an: Zbl 1201.47038
au: Heath, Matthew J.
ti: Compact failure of multiplicativity for linear maps between Banach algebras.
la: EN
so: Banach J. Math. Anal. 3, No. 2, 125-141, electronic only (2009).
py: 2009
dt: J
cc: *47B48 46H05 46J10 46H25
ut: Banach algebra; compact multilinear map; Banach extension
ab: 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.
rv: Nadia Boudi (Meknes)
ci: Zbl 0625.46059; Zbl 0652.46031; Zbl 0235.46068