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In her thesis The Brauer Group of Graded Continuous Trace $C^\ast$-Algebras (cf. Proposition 3.4), Ellen Maycock described the Brauer group of graded continuous trace $C^\ast$-algebras with spectrum a space $X$. It is isomorphic to the set $H^1(X;\mathbb{Z}/2)\times H^3(X;\mathbb{Z})$ with the group law $(a,b)\ast(a',b')=(a+a',b+b'+\beta(a\cup a'))$, where $\beta$ is the Bockstein homomorphism.

On the other hand, El-kaïoum Mohamed Moutuou, in his thesis Twisted groupoid $KR$-theory (cf. Proposition 4.9.3), showed that (up to replacing topological groupoids with spaces) the Brauer group of real graded $C^\ast$-algebras over $X$ is $H^0(X;\mathbb{Z}/8)\oplus(H^1(X;\mathbb{Z}/2)\times H^2(X;\mathbb{Z}/2))$ where the right hand summand has this funny group structure $(a,b)\ast(a',b')=(a+a',b+b'+(a\cup a'))$.

Both Maycock and Moutuou describe relationships between their Brauer groups and the graded Brauer groups of Donovan and Karoubi (see Graded Brauer groups and $K$-theory with local coefficients) and mention that the difference, essentially, is that they allow infinite dimensional algebras, which accounts for the lack of taking torsion in $H^3(X;\mathbb{Z})$ and maybe something about using Čech cohomology (I'm still a bit confused on this point). However, Maycock never seems to address the fact that her description does not see the $H^0(X;\mathbb{Z}/2)$ that Donovan and Karoubi's Brauer group does. What happened to the $H^0(X;\mathbb{Z}/2)$? For bonus points, what precisely would the "connected" version of Moutuou's group, namely $H^1(X;\mathbb{Z}/2)\times H^2(X;\mathbb{Z}/2)$, describe over $X$ in terms of $C^\ast$-algebras?

I think this is probably a minor discrepancy which is easily explained by experts in the field, but I am not familiar with the relevant literature, nor even the basic concepts, so I thought I'd see if anyone could help me understand.

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  • $\begingroup$ The first one looks complex and the second one looks real to me. In general a Brauer group should look something like $H^2(X, \mathcal{O}_X^{\times})$, yes? If we take real-valued functions we get something that looks like $H^2(X, \mathbb{R}^{\times}) \cong H^2(X, \mathbb{Z}_2)$ and if we take complex-valued functions we get something that looks like $H^2(X, \mathbb{C}^{\times}) \cong H^3(X, \mathbb{Z})$. Then these are related by a complexification map corresponding to the Bockstein. $\endgroup$
    – Qiaochu Yuan
    Commented Jul 27, 2022 at 22:13
  • $\begingroup$ @QiaochuYuan yes. That's all true. Maycock's classifies Morita classes of complex C*-algebras and Moutuou's does the same for real C*-algebras. However, both also look a lot like the Donovan-Karoubi Brauer groups except that Maycock's is missing the $H^0(X;\mathbb{Z}/2)$ that it should have. $\endgroup$
    – Jonathan Beardsley
    Commented Jul 28, 2022 at 4:38
  • $\begingroup$ I should mention that there is a real geometric relationship between the C*-algebra picture and Donovan-Karoubi one given by going between C*-algebras with spectrum X and some kind of Hilbert bundle over X. I don't understand this story well, but it's not just some coincidental isomorphism. $\endgroup$
    – Jonathan Beardsley
    Commented Jul 28, 2022 at 4:41

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