  We show the existence of a unital subalgebra of the symmetric group algebra
linearly spanned by sums of permutations with a common peak set, which we call
the peak algebra. We show that this algebra is the image of the descent algebra
of type B under the map to the descent algebra of type A which forgets the
signs, and also the image of the descent algebra of type D. The peak algebra
contains a two sided ideal which is defined in terms of interior peaks. This
object was introduced in previous work by Nyman; we find that it is the image
of certain ideals of the descent algebras of types B and D introduced in
previous work of N. Bergeron et al. We derive an exact sequence involving the
peak ideal and the peak algebras of degrees $n$ and $n-2$. We obtain this and
many other properties of the peak algebra and its peak ideal by first
establishing analogous results for signed permutations and then forgetting the
signs. In particular, we construct two new commutative semisimple subalgebras
of the descent algebra by grouping permutations according to their number of
peaks or interior peaks. We discuss the Hopf algebraic structures that exist on
the direct sums of these spaces over $n\geq 0$ and explain the connection with
previous work of Stembridge; we also obtain new properties of his
descents-to-peaks map and construct a type B analog.
