to, Tsirelson's Hilbert spaces. Using several inductive limit techniques, we show that (i) any C⁎-subproduct system can be dilated to a C⁎-product system; (ii) admits unit, i.e., co-multiplicative family projections, assembled into C⁎-algebra, which comes equipped with one-parameter comultiplication-like homomorphisms. also discuss co-units systems, consisting families states, they correspond idempotent states associated C⁎-algebras. then use GNS construction obtain Tsirelson spaces from co-units, describe relationship between dilation co-unit. All these results are illustrated concretely at level commutative