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Math Help - homotopy

  1. #1
    Member Mauritzvdworm's Avatar
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    homotopy

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

    Show that \varphi_t=\psi_t\varphi:A\rightarrow K(H)\otimes_{*}A is a homotopy
    Last edited by Mauritzvdworm; July 23rd 2010 at 06:09 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 \varphi is easy
    \varphi(a+b)=p\otimes (a+b)=p\otimes a+p\otimes b

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

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

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

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

    now let a\in A and let us consider what is happening at t=0
    \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
    \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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