On the extension of the Toeplitz algebra

We study the $C^*$-extensions of the Toeplitz algebras with the assistance of isometric operators. It is shown that in the case when the Toeplitz algebra acts irreducibly, all such $C^*$-extensions generate the same algebra, i.e., there is no non-trivial extension of the Toeplitz algebra. We provide the examples of the non-trivial extensions of the Toeplitz algebra in the case when its representation is reducible.