TITLE: Who first identified the universal $C^*$-algebra generated by an idempotent of norm at most $C$?
QUESTION [9 upvotes]: 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?

REPLY [5 votes]: 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.
I am still interested in any earlier reference that might exist.