Isometries between injective tensor products of Banach spaces (Q1074828)

The author investigates isometric isomorphisms between completed injective tensor products of real Banach spaces. The main result is (Theorem 1): Suppose that K and H have strictly convex duals and that T:X\({\hat \otimes}_{\epsilon}K\to Y{\hat \otimes}_{\epsilon}H\) is an isometric isomorphism. Then there exist Banach spaces Z and \(X_ 2\) such that (up to isometries) \[ X=(Z{\hat \otimes}_{\epsilon}H)\oplus_{\infty}X_ 2,\quad Y=(Z{\hat \otimes}_{\epsilon}K)\oplus_{\infty}X_ 2. \] Moreover, the author gives a description of T. As an immediate corollary to Theorem 1, he obtains the ''cancellation law'' (Theorem 2): X\({\hat \otimes}_{\epsilon}H=Y{\hat \otimes}_{\epsilon}H\) and H has strictly convex dual implies \(X=Y.\) The basic tool in the proof of Theorem 1 is an appropriate partition of the set of extreme functionals ex \(B_{(X{\hat \otimes}_{\epsilon}K)'}\), which is known to be (*) ex \(B_{(X{\hat \otimes}_{\epsilon}K)'}=ex B_ X\), \(\otimes ex B_ K\), [cf. \textit{I. I. Tseitlin}, Mat. Zametki 20, 521--527 (1976; Zbl 0346.47039), or \textit{W. M. Ruess} and \textit{C. P. Stegall}, Math. Ann. 261, 535--546 (1982; Zbl 0501.47015)]. Departing from his partition and (*), the author performs an intricate construction which yields the spaces Z and \(X_ 2.\) [Reviewer's remark: The restriction to real Banach spaces in this paper is not necessary, since the characterization (*) is valid for complex spaces, too, cf. Tseitlin's paper or \textit{A. Lima} and \textit{G. Olsen}, Proc. Am. Math. Soc. 94, 437--440 (1985; Zbl 0581.47029)].