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Math Help - positive matrix

  1. #1
    Member Mauritzvdworm's Avatar
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    positive matrix

    Say we have an n-dimensional row vector a=(a_1,\dots,a_n) with entries in a C*-algebra A. We construct the matrix
    a^*a=<br />
\left(<br />
\begin{array}{lll}<br />
a_1^*a_1 &\dots  &a_1^*a_n \\<br />
\vdots & \ddots & \vdots\\<br />
a_n^*a_1 & \dots & a_n^*a_n<br />
\end{array}<br />
\right)<br />
    How do we show that this matrix is positive?

    Here is my idea:
    If we have a representation \varphi:A\rightarrow B(H) for some Hilbert space H, we can inflate this homomorphism to
    \varphi:M_n(A)\rightarrow M_n(B(H)) with norm \|T\|=\|\varphi(T)\| which makes M_n(A) into a C*-algebra.
    Furthermore we know that M_n(B(H)) is isomorphic to B(H^{(n)}) where H^{(n)}:=\oplus^n_{i=1}H.

    We know that \varphi(a^*a)^*=\varphi(a^*a), so it is self adjoint. Now is we can show that \langle \varphi(a^*a)x,x\rangle\geq 0 where x\in H^{(n)} then the matrix is positive.

    So
    \[<br />
\langle \varphi(a^*a)x,x\rangle=<br />
\langle \left(<br />
\begin{array}{lll}<br />
\varphi(a_1)^*\varphi(a_1) &\dots  &\varphi(a_1^*)\varphi(a_n) \\<br />
\vdots & \ddots & \vdots\\<br />
\varphi(a_n)^*\varphi(a_1) & \dots & \varphi(a^*_n)\varphi(a_n)<br />
\end{array}<br />
\right)x,x\rangle\\<br />
=\sum^{n}_{i=1}\sum^{n}_{j=1}\langle \varphi(a_i)^*\varphi(a_j)x_j,x_i\rangle<br />
\]

    so when i=j we have positivity, but what about the terms where i\neq j?

    Or am I completely on the wrong track?
    Last edited by Mauritzvdworm; October 16th 2010 at 04:04 AM. Reason: typo
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