  We construct a lift of Schur's Q-functions to the peak algebra of the
symmetric group, called the noncommutative Schur Q-functions, and extract from
them a new natural basis with several nice properties such as the positive
right-Pieri rule, combinatorial expansion, etc. Dually, we get a basis of the
Stembridge algebra of peak functions refining Schur's P-functions in a simple
way.
