dense subset of C*-algbra

**Mauritzvdworm** · August 23rd 2010, 12:38 AM

Show that the span of elements of the form $a^*b$ where both $a,b\in A$ which is a C*-albegra is dense in $A$

(I'm missing something simple...)

**Opalg** · August 23rd 2010, 08:33 AM

If the algebra is unital, with identity element e, then this result is obvious, because any element $a\in A$ is equal to $e^*a$ .

Even in the nonunital case, a C*-algebra always has an approximate unit consisting of positive elements in the unit ball. Given $a\in A$ and $\varepsilon>0$ , choose an element $e_\iota$ in the approximate unit such that $\|e_\iota a-a\|<\varepsilon$ . That shows that elements of the form $b^*a$ are dense in A (without even having to take the linear span of them).

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