A compact quantum semialgebra generated by an isometry

In this paper we construct a compact quantum semigroup structure on a Toeplitz algebra. We prove the existence of a subalgebra in the dual algebra isomorphic to the algebra of regular Borel measures on a circle with the convolution product. We also prove the existence of a Haar functionals in the dual algebra and in the subalgebra mentioned above. We show that this compact quantum semigroup contains a dense subalgebra with the structure of a weak Hopf algebra.