In his work on P-partitions, Stembridge defined the algebra of peak
functions Pi which is both a
subalgebra and a retraction of the algebra of quasi-symmetric functions. We
show that Pi is closed under
coproduct, and therefore a Hopf algebra, and describe the kernel of the
retraction. Billey and Haiman, in
their work on Schubert polynomials, also defined a new class of
quasi-symmetric functions --- shifted quasi-symmetric functions --- and we
show that
Pi is strictly contained in the
linear span Xi of shifted quasi-symmetric functions. We show that
Xi is a coalgebra, and compute the rank of the n-th graded component.