NetNews Usenet Archive 1993 #3/NN_1993_3.iso/spool/sci/math/18783

Internet Message Format | 1993-01-27 | 1.8 KB

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From: [e-mail] (Charles Yeomans)
Newsgroups: sci.math
Subject: Re: Smooth manifolds and function extensions
Message-ID: <[e-mail]>
Date: 26 Jan 93 16:14:03 GMT
Article-I.D.: austin.1993Jan26.161403.29059
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In article <[e-mail]>, [e-mail] (M.R.Watkins) writes:
>
> Given a manifold M with a smooth (C-infinity) structure, it is possible to
> define E(M), the linear space of all smooth functions on M.  Now is it
> possible to reconstruct the smooth structure (that is the atlas on M) from
> the knowledge of E(M) alone?
>
I believe the answer is no, in general.  Consider , in C^2, the two sets
X = unit ball z^2 + w^2 < 1 and Y = X - (0,0).  Let H(X), H(Y) be the spaces of
holomorphic functions on X, Y.  Hartogs' theorem implies that any function
holomorphic on Y extends (uniquely) to a holomorphic function on X.  Thus H(X)
is isomorphic to H(Y).  But X is very diferent from Y.

But for a manifold with a C-infinity structure, what you ask might be possible.
However, I think you need more structure on X, say that of a ring or algebra.
With this, you ought to be able to do the following:  Suppose E(M) has the
structure of a commutative R-algebra.  Let I be a maximal ideal in E(M). What
you'd like is for I to have the form {f in E(M) | f(p) = 0}, for some p
in M. This would be the hard part.  Given such a fact, you could reconstruct
M using the set of all maximal ideals.

I seem to recall that there are such theorems in Banach algebra.

Charles Yeomans