Example 3

Tag 00P9, permanent. Cite it as [Bourbaki, Tag 00P9].

Algebra, VIII, p. 436, Appendix 1. Algebras without Unit Element

Let $L^1(\mathbf{R})$ be the convolution algebra of the locally compact group $\mathbf{R}$ . Recall (cf. Int., VIII, §4, No. 5, p. 38) that $L^1(\mathbf{R})$ is the space of classes of functions on $\mathbf{R}$ that are integrable for the Lebesgue measure. The product of the classes of two functions $f$ and $g$ is the class of the function $f*g$ defined by the formula

(f*g)(s) =\int_{-\infty}^{+\infty}f(t)g(s-t)dt

for almost every $s\in \mathbf{R}$ . The algebra $L^1(\mathbf{R})$ is not unital. For any $a$ in $\mathbf{R}$ , denote by $\mathfrak{m}_a$ the set of elements $f$ of $L^1(\mathbf{R})$ satisfying

\int_{-\infty}^{+\infty}f(t)e^{-iat}dt= 0

By TS, II, §3, n $^o2$ , p. 252, théorème 1, the mapping $a\mapsto \mathfrak{m}_a$ is a bijection from $\mathbf{R}$ to the set of regular maximal ideals of the algebra $L^1(\mathbf{R}).*$

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Int., VIII, §4, No. 5, p. 38 not in this corpus: INT VIII
TS, II, §3, n$^o2$, p. 252 not in this corpus: TS II