  For a CM abelian extension $F/K$ of an arbitrary totally real number field
$K$, we construct the Stickelberger splitting maps (in the sense of \cite{Ba1})
for both the \'etale and the Quillen $K$--theory of $F$ and we use these maps
to construct Euler systems in the even Quillen $K$--theory of $F$. The
Stickelberger splitting maps give an immediate proof of the annihilation of the
groups of divisible elements $div K_{2n}(F)_l$ of the even $K$--theory of the
top field by higher Stickelberger elements, for all odd primes $l$. This
generalizes the results of \cite{Ba1}, which only deals with CM abelian
extensions of $\Bbb Q$. The techniques involved in constructing our Euler
systems at this level of generality are quite different from those used in
\cite{BG1}, where an Euler system in the odd $K$--theory with finite
coefficients of abelian CM extensions of $\Bbb Q$ was given. We work under the
assumption that the Iwasawa $\mu$--invariant conjecture holds. This permits us
to make use of the recent results of Greither-Popescu \cite{GP} on the \'etale
Coates-Sinnott conjecture for arbitrary abelian extensions of totally real
number fields, which are conditional upon this assumption. In upcoming work, we
will use the Euler systems constructed in this paper to obtain information on
the groups of divisible elements $div K_{2n}(F)_l$, for all $n>0$ and odd $l$.
It is known that the structure of these groups is intimately related to some of
the deepest unsolved problems in algebraic number theory, e.g. the
Kummer-Vandiver and Iwasawa conjectures on class groups of cyclotomic fields.
We make these connections explicit in the introduction.
