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- Newsgroups: sci.math
- Path: sparky!uunet!snorkelwacker.mit.edu!galois!riesz!jbaez
- From: jbaez@riesz.mit.edu (John C. Baez)
- Subject: What a weird group!!!
- Message-ID: <1992Oct15.045312.13089@galois.mit.edu>
- Sender: news@galois.mit.edu
- Nntp-Posting-Host: riesz
- Organization: MIT Department of Mathematics, Cambridge, MA
- Date: Thu, 15 Oct 92 04:53:12 GMT
- Lines: 45
-
- No, I'm not referring to sci.math. I'm referring to the Bohr
- compactification of the real numbers, something I just learned about.
- This is, of course, due not to Niels Bohr but to his brother Harald.
-
- From an exercise in a book: The Bohr compactification of the real
- numbers is a compact topological group which contains the real numbers
- as a dense set. The embedding of the reals into the Bohr group is not a
- homeomorphism.
-
- Well, actually that in itself is not so weird now that I think of it.
- One can embed the real numbers in the torus as a subgroup, namely a line
- of irrational slope, and then it's dense but the image of the reals in
- the torus with the induced topology is not homeomorphic to the reals
- with their usual topology. (I think this is what was meant by the
- exercise above, by the way.)
-
- But I bet the Bohr group is a lot weirder than the torus.
-
- Here's how you get it. Recall that there is a 1-1 correspondence
- between compact Hausdorff spaces and a certain class of algebras called
- C*-algebras. The map one way is easy. You just take your space X and
- form C(X), the algebra of continuous complex-valued functions on X.
- The algebras you get have certain properties and have been described
- axiomatically, which is the point of the definition of (commutative
- unital) C*-algebras. Then the Gelfand-Naimark theorem says that any
- algebra satisfying this definition really is C(X) for some compact
- Hausdorff X, and tells you how to "construct" X.
-
- Okay, so Bohr was interested in almost periodic functions on R.
- These may be defined as follows. Consider the space of functions on R
- that are finite linear combinations of functions exp(ikx). Then take
- the closure of this space in L^infty(R). These are the almost periodic
- functions. There's another definition too, that makes it clearer in
- exactly what sense they are "almost" periodic.
-
- The point here is that they clearly form a subalgebra of L^infty(R) and
- because this algebra is a closed subspace of L^infty(R) one easily
- checks that it is a (commutative unital) C*-algebra. Thus by
- Gelfand-Naimark there is some compact Hausdorff space X out there such
- that this algebra is really just C(X). This X is the Bohr group. Of
- course, one has to show it's a group. I imagine one simply extends the
- group structure from R to this group in a unique continuous manner,
- using the fact that R is dense in X.
-
- But what does this group *look like*?? That's what I'd like to know!!
-