Existence of contractive projections on preduals of JBW*-triples
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In 1965, Ron Douglas proved that if X is a closed subspace of an L1-space and X is isometric to another L1-space, then X is the range of a contractive projection on the containing L1-space. In 1977 Arazy-Friedman showed that if a subspace X of C1 is isometric to another C1-space (possibly finite dimensional), then there is a contractive projection of C1 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. © 2011 Hebrew University Magnes Press.