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Math Help - *-isomorphism

  1. #1
    Member Mauritzvdworm's Avatar
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    *-isomorphism

    I can see that it might be possible to identify B(\mathbb{C}^{n}) with M_{n}(\mathbb{C})

    However, I am finding some trouble showing that
    \varphi:B(\mathbb{C}^{n})\rightarrow M_{n}(\mathbb{C})
    is a *-isomorphism...?

    Can you think of a specific *-isomorphism \varphi which will identify B(\mathbb{C}^{n}) with M_{n}(\mathbb{C})?

    B(\mathbb{C}^{n}) is the set of all bounded linear operators form the Hilbert space \mathbb{C}^{n} to itself.

    M_{n}(\mathbb{C}) is the space of all n\times n matrices with complex entries.
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  2. #2
    Member Mauritzvdworm's Avatar
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    Here is an idea that might work

    let (e_n) the the standard basis for \mathbb{C}^n and take a vector b\in\mathbb{C}^n. We can write b=\beta_1e_1+\dots+\beta_ne_n and let T be an operator in B(\mathbb{C}^n), then
    Tb=T(\beta_1e_1+\dots+\beta_ne_n)=\beta_1Te_1+\dot  s+\beta_nTe_n

    and then use this to construct a matrix representation for T

    How then would we proceed to find such a matrix construction?
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  3. #3
    Member Mauritzvdworm's Avatar
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    Figured out how to do this (for those who are interested)

    let (e)_{i=1}^{n} be the standard orthonormal basis of \mathbb{C}^{n} and let T\in B(\mathbb{C}^{n})

    we can write any vector x\in\mathbb{C}^{n} as x=x_1e_1+\dots+x_ne_n where the x_i`s are scalars, so

    Tx=x_1Te_1+\dots+x_nTe_n but Te_j\in\mathbb{C}^{n} so we can write it as follows

    Te_j=t_{j1}e_1+t_{j2}e_2+\dots+t_{jn}e_{n} for every j

    this then enables us to make to following connection
     <br />
\left( \begin{array}{cccc}<br />
t_{11} & t_{12} & \dots & t_{1n} \\<br />
t_{21} & t_{22} & \dots & t_{2n} \\<br />
\vdots & \vdots & \ddots& \vdots \\<br />
t_{n1} & t_{n2} & \dots & t_{nn}\end{array} \right)<br />
\left( \begin{array}{c}<br />
x_1\\<br />
x_2\\<br />
\vdots\\<br />
x_n<br />
\end{array}\right)<br />
=Tx<br />

    we can then use this to define the mapping \varphi:B(\mathbb{C}^{n})\rightarrow M_n(\mathbb{C}) as follows

    \varphi(T)=<br />
\left( \begin{array}{cccc}<br />
t_{11} & t_{12} & \dots & t_{1n} \\<br />
t_{21} & t_{22} & \dots & t_{2n} \\<br />
\vdots & \vdots & \ddots& \vdots \\<br />
t_{n1} & t_{n2} & \dots & t_{nn}\end{array} \right)

    it is easy to thow that this choice then leads to a *-isomorphism.
    Last edited by Mauritzvdworm; March 4th 2010 at 12:54 PM. Reason: typing error
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