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By de la Harpe P., Jones V.

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To see this, one introduces the right regular representation v of ; in `2(;) de ned by (v ) ( 0 ) = ( 0 ) for all 0 2 ; and 2 `2(;): The operators u and v commute for all 0 2 ;: Consider a b 2 Cred(;) such that (a) = (b): As a and b commute with the v 's, one has R C 0 a ( ) = av ( e ) = v a ( e ) = v b ( e ) = bv ( e) = b ( ) for all 2 ; and this shows that a = b: It follows that any a 2 Cred(;) can be written as a sum a= X 2; u: Typeset by AMS-TEX 1 2 4. ABSTRACT C -ALGEBRAS AND FUNCTIONAL CALCULUS We leave it to the reader to check that k (a)k`2(;) kak for all a 2 Cred (;): There isn't any simple condition on a family ( ) 2; of complex numbers which is both necessary and P su cient for the sum 2; u to represent an operator in Cred(;:) A particular case of this statement is that there isn't any simple condition on a trigonometric series which is both necessary and su cient for the series to be the Fourier series of a continuous function.

ABSTRACT C -ALGEBRAS AND FUNCTIONAL CALCULUS 0 1] ! A (ii) If there exists a continuous path of projections t 7! et , show that the same conclusion holds. (iii) Suppose moreover that there exists a tower A1 A2 ::: of sub-C -algebras of A S such that A1 = n 0 An is dense in A (examples will be discussed in Chapter 6). For each projection e 2 A show that there exists a unitary element u 2 A such that ueu 2 A1: (iv) Suppose moreover that 1 2 A1: For each unitary u 2 A and for each > 0 show that there exists a unitary v 2 A1 such that kv ; uk < : Indications.

In case A has no unit, the spectrum (a) of a 2 A is understood in the algebra A~ obtained from A by adding a unit. 41. Lemma. Let A be a C -algebra with unit. (i) For a 2 Asa such that kak 1 one has a 2 A+ () k1 ; ak 1: (ii) The set A+ is closed in Asa : Proof. 21. 14. Then (a) = 1 + (a ; 1) 0 2] + so that a 2 A+ : (ii) For a 2 Asa one has a 2 A+ () kkak ; ak kak by (i), so that A+ is closed in Asa: R R 20 4. 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.

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


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