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 nonzero projection (so $YY^{\star} \neq id$, see comments).

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 associated operator is non-normal; it is a generalization of the operator $S^{\star}$ of this post. These questions were inspired by the Collatz graph, in particular, the connected component of an eventual counter-example of Collatz Conjecture going to infinity by the Collatz map iteration (such a connected component would be a trivalent directed tree). The Collatz map induces a shift operator which generates an operator algebra (${\rm C}^{\star}$ or ${\rm W}^{\star}$). I wonder whether this operator algebra is interesting in its own right or for an application to number theory.

$\newcommand{\id}{\mathrm{id}}$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 nonzero projection (so $YY^{\star} \neq \id$, see comments).

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 associated operator is non-normal; it is a generalization of the operator $S^{\star}$ of this post. These questions were inspired by the Collatz graph, in particular, the connected component of an eventual counter-example of Collatz Conjecture going to infinity by the Collatz map iteration (such a connected component would be a trivalent directed tree). The Collatz map induces a shift operator which generates an operator algebra (${\rm C}^{\star}$ or ${\rm W}^{\star}$). I wonder whether this operator algebra is interesting in its own right or for an application to number theory.

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 nonzero projection (so $YY^{\star} \neq id$, see comments).

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 associated operator is non-normal; it is a generalization of the operator $S^{\star}$ of this post. These questions were inspired by the Collatz graph, in particular, the connected component of an eventual counter-example of Collatz Conjecture going to infinity by the Collatz map iteration (such a connected component would be a trivalent directed tree). The Collatz map induces a shift operator which generates an operator algebra (${\rm C}^{\star}$ or ${\rm W}^{\star}$). I wonder whether this operator algebra is interested for itself or for an application to number theory.

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 nonzero projection (so $YY^{\star} \neq id$, see comments).

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 associated operator is non-normal; it is a generalization of the operator $S^{\star}$ of this post. These questions were inspired by the Collatz graph, in particular, the connected component of an eventual counter-example of Collatz Conjecture going to infinity by the Collatz map iteration (such a connected component would be a trivalent directed tree). The Collatz map induces a shift operator which generates an operator algebra (${\rm C}^{\star}$ or ${\rm W}^{\star}$). I wonder whether this operator algebra is interesting in its own right or for an application to number theory.