Abstract. We discuss the order-theoretic properties of $CM$-ideals in matricially order smooth $\infty $-normed spaces. We study the relation between $CM$-ideals and $CL$-summands in the matrix duality setup. We introduce the notion of $L^{1}$-matricial split faces in an $L^{1}$-matricially normed space and characterize $CM$-ideals in a matricially order smooth $\infty $-normed space $V$ in terms of the $L^{1}$-matricial split face of the $L^{1}$-matrix convex set $\{Q_{n}(V)\}$.

DOI: 10.14232/actasm-019-259-6

AMS Subject Classification (1991): 46B40; 46L05, 46L30

Keyword(s): operator spaces, operator systems, $CM$-ideals, split faces

Received January 25, 2019 and in final form May 29, 2019. (Registered under 9/2019.)