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Thread: Positive Elements in a C*-Algebra

  1. #1
    Junior Member
    Mar 2009

    Question Positive Elements in a C*-Algebra

    Let  \cal{A} = \cal{B}(l^2), let  a = the \;unilateral \;shift \;on \;l^2, and let  b=a^* . Show that  \sigma(ab) \neq \sigma(ba)

    Given  \cal{A}  =  \cal{B}(l^2) is a C* -Algebra where for each operator T in B(l^2), T* = is the adjoint of T is that true? or only for B(H)?
    Last edited by Nusc; Mar 24th 2009 at 07:02 PM.
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  2. #2
    MHF Contributor
    Opalg's Avatar
    Aug 2007
    Leeds, UK
    Quote Originally Posted by Nusc View Post
    Let  \mathcal{A} = \mathcal{B}(l^2), let  a = \text{the unilateral shift on }l^2, and let  b=a^* . Show that  \sigma(ab) \neq \sigma(ba)
    The adjoint of the (forwards) unilateral shift is the backwards unilateral shift. So the product ba is the identity (if you shift forwards and then backwards you get back to where you started). But ab is not invertible because the backwards shift kills off the first basis vector. Thus 0\in\sigma(ab) but 0\notin\sigma(ba).

    In fact, 0 is the only number that can be in the spectrum of st but not in the spectrum of ts (where s, t are elements of a C*-algebra). There is a theorem which says that \sigma(st)\cup\{0\} = \sigma(ts)\cup\{0\}.
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