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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    $\begingroup$ Is there a notion of "condensed Banach algebra" that doesn't involve a norm function? Otherwise I'm not sure what's the problem with the two conditions you are mentioning. $\endgroup$ Mar 13 at 20:31
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    $\begingroup$ The norm play a fairly essential role in the theory of C* algebra so I'm not sure a notion corresponding to just topological vector space is sufficient here. For e.g. injective morphisms of C*-algebras are automatically isometric - I'm not sure one can define a notion of C* algebra without including the norm function as part of the definition - at which point there is no difficulty including the two conditions you are talking about explicitly. $\endgroup$ Mar 13 at 21:03
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    $\begingroup$ Though there are some way to characterize C*-algebra in a way not involving the norm, exploiting the fact that the norm can be defined using the spectral radius of $x^* x$. see for e.g. "a characterization of C* algebras" by Hennings , proceedings of the Edinburgh Mathematical Society (1987) 30, 445-4S3 (cambridge.org/core/services/aop-cambridge-core/content/view/…) $\endgroup$ Mar 13 at 21:06
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    $\begingroup$ I thought a "condensed XYZ" is just a sheaf on the pro-etale site in the category of XYZs. Then a condensed C* algebra would be a sheaf in the category of C* algebras. $\endgroup$
    – Echo
    Mar 13 at 21:17
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    $\begingroup$ @Echo I am not asking for the condensed C-star algebra, but the condensed analogue eg the condensed analogue of top vector space is condensed vector space and not condensed top vector space. $\endgroup$ Mar 13 at 21:22


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