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I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu.

To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All I know about $C^*$-algebras is their definition and one or two results.

Can some one suggest me some other reference where there is some (partially) detailed explanation of appearance (and necessity) of $C^*$-algebras in the study of Lie groupoids/differentiable Stacks?

Is there any set up of special case of Lie groupoids, say manifolds, where the appearance of $C^*$-algebras is already a standard notion? Any references for this would also be very useful.

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  • $\begingroup$ It is surprising that there is no tag for $C^*$-algebras. $\endgroup$
    – Praphulla Koushik
    Commented Oct 12, 2019 at 13:40
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    $\begingroup$ Praphulla Koushik: the tag is c-star-algebras (one should avoid asterisks as they are usually reserved as special characters by parsers) $\endgroup$
    – Yemon Choi
    Commented Oct 13, 2019 at 1:21
  • $\begingroup$ @YemonChoi thank you for the tag suggestion... edited accordingly $\endgroup$
    – Praphulla Koushik
    Commented Oct 13, 2019 at 3:00
  • $\begingroup$ @mike miller : As there is a restriction on number of tags, I have to remove that operator algebras tag and add c star algebras $\endgroup$
    – Praphulla Koushik
    Commented Oct 13, 2019 at 3:01

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  1. Consider first the case of a manifold $M$ seen as the space of units of the groupoid structure $M\to M$ where $s=t=id_M$ is the projection and no pairs are composable, so that you have only identities. Then the corresponding groupoid $C^*$-algebra is nothing but the standard $C^*$-algebra of continuous functions on $M$.
  2. Take now $G$ to be a (Lie) group seen as a groupoid over a point. Then the groupoid $C^*$-algebra is the group algebra of $G$ which can be seen as a completion of the universal enveloping algebras of the Lie algebra $\mathfrak g=Lie(G)$.

Generally speaking the groupoid $C^*$ algebra can be considered as a mixture of this two aspects: continuous functions on units plus additional informations on group of symmetries resting over orbits. Examples that can be thought of this kind: orbifolds (can be seen as groupoids), group actions, foliations. To each such object you may associate a Lie groupoid and a corresponding $C^*$-algebra which carries informations on orbits (points of the orbifold,orbits of the action, leaves) and their internal symmetries (discrete group over singular points, isotropy of orbits, holonomy of leaves). This is why the groupoid $C^*$-algebra is sometimes referred to as a desingularization of a singular space, such as the space of leaves or the space of orbits.

To any topological groupoid + a choice of Haar system you can build a groupoid $C^*$-algebra. If your groupoid is Lie you basically have a canonical choice of Haar system essentially unique.

There is a very nice paper by Ieke Moerdijk on Orbifolds as groupoids that you can easily find online. Also searching for "foliation C^* algebra" will provide you way too many refs. One I like is the following http://folk.uio.no/rognes/higson/zurich.pdf

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  • $\begingroup$ Sir, thanks for the response.. I understand the first point... For the second point, I only know group ring/algebra of a discrete group... The corresponding definition for Lie group is interesting :) I will read more about it... Yes, it makes perfect sense that how we associate $C^*$-algebras for manifolds seen as Lie groupoids and Lie groups seen as Lie groupoids.. :) $\endgroup$
    – Praphulla Koushik
    Commented Feb 26, 2020 at 0:42
  • $\begingroup$ Your comment about associating a $C^*$-algebra for an arbitrary Lie groupoid is almost clear.. I just need to read it for some more times.. I knew what it means to say a Haar measure on a (LC)topological group and knew that upto (mul.constant) there exists unique (left) measure on a Lie group... I have not seen the notion of Haar measure on Lie groupoid.. I will read about that :) $\endgroup$
    – Praphulla Koushik
    Commented Feb 26, 2020 at 0:51
  • $\begingroup$ I have seen the paper Orbifolds as groupoids multiple times and now also I looked at it; it does not contain anything about C^*-algebras. Was it for some other purpose you suggested that paper :O Thanks for the Nigel Higson's lecture notes.. I will read and ask if I have any further questions :) $\endgroup$
    – Praphulla Koushik
    Commented Feb 26, 2020 at 1:00

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