Example

Let $X = [0,1]$ and let $n\geqslant 0$ be an integer. Let B be the algebra of functions $f: X\rightarrow \mathbf{C}$ admitting continuous derivatives on $[0,1]$ up to order $n$, endowed with the norm considered in example 4 of I, p. 18. Then B is a full involutive subalgebra of $\mathscr{C}(X)$ separating the points of X, hence $\mathsf{X}(B)$ is identified with X.

We consider the logarithm function as defined on $\mathbf{R}_+$ and with values in $\mathbf{R}\cup \{-\infty \}$ by setting log(0) $=-\infty$.