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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Splittings of Banach spaces
induced by Clifford algebras


Authors: N. L. Carothers, S. J. Dilworth and David Sobecki
Journal: Proc. Amer. Math. Soc. 128 (2000), 1347-1356
MSC (1991): Primary 46B20
Published electronically: October 18, 1999
MathSciNet review: 1670426
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Abstract | References | Similar Articles | Additional Information

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.


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Additional Information

N. L. Carothers
Affiliation: Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43402
Email: carother@math.bgsu.edu

S. J. Dilworth
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: dilworth@math.sc.edu

David Sobecki
Affiliation: Department of Mathematics, Miami University, Hamilton, Ohio 45014
Email: sobeckdm@muohio.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05374-5
PII: S 0002-9939(99)05374-5
Received by editor(s): June 19, 1998
Published electronically: October 18, 1999
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society