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In this paper we construct a compact quantum semigroup structure on the Toeplitz algebra T . The existence of a subalgebra, isomorphic to the algebra of regular Borel's measures on a circle with convolution product, in the dual algebra T * is shown. The existence of Haar functionals in the dual algebra and in the above-mentioned subalgebra is proved. Also we show the connection between T and the structure of weak Hopf algebra.

A C * -algebra generated by a commuting family of isometries is a natural generalization of the Toeplitz algebra. We study the * -automorphisms and invariant ideals of the C * -algebra generated by a semigroup.

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