The category of $C^*$-algebras is not abelian (a "proof" that it is pre-abelian can be found here, but it does not seem correct; I can't find any authoritative sources). However, it's possible to do K-theory over the category of $C^*$-algebras, and one can often directly use tricks and techniques from homological algebra, especially when working with exact sequences. Sometimes people even use directly definitions and terms in homological algebra that are not really well-defined if the underlying category is not abelian.

The question is, why does this "just work"? Is it that the category of $C^*$-algebras have some good properties that allow one to use techniques from homological algebra, or is it that a lot of homological algebra don't really require the category one is working in to be abelian (e.g., pre-abelian is actually enough)?

I apologize if this question is not research-level enough.

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    $\begingroup$ The 6-term exact sequence is proved using techniques in homological algebra, IIRC. Another example would be the proof of $K_0(\mathcal{C}_0) \oplus K_0(\mathcal{C}_1) \cong K_0(\mathcal{C}_0 \oplus \mathcal{C}_1)$ which uses the five lemma. $\endgroup$ – xuq01 Jun 18 '20 at 11:40
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    $\begingroup$ One should pay attention to the comments on the linked math.so post: the hom sets for $C^*$-algebras (that is, $*$-homomorphisms) do not form an abelian group, and so the category is not additive. $\endgroup$ – Matthew Daws Jun 18 '20 at 13:09
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    $\begingroup$ I think I disagree with you assessment that these are techniques specific to abelian categories. The category of spaces is even 'less abelian' than the category of C* algebra, and most of these results still holds for many invariant on the category of spaces (the K-theory of C* algebra is partly inspired from the K-theory of spaces...) To me these are methods from abstract homotopy theory (which happen to includes homological algebra) $\endgroup$ – Simon Henry Jun 18 '20 at 17:27
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    $\begingroup$ A long long time ago I remember seeing the assertion that the category of Cstar-algebras and star-homomorphisms is semi-abelian in the sense of Borceux and Bourn, ( ncatlab.org/nlab/show/semi-abelian+category ). I don't know if this is really relevant to K-theory, E-theory, KK-theory and so on $\endgroup$ – Yemon Choi Jun 18 '20 at 22:40
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    $\begingroup$ The semi-abelianness of $\mathrm{C}^*\mathrm{Alg}$ is shown in Semi-abelian monadic categories by Gran and Rosický. Since $\mathrm{C}^*\mathrm{Alg}$ is monadic, has finite coproducts and a zero object, semi-abelianness boils down to inheriting the split short five lemma from the category of general *-algebras. $\endgroup$ – Cameron Zwarich Jun 20 '20 at 14:07

((this is not an answer, but I want to be able to delete my post, that's why do not use comments:

$KK$-theory is an abelianization of the $C^*$-category: it is the universal matrix-stable (more precisely: compact-operators-stable), homotopy-invariant, split-exact category formed from the $C^*$-cagtegory. Hence it is additive and abelian. If you refer to $K$-theory (recall: $K(A) = KK(\mathbb{C},A)$), then it is clear.

You can also other categories of algebras than $C^*$ make in this way abelian.))


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