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Splittings of Banach spaces induced by Clifford algebras

Authors:	N L Carothers S J Dilworth David Sobecki

Institution:	Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43402 ; Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208 ; Department of Mathematics, Miami University, Hamilton, Ohio 45014

Abstract:	Let be an infinite-dimensional Hilbert space of density character $\mathfrak{m}$ . By representing as a module over an appropriate Clifford algebra, it is proved that 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 is an arbitrary nonzero Banach space.

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