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Thread: homotopy

  1. #1
    Member Mauritzvdworm's Avatar
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    homotopy

    $\displaystyle \varphi:A\rightarrow K(H)\otimes_{*}A,a\mapsto p\otimes a$
    show that the above map is a *-homomorphism with A a C*-algebra, H a Hilbert space and p a rank-one projection in K(H).

    If we let $\displaystyle \psi_t=\text{Ad}u_t\otimes_{*}\text{Id}_{A}:K(H)\o times_{*}A\rightarrow K(H)\otimes_{*}A$
    where $\displaystyle (\text{Ad}u)(x)=uxu^*$ and $\displaystyle u_t=e^{itv}$ with v a self-adjoint operator in B(H)

    Show that $\displaystyle \varphi_t=\psi_t\varphi:A\rightarrow K(H)\otimes_{*}A$ is a homotopy
    Last edited by Mauritzvdworm; Jul 23rd 2010 at 05:09 AM. Reason: missing t in def of u_t
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  2. #2
    Member Mauritzvdworm's Avatar
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    linearity of $\displaystyle \varphi$ is easy
    $\displaystyle \varphi(a+b)=p\otimes (a+b)=p\otimes a+p\otimes b$

    so is multiplicativity
    $\displaystyle \varphi(ab)=p\otimes ab=pp\otimes ab=(p\otimes a)(p\otimes b)$

    so is preservation of the adjoint
    $\displaystyle \varphi(a^*)=p\otimes a^*=p^*\otimes a^*=(p\otimes a)^*$
    This shows that $\displaystyle \varphi$ is indeed a *-homomorphism

    To show that $\displaystyle \varphi_t$ is a homotopy we need to be a little more creative

    suppose $\displaystyle q\in K(B)$ in another rank-one projection then there exits unitary element $\displaystyle u\in B(H)$ such that $\displaystyle q=upu*$

    now let $\displaystyle a\in A$ and let us consider what is happening at t=0
    $\displaystyle \varphi_0(a)=\psi_0\varphi(a)=(\text{Id}_{K(H)}\ot imes_{*}\text{Id}_A)(p\otimes a)=p\otimes a=\varphi(a)$

    now consider what is happening at t=1
    $\displaystyle \varphi_1(a)=\psi_1\varphi(a)=(\text{Ad}u\otimes_* \text{Id}_A)(p\otimes a)=upu^*\otimes_* a=q\otimes a$
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