Infinite-dimensional Compact Quantum Semigroup
  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. 
Marat A. Aukhadiev, Suren A. Grigoryan, Ekaterina V. Lipacheva
math.QA math.FA math.OA