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Math Help - constructing a C*-algebra

  1. #1
    Member Mauritzvdworm's Avatar
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    constructing a C*-algebra

    Is it possible to construct a C*-algebra \mathcal{A} such that there exists x\in\mathcal{A} which cannot be decomposed into x=x_1x_2 with x_1\geq0 and x_2 a partial isometry?

    It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some B(H) and any operator T\in B(H) can be decomposed using the polar decomposition, or am I missing something?
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  2. #2
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    Quote Originally Posted by Mauritzvdworm View Post
    Is it possible to construct a C*-algebra \mathcal{A} such that there exists x\in\mathcal{A} which cannot be decomposed into x=x_1x_2 with x_1\geq0 and x_2 a partial isometry?

    It does not seem so, since any C*-algebra can be seen as a C*-subalgebra of some B(H) and any operator T\in B(H) can be decomposed using the polar decomposition, or am I missing something?
    The decomposition certainly works in B(H), giving a positive operator and a partial isometry in B(H). The positive operator will be in \mathcal{A}, but the partial isometry does not necessarily belong to \mathcal{A}.

    For example, if \mathcal{A} is the commutative C*-algebra of continuous functions on [1,1] and x is the function x(t) = t, then the positive part x_1 will be the function x_1(t) = |t|. But the partial isometry will be the function x_2, where x_2(t) is 1 when x<0 and +1 when x>0. Obviously x_2 is not continuous and so cannot belong to \mathcal{A}.
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  3. #3
    Member Mauritzvdworm's Avatar
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    That is what I thought, interestingly in the case of von Neumann algebras the partial isometry is part of the von Neumann algebra. Thank you.
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