Splittings of Banach spaces induced by Clifford algebras

Abstract:

Let $H$ be an infinite-dimensional Hilbert space of density character $\mathfrak {m}$. By representing $H$ as a module over an appropriate Clifford algebra, it is proved that $H$ possesses a family $\{A_{\alpha }\}_{\alpha \in \mathfrak {m}}$ of proper closed nonzero subspaces such that \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 $L_{p}$ spaces and for $c_{0}(X)$ and $\ell _{p}(X)$ ($1 \le p \le \infty$) when $X$ is an arbitrary nonzero Banach space.