I am wondering whether there exists a proof of the classical Borsuk Ulam theorem for the Euclidean n-sphere, $n>2$ that is based only on the theory of Banach algebras. I checked on MR but had no success.
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2$\begingroup$ Why do you expect such prove to exist? $\endgroup$– Fedor PetrovCommented Sep 19, 2015 at 10:14
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2$\begingroup$ because of MR2898039 Taghavi, Ali(IR-DUBSMC) A Banach algebraic approach to the Borsuk-Ulam theorem. (English summary) Abstr. Appl. Anal. 2012, Art. ID 729745, 11 which gives this for n=2. $\endgroup$– rayCommented Sep 19, 2015 at 12:12
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2$\begingroup$ What do you mean "only on the theory of Banach algebras"? My intuition is that any "Banach-algebraic proof" of a theorem in algebraic topology will just be hiding the ingredients of the usual algebraic topology proof inside $\endgroup$– Yemon ChoiCommented Sep 19, 2015 at 15:31
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1$\begingroup$ Ali, my intuition does not come from examples, but from the lack of examples (and from 10+ years working on Banach algebras and reading people's proofs of facts about Banach algebras). I guess at heart I believe this is a meta-principle about "conservation of information" - the dictionary between topology and (certain kinds of) Banach algebra is to a large extent formal, and if one plans to use K-theory of BAs then this already uses topological arguments in proving its fundamental properties $\endgroup$– Yemon ChoiCommented Oct 18, 2015 at 15:42
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$\begingroup$ See my comments to Igor Rivin's (interesting) answer. $\endgroup$– Yemon ChoiCommented Oct 18, 2015 at 15:43
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2 Answers
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Yes, see the recent paper of Benjamin Passer.
answered Sep 19, 2015 at 12:52
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$\begingroup$ I have not read the paper in question, but can we be sure that the proof there is not using essentially the same algebraic topology as in the BU theorem? $\endgroup$ Commented Sep 19, 2015 at 15:30
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$\begingroup$ @YemonChoi Can one be sure of anything in these troubled times? On first glance, it looks quite different, and proves a more general result (though those of us who don't care much about non-commutative geometry might not care). $\endgroup$ Commented Sep 19, 2015 at 15:33
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$\begingroup$ Well, Mike Batt assures me that there are 9 million bicycles in Beijing, that's a fact that no one can deny. Sadly, he doesn't seem to have said anything about alternative proofs of Borsuk--Ulam. $\endgroup$ Commented Sep 19, 2015 at 18:51
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2$\begingroup$ Fair point. BTW, I am speaking from the perspective of a professional Banach algebraist. My concern is that some claims to "prove Topological Result X using techniques from Banach algebras" use, in the various lemmas or subresults, facts which turn out to be more or less the same as those used in the usual proofs, and therefore the informational content of these results is (X and Y) $\implies$ X. $\endgroup$ Commented Sep 19, 2015 at 20:07
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1$\begingroup$ @YemonChoi I understood your point, and I agree that "only" is a loaded word. In addition, the "standard" proof of the BU theorem is short enough that there is always a risk of embedding it without us (or the author, for that matter) noticing. I also agree that this should be of interest to Banach Algebraists, such as yourself... $\endgroup$ Commented Sep 20, 2015 at 8:27
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Not a direct answer to the question but somehow related to the question:
Apart from Benjamin Passer's paper, Here are two other papers about non commutative Borsuk Ulam theorem:
answered Oct 18, 2015 at 13:52