  In this article, we use Exel's construction to associate a C*-algebra to
every shift space. We show that it has the C*-algebra defined in [Carlsen and
Matsumoto: Some remarks on the C*-algebras associated with subshifts] as a
quotient, and possesses properties indicating that it can be thought of as the
universal C*-algebra associated to a shift space. We also consider its
representations, relationship to other C*-algebras associated to shift spaces,
show that it can be viewed as a generalization of the universal Cuntz-Krieger
algebra, discuss uniqueness and a faithful representation, provide conditions
for it being nuclear, for satisfying the UCT, for being simple, and for being
purely infinite, show that the constructed algebras and thus their K-theory,
$K_0$ and $K_1$, are conjugacy invariants of one-sided shift spaces, present
formulas for those invariants, and also present a description of the structure
of gauge invariant ideals.
