We introduce Hilbert algebras I(P, w) as analogues of the combinatoria
l incidence algebras of Doubilet, Rota and Stanley. This leads to inte
resting Banach algebra contexts in which to view certain important com
binatorial functions such as the zeta and Mobius functions. The automo
rphisms and derivations of these Hilbert algebras are completely descr
ibed. The algebras I(P, w) are used to construct a broad new class of
reflexive strictly cyclic operator algebras on Hilbert space.