By de la Harpe P., Jones V.

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As a vector space, set A~ = A: The involution is de ned on A~ by ( a) = and the product by ; a ( a)( b) = ( b + a + ab): In particular A is a two-sided ideal in A~ which is invariant by the involution. The algebra A~ has a unit e~ = (1 0): In case A itself has a unit e then A~ is the product of its two-sided ideals (~e ; e) and A: If A is moreover an involutive Banach algebra, then A~ is also an involutive Banach algebra for the norm de ned by k( a)k = j j + kak : However, this choice of a norm on A~ is often not the best one: for example, if A is a C -algebra, it does not make A~ a C -algebra.

42. Lemma. Let A be a complex algebra with unit and let a b 2 A: Then C (ab) f0g = (ba) f0g: R R In particular, if A is a C -algebra and if b 2 A then (bb ) + if and only if (b b) +: Proof. Let 2 be such that ; ab has an inverse, say x. 43. Proposition. Let A be a C -algebra. For each a 2 Asa the three following ;1 R properties are equivalent (i) (a) + namely a 2 A+ (ii) there exists b 2 A such that a = b b (iii) there exists b 2 Asa such that a = b2: Moreover (iv) A+ is a closed convex cone in Asa (v) A+ \ (;A+) = f0g: Proof.

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.

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

by William

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