Let $H$ be a Hilbert space, and $X \in B(H)$ a proper isometry (i.e. $X^{\star}X = id$ and $XX^{\star} \neq id$). [Coburn's theorem][1] states that ${\rm C}^{\star}(X)$, the ${\rm C}^{\star}$-algebra generated by $X$,  is the [Toeplitz algebra][2].  

*Question*: Can we extend $X^{\star}X = id$ to $X^{\star}X = id+p$, with $p \in B(H)$ a projection?  
If ${\rm C}^{\star}(X)$ is then not always the Toeplitz algebra, is it still a ${\rm C}^{\star}$-algebra of type ${\rm I}$? 

*Motivation*: I'm mainly interested in a shift on a trivalent directed tree with one parent and one or two children for each vertex, such that the set of vertices with one child and the set of vertices with two children are both infinite (it is a generalization of the operator $S^{\star}$ of [this post][3]).


  [1]: https://doi.org/10.1090%2FS0002-9904-1967-11845-7
  [2]: https://en.wikipedia.org/wiki/Toeplitz_algebra
  [3]: https://mathoverflow.net/q/288941/34538