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?


1 Answer 1


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.

  • 3
    $\begingroup$ Shouldn't I lose point here, for having an intermittent memory and answering my own question? $\endgroup$ Dec 20, 2013 at 16:31

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