Proposition 10

Let A be a unital stellar algebra. Suppose that for every pair $(x, y)$ of permutable elements of A, the condition $xy= 0$ implies that $x= 0$ or $y= 0$. Then $A =\mathbf{C}\cdot 1$.

If A is not equal to $\mathbf{C}\cdot 1$, there exists a hermitian element $x$ in A which does not belong to $\mathbf{C}\cdot 1$ (Lemma 2 of I, p. 96). Let B be the unital involutive subalgebra of A generated by $x$. It is commutative, and isomorphic to $\mathscr{C}$ (Sp$_A(x)$) (I, p. 110, Remark). Since $x$ is not scalar, its spectrum in B is not reduced to a single element (Example 1 of I, p. 110). There therefore exist continuous nonzero functions $f$ and $g$ on Sp$_A(x)$ such that $f g= 0$ (Lemma 11). The elements $f(x)$ and $g(x)$ of A are nonzero, commuting, and satisfy $f(x)g(x) = 0$ in A.