an:05702393 Zbl 1201.47038 Heath, Matthew J. Compact failure of multiplicativity for linear maps between Banach algebras. EN Banach J. Math. Anal. 3, No. 2, 125-141, electronic only (2009). ISSN 1735-8787/e 2009

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*47B48 46H05 46J10 46H25 Banach algebra; compact multilinear map; Banach extension Let $\cal A$ and $\cal B$ be Banach algebras and let $T: \cal A \to \cal B$ be a linear map. Set $$S_T (a, b)= T(ab)- T(a) T(b), \quad a, b \in \cal A.$$ ? For $\delta > 0$, the map $T$ is called $\delta$-multiplicative if $ \| S_T \| < \delta$. In {\it B.\,E.\thinspace Johnson} [J.~Lond.\ Math.\ Soc., II.\ Ser.\ 34, 489--510 (1986; Zbl 0625.46059)] and [J.~Lond.\ Math.\ Soc., II.\ Ser.\ 37, No.\,2, 294--316 (1988; Zbl 0652.46031)], pairs $(\cal A, \cal B)$ which are AMNM (almost multiplicative bounded linear maps are near multiplicative bounded linear maps) were investigated. Since then, many authors have contributed to the study of these algebras. ? ? In the paper under review, the author considers other concepts of smallness of $S_T$. First of all, he introduces notions of compactness and weak compactness for multilinear maps from a product of normed spaces to a topological space. As pointed out by the author, compact multilinear maps were also considered in the normed case by {\it N.\,Krikorian} [Proc.\ Am.\ Math.\ Soc.\ 33, 373--376 (1972; Zbl 0235.46068)]. Now a map $T$ is said to be a cf-homomorphism (compact from homomorphism) if $S_T$ is compact. ? ? On the other hand, $T$ is called semi-cf-homomorphism if, for each $a \in \cal A$, $S_T (a, \cdot)$ and $S_T (\cdot, a)$ are compact linear maps. In a similar way, the author defines weakly compact, $n$-dimensional (resp., semi weakly compact, semi $n$-dimensional) from homomorphism. ? ? The author studies general properties of such maps. Moreover, he gives a characterization of some Banach function algebras where such maps are automatically multiplicatives. Finally, the paper is concluded with generalizations of some results in the Hochschild-Kamowitz cohomology theory. Nadia Boudi (Meknes) Zbl 0625.46059; Zbl 0652.46031; Zbl 0235.46068