A note on the greatest crossnorm

Abstract

Schattenhas shown [5, Lemma 2, p. 323; 6, Lemma 3.7, p. 55] that, if 91 is a closed subspace of a Banacli space Z, and there is a projection of e onto 9Y with bound unity, then the greatest crossnorm on the tensor product 5 091 is an extension of the greatest crossnorm on PE) 0 for any Banach space 9. Now it is known that there is a projection with bound unity of the second conjugate 58 of a Banach space e onto 5o (the canonical image of e in 5 * *) for conjugate spaces e8 and for some others [3, p. 580], though not for all Banach spaces (cf. [7]). For such spaces, then, the greatest crossnorm on 5 * *091 is an extension of the greatest crossnorm on o E0 91. The purpose of this note is to show that the restriction to such spaces is unnecessary. (N.B. e is sometimes embedded in 5** by identifying it with 0o.)