Pure states on O-d

We study representations of the Cuntz algebras O-d and their associated dec ompositions. In the case that these representations are irreducible, their restrictions to the gauge-invariant subalgebra UHFd have an interesting cyc lic structure. If S-i, 1 less than or equal to i less than or equal to d, a re representatives of the Cuntz relations on a Hilbert space H, special att ention is given to the subspaces which are invariant under S-i*. The applic ations include wavelet multiresolutions corresponding to wavelets of compac t support (to appear in the later paper [8]), and finitely correlated state s on one-dimensional quantum spin chains.