  Generalizations of the *-product (e.g. n-ary *_n operations) appear in
various places in the discussion of noncommutative gauge theories. These
include the one-loop effective action of noncommutative gauge theories, the
couplings between massless closed and open string modes, and the Seiberg-Witten
map between the ordinary and noncommutative Yang-Mills fields. We propose that
the natural way to understand the *_n operations is through the expansion of an
open Wilson line. We establish the connection between an open Wilson line and
the *_n operations and use it to: (I) write down a gauge invariant effective
action for the one-loop F^4 terms in the noncommutative N=4 SYM theory; (II)
find the gauge invariant couplings between the noncommutative SYM modes and the
massless closed string modes in flat space; (III) propose a closed form for the
Seiberg-Witten map in the U(1) case.
