Complemented subspaces of products of Banach spaces

Abstract:

It is proved that: (i) every complemented subspace in an infinite product of ${L_1}$-predual Banach spaces $\prod \nolimits _{i \in I} {{X_i}}$ is isomorphic to $Z \times {{\mathbf {K}}^\mathfrak {m}}$, where $\dim {\mathbf {K}} = 1,\;\mathfrak {m} \leqslant \operatorname {card} I$ and $Z$ is isomorphic to a complemented subspace of $\prod \nolimits _{i \in J} {{X_i},\;J \subseteq I,\;Z}$ contains an isomorphic cop[ill] of $c_0^{\operatorname {card} J}$; (ii) every injective lcs (in particular, Fréchet) is of the form $Z \times {{\mathbf {K}}^\mathfrak {m}},\;\dim {\mathbf {K}} = 1$, where $Z$ has a fundamental family of seminorms of the cardinality $\tau$ and $Z$ contains an isomorphic copy of $l_\infty ^\mathfrak {r}$ (this is a generalization of Lindenstrauss’ theorem on injective Banach spaces); (iii) whenever $X \simeq {l_p},\;1 \leqslant p \leqslant \infty$, or $X \simeq {c_0}$, then every complemented subspace in a power ${X^\mathfrak {m}}$ ($\mathfrak {m}$ is an arbitrary cardinal number) is isomorphic to ${X^\mathfrak {r}} \times {{\mathbf {K}}^\mathfrak {s}},\;\mathfrak {r} + s \leqslant \mathfrak {m}$ (a generalization of the results due to Lindenstrauss and Pełczyński for $\mathfrak {m} = 1$).