Example

Let us take up again the examples of n$^o2$ of I, p. 17.

The algebra of continuous functions with complex values tending to 0 at infinity on a locally compact space X (example 3 of I, p. 17) is regular (cf. I, p. 36, example 1).

The algebra of $n$ times differentiable functions on $[0,1]$ (example 4 of I, p. 18) is regular (cf. I, p. 36, example 2).

If G is a commutative locally compact group and $\mu$ a Haar measure on G, then the algebra $L^1(G, \mu)$ (example 7 of I, p. 19) is regular (cf. II, p. 219, cor. 2).

The algebra of functions which are continuous in the disk $|z|\leqslant 1$ and analytic in the interior (example 9 of I, p. 20) is not regular (cf. I, p. 193, exerc. 6).