Existence of contractive projections on preduals of JBW*-triples

Abstract

In 1965, Ron Douglas proved that if X is a closed subspace of an L 1 -space and X is isometric to another L 1 -space, then X is the range of a contractive projection on the containing L 1 -space. In 1977 Arazy-Friedman showed that if a subspace X of C 1 is isometric to another C 1 -space (possibly finite dimensional), then there is a contractive projection of C 1 onto X. In 1993 Kirchberg proved that if a subspace X of the predual of a von Neumann algebra M is isometric to the predual of another von Neumann algebra, then there is a contractive projection of the predual of M onto X.We widen significantly the scope of these results by showing that if a subspace X of the predual of a JBW * -triple A is isometric to the predual of another JBW * -triple B, then there is a contractive projection on the predual of A with range X, as long as B does not have a direct summand which is isometric to a space of the form L ∞ (Ω, H), where H is a Hilbert space of dimension at least two. The result is false without this restriction on B.

Extending representations of Banach algebras to their biduals

“…It follows that for each p, the class of L p -spaces (of QL p -spaces, of SL pspaces, of QSL p -spaces) is closed under passing to 1-complemented subspaces. For an overview of classes of Banach spaces with this property, see [NR11].…”

Completely contractive projections on operator algebras

“…the subcategory of TRO's and completely contractive projections is projectively stable by Youngson's theorem [50]. In the cited pages of [44] the concept of a 'projectively rigid category' is discussed. The associated question for us would be if the preduals of dual operator algebras are projectively (completely) rigid in their sense.…”