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The most function of those lectures is first to in short survey the elemental con­ nection among the illustration thought of the symmetric staff Sn and the speculation of symmetric capabilities and moment to teach how combinatorial equipment that come up certainly within the conception of symmetric features result in effective algorithms to precise a number of prod­ ucts of representations of Sn by way of sums of irreducible representations.

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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.