  We introduce notions of compactness and weak compactness for multilinear maps
from a product of normed spaces to a normed space, and prove some general
results about these notions. We then consider linear maps $T:A\to B$ between
Banach algebras that are "close to multiplicative" in the following senses: the
failure of multiplicativity, defined by $S_T(a,b)=T(a)T(b)-T(ab)$, is compact
[respectively weakly compact]. We call such maps cf-homomorphisms [respectively
wcf-homomorphisms]. We also introduce a number of other, related definitions.
We state and prove some general theorems about these maps when they are
bounded, showing that they form categories and are closed under inversion of
mappings and we give a variety of examples. We then turn our attention to
commutative $C^*$-algebras and show that the behaviour of the various types of
"close-to-multiplicative" maps depends on the existence of isolated points.
Finally, we look at the splitting of Banach extensions when considered in the
category of Banach algebras with bounded cf-homomorphisms [respectively
wcf-homomorphisms] as the arrows. This relates to the (weak) compactness of
2-cocycles in the Hochschild-Kamowitz cohomology complex. We prove "compact"
analogues of a number of established results in the Hochschild-Kamowitz
cohomology theory.
