/sci/ - Science & Math

We start with a Hilbert space [math]\mathcal{H}[/math]. Then, we take the free associative complex algebra over the bras and kets and quotient out [math]\langle \phi | * | \psi \rangle \sim Id \langle \phi , \psi \rangle[/math] , where [math]*[/math] is the product in the algebra and the right side is just the inner product.
Is there a name for this construction? The "Dirac Bra-Ket algebra" or something?
Is it useful for anything other than reducing the abuse of notation in QM?