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Thread: positive matrix

  1. #1
    Member Mauritzvdworm's Avatar
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    positive matrix

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

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

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

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

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

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