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Let $X$ be a (separated) complex algebraic variety. Then we can view its analytification $\newcommand\topo{\text{top}}X^{\topo}$ as a locally compact Hausdorff space. I wonder whether the same construction exists in the noncommutative world?

Let me phrase a slightly more precise version. Our noncommutative analogues of complex algebraic varieties are $\mathbb C$-linear DG-categories. Our noncommutative analogues of topological spaces are non-unital $C^*$-categories. Thus our question amounts to asking a "forgetful" functor which carries $\mathbb C$-linear DG-categories $\mathcal C$ to $C^*$-categories, denoted by $\mathcal C^{\topo}$. I hope that the following two properties are satisfies:

  1. If the input $\mathcal C$ is the DG-category $\DeclareMathOperator\Perf{Perf}\Perf(X)$ of perfect complexes on $X$ for a complex algebraic variety $X$, then the output $\mathcal C^{\topo}$ is the non-unital $C^*$-algebra $C(X^{\topo},\mathbb C)$ of continuous complex-valued functions on $X^{\topo}$;
  2. The operator $K$-theory of $\mathcal C^{\topo}$ should be the same as the topological $K$-theory of $\mathcal C$ à la Blanc.
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  • $\begingroup$ Are there some finiteness assumptions on your linear DG categories? Otherwise, even in the commutative setting, I'd be a bit surprised to get (locally) compact spaces $\endgroup$
    – Yemon Choi
    Commented yesterday
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    $\begingroup$ @YemonChoi This is a question to understand what is already known. For finiteness assumptions, I would not add on the source, but potentially enlarge the target a bit (I am not sure whether $C^*$-categories are general enough). However, if there is an answer for smooth proper categories, it is also desirable (I do not know how to impose finiteness assumption to incorporate non-smooth but finite type situations). $\endgroup$
    – Z. M
    Commented 23 hours ago
  • $\begingroup$ My knowledge of algebraic geometry is inadequate, but presumably there is some kind of finiteness assumption inside the definition of a complex algebraic variety, in the same way that differentiable manifolds are normally assumed to be modelled on finite-dimensional Euclidean space. This is why "linear DG category" seems much broader to me than "complex algebraic variety". After all, one could take some unpopular additive category such as the category of Banach spaces and bounded linear maps and then take chain complexes to get a linear DG category... $\endgroup$
    – Yemon Choi
    Commented 18 hours ago
  • $\begingroup$ (My point is that the "noncommutative topology dictionary" beloved by those who work with Cstar algebras starts to lose persuasiveness/validity if one goes outside the locally compact setting.) $\endgroup$
    – Yemon Choi
    Commented 18 hours ago
  • $\begingroup$ @YemonChoi Usual analytification is only defined for complex schemes locally of finite type. However, I do not think it essential in the following way: suppose that we have a filtered diagram $(R_i)$ of commutative $\mathbb C$-algebras of finite type with colimit $R$. Then we can define the ring of "continuous functions on its analytification" as the filtered colimit of continuous functions on analytified $\operatorname{Spec}R_i$ — there might be no topological space such that it is the ring of continuous functions on that space on the nose, but such an Ind-$C^*$-algebra still makes sense. $\endgroup$
    – Z. M
    Commented 13 hours ago

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