The classical definition of a $C^{\ast}$-algebra is a Banach algebra with an isometric antilinear involution map $a \mapsto a^\ast$. What would be a good definition for a condensed $C^{\ast}$-algebra? How to make sense of properties like $\|a \cdot a^{\ast}\| = \|a\|^2$ and $\|a \cdot b\| \leq \|a\| \cdot \|b\|$?

Edit: Following the comment by Prof Simon Henry. A definition of condensed Banach algebra not using the norm would be nice.

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