Two methods of constructing infinitely many isomorphically distinct
£p-spaces have been published. In this article we show that these construc-
tions yield very different spaces and in the process develop methods for dealing
with these spaces from the isomorphic viewpoint. We use these methods to give
a complete isomorphic classification of the spaces Rp, a uj\ , constructed by
Bourgain, Rosenthal, and Schechtman and to show that Xp ® Xp is not isomor-
phic to a complemented subspace of any Rp. This latter result is a consequence
of a general result concerning complemented embeddings of Xp S Xp into in-
dependent sums which shows that the tensor product cannot be broken into a
(p, 2)—sum. As a technical tool we also develop methods for dealing with gliding
hump arguments in spaces with bases which have many subsequences which span
an isomorph of the space.
1991 Mathematics Subject Classification. Primary 46B20 Secondary 46E30.
Key words and phrases, tensor product, projection, complemented, ordinal index.
Research supported in part by NSF grant DMS-9301506
Received by the editor April 17. 199G.
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