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Thread: Simple C*-algebra

  1. #1
    Member Mauritzvdworm's Avatar
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    Simple C*-algebra

    Suppose we have a simple C*-algebra, say $\displaystyle A$. Show that $\displaystyle M_n(A)$ is also simple.

    My first guess would be to try and obtain some contradiction by assuming $\displaystyle M_n(A)$ has some closed ideal $\displaystyle J$ other than $\displaystyle 0$ or $\displaystyle M_n(A)$ and then find that $\displaystyle A$ will then have some closed ideal other than $\displaystyle 0$ or $\displaystyle A$ which will conclude the proof.

    However, I'm having some trouble with the particulars...

    We can suppose that there is some sequence $\displaystyle \{T^{(m)}\}\in J$ that converges to $\displaystyle T\in J$. Then we have $\displaystyle RT^{(m)},T^{(m)}R\in J$ and also $\displaystyle RT,TR\in J$ for every $\displaystyle R\in M_n(A)$.

    How can we now use this to construct a closed ideal in $\displaystyle A$? (If it is at all possible)
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  2. #2
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    Let $\displaystyle e_{ij}\ (1\leqslant i,j\leqslant n)$ denote the matrix units in $\displaystyle M_n(A)$. Show that $\displaystyle e_{11}Je_{11}$ is a closed ideal in $\displaystyle A$ (where $\displaystyle A$ is identified with $\displaystyle e_{11}M_n(A)e_{11}$). Hence $\displaystyle e_{11}Je_{11} = e_{11}M_n(A)e_{11}$ (unless $\displaystyle e_{11}Je_{11} = \{0\}$, in which case show that $\displaystyle J=\{0\}$). Deduce that $\displaystyle e_{ij}Je_{ij} = e_{ij}M_n(A)e_{ij}$ for all i,j, and so $\displaystyle J = M_n(A)$.
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  3. #3
    Member Mauritzvdworm's Avatar
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    I can show the first part, but I'm not sure how to attack the part where I need to show that $\displaystyle e_{ij}M_n(A)e_{ij}=e_{ij}Je_{ij}$?
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  4. #4
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    Quote Originally Posted by Mauritzvdworm View Post
    I can show the first part, but I'm not sure how to attack the part where I need to show that $\displaystyle e_{ij}M_n(A)e_{ij}=e_{ij}Je_{ij}$?
    The (1,1)-element of the matrix $\displaystyle x$ is the (i,j)-element of the matrix $\displaystyle e_{i1}xe_{1j}$. If an element $\displaystyle x\in J$ can have an arbitrary element of $\displaystyle A$ in its (1,1)-position, then $\displaystyle e_{i1}xe_{1j}$ (which is also an element of $\displaystyle J$) can have an arbitrary element of $\displaystyle A$ in its (i,j)-position.
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