Show that the span of elements of the form where both which is a C*-albegra is dense in

(I'm missing something simple...)

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- Aug 23rd 2010, 01:38 AMMauritzvdwormdense subset of C*-algbra
Show that the span of elements of the form where both which is a C*-albegra is dense in

(I'm missing something simple...) - Aug 23rd 2010, 09:33 AMOpalg
If the algebra is unital, with identity element e, then this result is obvious, because any element is equal to .

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