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2 2 ? 2 2 0 0 ? 2 0 2 ? 3. ii). (ii) Show that the Calkin algebra B(H)=K(H) is simple (more on this in Har]). (iii) For any a 2 K(H) it is known that there exists a two-sided ideal J in B(H) such that a 2 J $ K(H) (see Sal]). b. 17. De nition. A bounded operator a : H ! H is Hilbert-Schmidt if it is compact 0 and if the series ( j )j J of the eigenvalues of a a is summable. 18. Lemma. Let a : H ! H be a bounded operator. Let ( j )j J be an orthonormal basis of H and let ( k )k negative real numbers ka k j 0 2 K be an orthonormal basis of ka k 2 j 2J 2 0 ;jh j a ij2 k j j 2 k H : The three families of non k2K J k2K 2 are simultaneously summable or not.

36. Conversely, let : Y ! X be a surjective continuous map of Y onto some compact space X: Then AX = b 2 C (Y ) j there exists a 2 C (X ) such that b = a is a sub-C -algebra of B containing the unit. Thus there is a bijective correspondance between unital sub-C -algebras of B and compact quotients of Y: 4. 39. Proposition. Let A B be two commutative C -algebras with units and let : A ! B be a morphism of C -algebras such that (1A ) = 1B : Then there exists a closed subset Z of the character space X (A) of A and a surjective continuous map : X (B) !

I) Let a 2 A be a normal element. Then T (a) = kak : In particular, if a 2 A is self-adjoint, then one at least of kak ; kak is in the spectrum (a): (ii) Let H be a Hilbert space and let a 2 B(H) be a self-adjoint element. Set m(a) = inf h ja i 2H k k 1 then M (a) = sup h ja i : and 2H k k 1 m(a) M (a)]: Proof. (i) Assume rst that a is self-adjoint. 1 2;k = kak : 4. ii that at least one of the numbers kak ;kak is in (a): Assume now that a is normal. 18. 18. 22. Corollary. 23. Remark. There are elements a 2 A such that (a) < kak : The simplest example is probably the nilpotent operator a = 00 10 2 B( 2 ): Another example is the Volterra C integration operator V de ned on L2 ( 0 1]) by (V f )(x) = Z 0 x f (t)dt: It is a quasi-nilpotent operator, namely one with spectral radius equal to zero.

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An introduction to C-star algebras by de la Harpe P., Jones V.

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