[math] \newcommand{\mathds}{\mathbb}[/math]

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Develop a theory of [math]*[/math]-algebras [math]A[/math] for which the quantity

[[math]] ||a||=\sqrt{\sup\left\{\lambda\in\mathbb C\Big|aa^*-\lambda\notin A^{-1}\right\}} [[/math]]

defines a norm, for the elements [math]a\in A[/math].