Abstract
In this paper factorization techniques are introduced into the study of tensor product mappings to complete and improve on some results obtained by the author in an earlier paper [Tensor product mappings, Math. Ann. 188 (1970), 1-12. MR 44 #2052]. The main results are as follows: Let $\alpha$ be any $\otimes$-norm. Then (i) if S is absolutely summing and T is an integral operator then $S{ \otimes _\alpha }T$ is absolutely summing, (ii) if S is quasi-nuclear and T is nuclear then $S{ \otimes _\alpha }$ T is quasi-nuclear, (iii) if S and T are integral operators then $S{ \otimes _\alpha }T$ is integral. That the results (i) and (ii) are essentially the best possible was shown by examples in the earlier quoted paper. Also, the methods developed in this paper yield a much simpler proof of the main result of the earlier paper.
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