  In this paper we construct a compact quantum semigroup structure on the
Toeplitz algebra $\mathcal{T}$. The existence of a subalgebra, isomorphic to
the algebra of regular Borel's measures on a circle with convolution product,
in the dual algebra $\mathcal{T}^*$ 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 $\mathcal{T}$ and the structure of weak Hopf
algebra.

Quantum Algebra;Functional Analysis;Operator Algebras