Splittings of Banach spaces induced by Clifford algebras

Abstract

Let H H be an infinite-dimensional Hilbert space of density character m \mathfrak {m} . By representing H H as a module over an appropriate Clifford algebra, it is proved that H H possesses a family { A α } α ∈ m \{A_{\alpha }\}_{\alpha \in \mathfrak {m}} of proper closed nonzero subspaces such that d ( S A α , S A β ) = d ( S A α ⊥ , S A β ) = d ( S A α ⊥ , S A β ⊥ ) = 2 − 2 ( α ≠ β ) . \begin{equation*}d(S_{A_{\alpha }},S_{A_{\beta }})=d(S_{A^{\perp }_{\alpha }},S_{A_{\beta }}) =d(S_{A^{\perp }_{\alpha }},S_{A^{\perp }_{\beta }})=\sqrt {2-\sqrt 2}\qquad (\alpha \ne \beta ).\end{equation*} Analogous results are proved for L p L_{p} spaces and for c 0 ( X ) c_{0}(X) and ℓ p ( X ) \ell _{p}(X) ( 1 ≤ p ≤ ∞ 1 \le p \le \infty ) when X X is an arbitrary nonzero Banach space.