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Math Help - isomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    isomorphism

    Let K=K(H) with H a infinite dimensional, separable Hilbert space. Let (e_n)^{\infty}_{n=1} be an orthonormal basis for H. Let e_{ij} be an operator in B(H) defined by e_{ij}(x)=\langle x,e_j\rangle e_i
    set p_n=\sum^{n}_{j=1}e_{jj} now show that the following map is a *-isomorphism (bijective *-homomorphism)

    \psi_{n}:M_n(A)\rightarrow p_nKp_n\otimes A, (a_{r,s})\mapsto\sum^{n}_{i,j=1}e_{ij}\otimes a_{r,s}
    Last edited by mr fantastic; July 19th 2010 at 04:18 AM.
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  2. #2
    Member Mauritzvdworm's Avatar
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    I managed to show that \psi_n is linear, multiplicative and a bijection (though it took some work)

    The only missing part is to show that \psi(a^*)=\psi(a)^*
    What does e^*_{ij} look like?
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  3. #3
    Member Mauritzvdworm's Avatar
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    Let y\in H then consider
    \langle e^{*}_{ij}y,x \rangle=\langle y,e_{ij}x \rangle=\langle y,\langle e_j,x \rangle e_i \rangle=\langle e_j,x \rangle \langle y,e_i \rangle=\langle \langle e_i,y \rangle e_j,x \rangle

    so e^{*}_{ij}=e_{ji}

    Then everything works out
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