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I note that Zoran has started an entry pro-C-star-algebra. I was wondering if inverse limits in topological -algebras are exact. If not then taking the limit seems a strange thing to do. It would be better to handle the pro-object as such. I.e. within the pro-category. Zoran, can you enlighten me? :-) I suspect that if the C-algebras are finite dimensional as vecor space then there would be not much difference… any thoughts?
It seems not. Regarding that the Yoneda embedding is continuous, whenever the limit of $C^\ast$ exists within $C^\ast$-algebras, it would of course coincide with the pro-object; but there are such examples when the limit exists in $C^\ast$ but the corresponding pro-$C^\ast$-algebra (in the sense of Phillips, hence the limit within topological *-algebras) is not representable. The “sheaves of $C^\ast$-algebras” in the sense of Ara dwell on this difference. The limits within $C^\ast$ kind of truncate unbounded information. If I understand the point, this is then obviously very important for the noncommutative geometry, I think, but I am not yet understanding the technical side of the story (in process, I hope).
An interesting thing about pro-$C^\ast$-algebras is that there is a an extension of Gelfand duality to the duality between commutative pro-$C^\ast$-algebras and compactly generated Hausdorff spaces in the sense of Steenrod. One should point out the papers of Dubuc and Porta on convenient categories of topological algebras to see the background to this.
Zoran: as you probably know, according to WIkipedia compactly generated spaces were studied by Hurewicz (and are in Kelley’s book as well) so predate Steenrod’s interest in them.
My point was that profinite groups work well because the pro-object / projective system gives the limit group and the limit group returns the projective system as asystem of finite quotients. If you go away from pro-finite, of course, that property fails and usually the ’right’ object is the pro-object rather than the limit, because of non-exactness of theinverse limit. Looking at the homotopy theory of the simplicial profinite spaces that arise in alg. geom. (commutative, I’m afradi, at least for the moment) the proofs simplify enormously if one replaces the limiting space by that corresponding pro-obnject. The duality between pro-objects and ind-objects then extends to give duality results in this setting. If one then looks at shape and strong shape and the duality between strong shape and proper homotopy, one is possibly seeing the same phenomenon (perhaps replace ’the same’ by ’similar’ here!), but in a setting nearer to $(\infty,1)$-categorical considerations.
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