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.)
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