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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)