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Using the device of the tensor product of      C ∗  {C^ \ast }      -algebras, we study the      C ∗  {C^ \ast }      -algebra generated by the Toeplitz operators on the quarter-plane. We obtain necessary and sufficient conditions for such an operator to be Fredholm, but show in this case that not all such operators are invertible.

On the 𝐶*-algebra of Toeplitz operators on the quarterplane