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

    Given $\displaystyle \cal{A} = \cal{B}$(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 06:02 PM.
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  2. #2
    MHF Contributor
    Opalg's Avatar
    Aug 2007
    Leeds, UK
    Quote Originally Posted by Nusc View Post
    Let $\displaystyle \mathcal{A} = \mathcal{B}(l^2)$, let $\displaystyle a = \text{the unilateral shift on }l^2$, and let $\displaystyle b=a^* $. Show that $\displaystyle \sigma(ab) \neq \sigma(ba) $
    The adjoint of the (forwards) unilateral shift is the backwards unilateral shift. So the product $\displaystyle ba$ is the identity (if you shift forwards and then backwards you get back to where you started). But $\displaystyle ab$ is not invertible because the backwards shift kills off the first basis vector. Thus $\displaystyle 0\in\sigma(ab)$ but $\displaystyle 0\notin\sigma(ba)$.

    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 $\displaystyle \sigma(st)\cup\{0\} = \sigma(ts)\cup\{0\}$.
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