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 states that ${\rm C}^{\star}(X)$, the ${\rm C}^{\star}$-algebra generated by $X$, is the Toeplitz algebra.

We are interested in an extension of Coburn's theorem. Consider $Y \in B(H)$ such that $Y^{\star}Y = id+p$, with $p \in B(H)$ a projection, and $YY^{\star} \neq id$.

Question: Is ${\rm C}^{\star}(Y)$ always the Toeplitz algebra? If not, 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).