Exercise 19, § 6

Let A be a star algebra. An element $x$ is said to be systematically positive if $x\in A_+$ and if $xAx$ is dense in A.

a) Now prove that a commutative star algebra A contains a systematically positive element if, and only if, there exists a locally compact topological space X countable at infinity X such that A is isomorphic to $\mathscr{C}_0(X)$.

b) If A is unital, then an element $x$ of A is systematically positive if, and only if, $x$ is positive and invertible.

c) Let $x\in A$ be a systematically positive element such that $\|x\|\leqslant 1$. Prove that the sequence $(x^{1/n})_{n\geqslant 1}$ is an increasing approximate unit in A.

d) A star algebra A contains a systematically positive element if, and only if, there exists a sequence $(x_n)_{n\geqslant 1}$ in A which forms an approximate unit of A.