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Representation Theory

  1. May 17, 2006 #1
    If I have a faithful nondegenerate representation of a C*-algebra, A:

    [tex]\pi\,:\,A \rightarrow B(\mathcal{H})[/tex]

    where [itex]B(\mathcal{H})[/itex] is the set of all bounded linear operators on a Hilbert space. And just suppose that [itex]a\geq 0 \in A[/itex]. How is the fact that a is positive got anything to do with [itex]\pi(a)[/itex] being positive?

    Apparantly there is an if and only if relationship!? How does one begin to prove something like [itex]a\geq 0 \Leftrightarrow \pi(a) \geq 0 \in B(\mathcal{H})[/itex]?
     
  2. jcsd
  3. May 17, 2006 #2

    matt grime

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    Define positive for me again for a C* alg.
     
  4. May 17, 2006 #3
    Let [itex]A[/itex] be a C*-algebra, and [itex]a,b \in A[/itex]. Then [itex]a\geq b[/itex] if [itex]a-b[/itex] has the form [itex]c^*c[/itex] for some [itex]c\in A[/itex]. In particular, [itex]a[/itex] is positive if [itex]a = c^*c[/itex] for some [itex]c \in A[/itex].
     
  5. May 18, 2006 #4

    matt grime

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    Well, one implication is obvious a positive implies pi(a) positive since pi is a *-homomorphism.

    Conversely, hmm, well, faithfulness and nondegeneracy must come into it somewhere.
     
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