Splittings of Banach spaces induced by Clifford algebras

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

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.