Proposition 7

Let A be a star algebra. Every positive linear form $\lambda$ on A is continuous.

Let us first prove that $\lambda$ is bounded on the intersection of $A_+$ and the unit ball of A. Suppose this is not so. Then there exists a sequence $(x_n)_{n\geqslant 1}$ in $A_+$ such that $\|x_n\|\leqslant 1$ and $\lambda (x_n)\geqslant n$ for every integer $n\geqslant 1$. The series with general term $n^{-2}x_n$ converges to an element $x$ of A (cf. TG, IV, p. 33, Example 3). For every integer $N\geqslant 1$, since

$+\infty$

$\sum\frac{1}{n^2}x_n\in A_+$,

$n=N+1$

(I, p. 116, Prop. 14), it follows that

N N $+\infty$

$\sum\frac{1}{n}\leqslant \sum\frac{1}{n^2}\lambda (x_n) =\lambda (x)-\lambda (\sum\frac{1}{n^2}x_n)\leqslant \lambda (x)$,

$n=1n=1n=N+1$

which is absurd (TG, IV, p. 33, Example 4).

Since every element of the unit ball of A is a linear combination with coefficients bounded by 1 of at most four positive elements of A of norm $\leqslant 1$(I, p. 96, Lemma 2 and formula (4) of I, p. 117), we conclude that $\lambda$ is continuous.

There exist involutive Banach algebras which admit positive linear forms that are not continuous (Exercise 3, (a) of V, p. 493).