By de la Harpe P., Jones V.
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Extra resources for An introduction to C-star algebras
Let A be a commutative C -algebra with unit and let a 2 A: Assume that a generates A: Then the mapping ( X (A) ;! 7;! (a) (a) is an homeomorphism. Proof. 30. To show that the mapping is injective, consider 1 2 2 X (A) such that 1(a) = 2(a): The set f b 2 A j 1(b) = 2(b) g is a sub-C algebra of A which contains 1 it is therefore A itself, and 1 = 2 : Hence the mapping is a homeomorphism. 33. Ideals. Let X be a locally compact space and let A = Co (X ) be the corresponding abelian C -algebra. g. Bou], chap.
Exercice. Structure of the generic Temperley-Lieb algebras. P. de la Harpe and V. Jones, July 1995. CHAPTER 3. COMPACT AND HILBERT-SCHMIDT OPERATORS Let H and H denote Hilbert spaces. a. 1. De nition. A bounded operator a : H ! H is nite rank if the image a(H) is nite 0 dimensional. 2. Lemma. For any bounded operator a : H ! H one has Im(a ) = Ker(a) Ker(a ) = Im(a) : 0 0 ? Proof. It is straightforward to check that Im(a ) Ker(a) and also that (Im(a )) Ker(a) namely that Im(a ) Ker(a) : The rst equality follows.
14. Lemma. Let a be a compact self-adjoint operator on H: Then one at least of the numbers kak ;kak is an eigenvalue of a: Proof. 1 j j a(a )i 2 f1 ;1g: j Upon replacing ( j )j N by a subsequence, one may assume by compacity of a that the sequence (a ( j ))j N converges to some 2 H(1): Then 2 2 h j a i 2 f1 ;1g and the proof follows from the previous observation. 15. Proposition. Let a be a compact self-adjoint operator on H: Then there exists an orthonormal basis ( j )j J consisting of eigenvectors of a and the corresponding sequence ( j )j J of eigenvalues converges to 0: Proof.