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C*-Algebras and Operator Theory: 2.4 Compact Hilbert Space Operators

January 18, 2011 in C*-Algebras & Operator Theory, Research | 2 comments

This starts on P.58 rather than p.53. I underline my own extra explanations, calculations, etc. The notation varies on occasion from Murphy to notation which I prefer.

H is always a Hilbert space. H^1 is the set of unit vectors.

If P is a finite-rank projection on H, then the C*-algebra A=PB(H)P is finite dimensional. To see this, write P=\sum_{j=1}^ne_j\otimes e_j, where e_1,\dots,e_n\in H. If T\in B(H), then

PTP=\sum_{j,k=1}^n(e_j\otimes e_j)T(e_k\otimes e_k)=\sum_{j,k=1}^n\langle Te_k,e_j\rangle (e_j\otimes e_k)

Hence, A\subset <e_j\otimes e_k>, (j,k=1,\dots,n) (*), and therefore is finite dimensional.

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