3

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.

Ali Taghavi
  • 312
ray
  • 687
  • 2
    Why do you expect such prove to exist? – Fedor Petrov Sep 19 '15 at 10:14
  • 2
    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. – ray Sep 19 '15 at 12:12
  • 2
    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 – Yemon Choi Sep 19 '15 at 15:31
  • 1
    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 – Yemon Choi Oct 18 '15 at 15:42
  • See my comments to Igor Rivin's (interesting) answer. – Yemon Choi Oct 18 '15 at 15:43
  • @YemonChoi To what extent you are sure on your intuition that "Any Banach algebraic proof of a (generalization of a )theorem in Algebraic topology is just a hiding the inggredients of the usual algebraic topology proof"? – Ali Taghavi Feb 11 '16 at 05:29

2 Answers2

4

Yes, see the recent paper of Benjamin Passer.

Igor Rivin
  • 95,560
  • 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? – Yemon Choi Sep 19 '15 at 15:30
  • @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). – Igor Rivin Sep 19 '15 at 15:33
  • 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. – Yemon Choi Sep 19 '15 at 18:51
  • @YemonChoi The OP asked for a proof of Borsuk-Ulam using Banach algebras, extending the proof of Ali Taghavi in the paper he refers to. The reference given gives such a proof, stating explicitly that he extends the work of Ali Taghavi. What more do you want? It is conceivable that proof theorists will show that the argument is logically equivalent, which will be interesting, since all of the words are completely different. – Igor Rivin Sep 19 '15 at 20:00
  • 2
    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. – Yemon Choi Sep 19 '15 at 20:07
  • That said, the paper of Passer does seem to be doing something more fruitful than what I hypothesized in the previous comment. I note that the original post actually asks for a proof of Borsuk-Ulam "based only on the theory of Banach algebras" and it is not clear to me what only means. K-theory for Banach algebras already has arguments about homotopy of idempotents built into it, so it is debatable whether one is really avoiding the ingredients of the usual proofs. Anyway, regardless, I guess it is good that you bring Passer's paper to our attention. – Yemon Choi Sep 19 '15 at 20:07
  • 1
    @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... – Igor Rivin Sep 20 '15 at 08:27
2

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:

http://arxiv.org/abs/1502.05756

http://arxiv.org/abs/1109.2991

Ali Taghavi
  • 312