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Extension of completely bounded \(C^*\)-module homomorphisms. (English) Zbl 0535.46003
Operator algebras and group representations, Proc. int. Conf., Neptun/Rom. 1980, Vol. II, Monogr. Stud. Math. 18, 238-250 (1984).
[For the entire collection see Zbl 0515.00017.]
Continuing earlier research concerning generalizations of the Hahn-Banach theorem [as in J. Funct. Anal. 40, 127-150 (1981; Zbl 0495.46005)] the author presents the following results: Let \({\mathfrak A}\) be an injective \(C^*\)-algebra, \({\mathfrak B}\subseteq {\mathfrak A}\) a unital \(C^*\)- subalgbra, E a matricial normed \({\mathfrak B}\)-bimodule [resp. left \({\mathfrak B}\)-module], \(F\subset E\) a \({\mathfrak B}\)-bi-[left] submodule and \(\phi\) :\(F\to {\mathfrak A}\) a completely bounded \({\mathfrak B}\)-module homomorphism. Then there exists a completely bounded \({\mathfrak B}\)-bi-[left] module homomorphism \({\tilde \phi}\):\(E\to {\mathfrak A}\), which extends \(\phi\) preserving the cb-norm. There are also remarks on special cases and results, asserting that in certain situations bounded module- homomorphisms are completely bounded. At the end the author gives a short proof of a (sharpened version of an) extension theorem due to M. Takesaki [Kodai Math. Sem. Reports 12, 1-10 (1960; Zbl 0091.106)], covering the case of a commutative \(AW^*\)-algebra \({\mathfrak A}\), and normed \({\mathfrak B}\)-bi-submodules E and F.
Reviewer: H.G.Feichtinger

MSC:
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46L05 General theory of \(C^*\)-algebras
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)