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Math Help - dense subset of C*-algbra

  1. #1
    Member Mauritzvdworm's Avatar
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    dense subset of C*-algbra

    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...)
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  2. #2
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    Opalg's Avatar
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    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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