Contractively complemented subspaces of pre-symmetric spaces

الأصل · EN

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.