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From Gelfand-Neumark Theorem, we know that topological properties of a compact Hausdorff space $X$ are encoded in the abelian $C^*$-algebra of continuous complex-valued functions on $X$ (with $||f||= \sup |f|$). For example, the existence of non-trivial idempotents would imply that the space $X$ is not connected.

Is there such an algebraic description to determine whether $X$ is simply connected or not ?

nsoum
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2 Answers2

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I am not really an expert in non-commutative point-set/algebraic topology (but I used to think about it a decent amount). So take my answer with that in mind.

So the silly answer is yes and the more interesting answer is no.

Specifically, The Gelfand correspondence sets up a contravariant equivalence of categories between compact Hausdorff space with continuous maps and unital abelian $C^*$-algebras with $*$-homomorphisms. So through this you can unwind (wind up?) the definition of simply connected though this equivalence to get a characterization in terms of the associated $C^*$-algebras. So it would be something like $A$ is simply connected if every map $\phi:A\rightarrow C(S^1)$ is homotopic to the constant maps. Where $C(S^1)$ is continuous functions on the circle and note that homotopy does translate well into the $C^*$-algebraic framework.

However, this isn't particularly useful. In fact, what I think that you are asking is "is there a characterization which is useful for non-abelian $C^*$-algebras". This is the case with connectedness that you give above.

Here I think the answer is no and, in fact I will say something stronger. I think that there can be no such characterization (Here I'm being a little bolder). The reason is because of the contravariance of the equivalence. In particular, any topological invariant that comes from maps of some space $X$ (in the above case $S^1$) to the desired space $Y$ will not bear anything fruitful for general $C^*$-algebras. Because on the level of the algebras you get a map from $C(Y)\rightarrow C(X)$. So in order for this to work for any $C^*$-algebras it would in particular have to work for the non-abelian simple ones (these are largely the ones of interest). And they can't have any morphisms into abelian algebras.

Owen Sizemore
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  • What would be some good sources to learn about homotopy theory of $C^*$-algebras ? – nsoum Sep 23 '13 at 01:11
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    The notion of homotopy of maps does work well in $C^$-algebras. Though for the reasons above homotopy groups do not. I would think any book on the K-theory of C$^$-algebras would do. I learned what I know mostly from "K-theory for Operator Algebras" by Bruce Blackadar. – Owen Sizemore Sep 23 '13 at 13:40
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There are some works devoting to fundamental groups of non commutative spaces. For example:

https://arxiv.org/abs/math/0604508

https://arxiv.org/abs/1910.10127

https://www.math.ku.dk/english/calendar/events/babyncg012115/

Ali Taghavi
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