By de la Harpe P., Jones V.

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Example text

If ( a) = j j (a) < 1 it follows from the de nition above that the series P1 (i) n an is convergent, and its limit is (1 ; a);1 : n=0 ; If j j > (a) the previous claim implies that ; a = 1 ; ( );1 a is invertible. (ii) Let a 2 Ainv : For each b 2 A such that kb ; ak < a;1 ;1 the element ; b = a 1 ; a;1(a ; b) 4. ABSTRACT C -ALGEBRAS AND FUNCTIONAL CALCULUS ; is in Ainv because a;1 (a ; b) 1 a;1 ;1 one has moreover 2 b;1 ; a;1 = 7 a;1 (a ; b) < 1: Hence Ainv is open. If kb ; ak 1 X ; 1 X n=0 n=1 a;1 (a ; b) n a;1 ; a;1 a;1(a ; b) n a;1 ka ; bk 2 a;1 2 ka ; bk 1 ; ka (a ; b)k and it follows that a 7!

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. If they are summable, the three sums have the same value, which depends consequently only on a and not on the choosen basis. Proof. By Parseval's equality, one has ka k j 2 = X k 0 2K jh j a ij k and 2 j 0 ka k = X jh j a ij k 2 j0 k 2 j 0 2J for all j 2 J and k 2 K: If any of the families above is summable, one has X j 2J ka k j 2 = X j 2J k2K and the proof is complete.

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. (i) Set rst g = 1 ; e0 ; e1 + 2e0e1 : Check that e0g = ge1 and that g is invertible if ke1 ; e0k is small enough. Use functional calculus to de ne u = g(g g);1=2 check it solves (i), and that (ii) follows. (iii) Let x 2 A1 be such that x = x and such that ke ; xk is small enough, and let n 1 be such that x 2 An: Using functional calculus in An one nds a projection f 2 An such that kf ; ek is small, so that (iii) follows from (i).

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