Let $F/K$ be an abelian extension of number fields with $F$ either CM or
totally real and $K$ totally real. If $F$ is CM and the Brumer-Stark conjecture
holds for $F/K$, we construct a family of $G(F/K)$--equivariant Hecke
characters for $F$ with infinite type equal to a special value of certain
$G(F/K)$--equivariant $L$-functions. Using results of Greither-Popescu on the
Brumer-Stark conjecture we construct $l$-adic imprimitive versions of these
characters, for primes $l> 2$. Further, the special values of these $l$-adic
Hecke characters are used to construct $G(F/K)$-equivariant
Stickelberger-splitting maps in the $l$-primary Quillen localization sequence
for $F$, extending the results obtained in 1990 by Banaszak for $K = \Bbb Q$.
We also apply the Stickelberger-splitting maps to construct special elements in
the $l$-primary piece $K_{2n}(F)_l$ of $K_{2n}(F)$ and analyze the Galois
module structure of the group $D(n)_l$ of divisible elements in $K_{2n}(F)_l$,
for all $n>0$. If $n$ is odd and coprime to $l$ and $F = K$ is a fairly general
totally real number field, we study the cyclicity of $D(n)_l$ in relation to
the classical conjecture of Iwasawa on class groups of cyclotomic fields and
its potential generalization to a wider class of number fields. Finally, if $F$
is CM, special values of our $l$-adic Hecke characters are used to construct
Euler systems in the odd $K$-groups with coefficients $K_{2n+1}(F, \Bbb
Z/l^k)$, for all $n>0$. These are vast generalizations of Kolyvagin's Euler
system of Gauss sums and of the $K$-theoretic Euler systems constructed in
Banaszak-Gajda when $K = \Bbb Q$.Comment: 38 page