# Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?

So much is known about hermitian and non-hermitian idempotents in a $C^*$-algebra, that someone must have written down the following.

Theorem The universal $C^*$-algebra generated by one element $x$ subject to the relations $x^2=x$ and $\| x \| \leq C$ is isomorphic to $$\mathcal{I}_D= \left\{ f\in C\left([0,D],\mathbf{M}_{2}(\mathbb{C})\right) \left| f(0)\in\left[\begin{array}{cc} \mathbb{C} & 0\\ 0 & 0 \end{array}\right] \right. \right\}$$ where $D = \sqrt{C-1}$ and the isomorphism is specified by $$x \mapsto \left[\begin{array}{cc} 1 & 0\\ t & 0 \end{array}\right],$$

This is not hard to prove, and makes it easy to see why an idempotent is homotopic to a projection.

Is there a reference for this result, that explicitly mentions a universal $C^*$-algebra or that spells out the universal property?

I almost found it: Example 4.4 in "Presentations and Tietze transformations of C*-algebras" by Will Grilliette, New York J. Math. 18 (2012) 121--137. The generator in the concrete algebra is not explicit, and must not be $$\left[\begin{array}{cc} 1 & 0\\ t & 0 \end{array}\right]$$ as the length of the interval does not come out the same.