A Banach algebra generated by idempotents A Banach algebra generated by idempotents i.e. elements x such that x2 = x.

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In the following, we show that the Banach algebra C(X), where X is a compact Hausdorff space, with Card(X) > 1 ,is generated by idempotents iff X is totally disconnected.
Recall that a topological space is said to be totally disconnected if for every distinct x1, x2 Î X, there exist disjoint open sets G1 and G2 such that x1 Î G1, x2 Î G2 and X = G1 ÈG2.
If X is totally disconnected, x1 ¹ x2, x1 Î G1, x2 Î G2, X = G1 ÈG2, G1 ÇG2 = Æ, G1 and G2 are open, then the continuous function f(x) = {
1
x Î G1
0
x Î G2
separates x1 and x2. So the closed self-adjoint subalgebra generated by idempotent, by the Stone-Weierstrass theorem, is C(X).
Conversely, suppose that C(X) is generated by its idempotents. Let x1 and x2 belong to X. By Urysohn's lemma there exists a function f Î C(X) such that f(x1) = 1 and f(x2) = 0. Every element of the self-adjoint subalgebra generated by idempotents is of the form h = åi = 1kligi (§) for some idempotents gi and li Î C. Hence there is a sequence (hn) of elements of the form (§) such that hn ® f uniformly on X. So hn(x1) ® 1 and hn(x2) ® 0. Therefore there exists a number N such that |hN(x1)| > [1/2] and |hN(x2)| < [1/2]. So that x1 Î hN-1({z Î C; |z| > 1}) = G1, x2 Î hN-1({z Î C; |z| < 1}) = G2, X = G1 ÈG2, X = G1 ÇG2 = Æ. Thus X is totally disconnected.


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On 22 Feb 2001, 00:13.