By Courant R.

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**Extra info for Advanced methods in applied mathematics, lecture course**

**Example text**

Hence M is self-adjoint. = For any T E 1 G ( A J ) we define two linear functionals on Ad by T f ( X ) = tr T X , f T ( X ) = tr Xt*T, X E M. 7. (1) fT = T f for each T E ~ l ( d ~ ) . A/l)+. (3) Any fT, T E ~l(Ykd), is written as fT = fT1 -- fT2 + i(fTa -- fT4) whereby T j c G](A4)+,j = 1,... ,4. Proof. (1) Let T = E tn~n | ~ be a canonical representation of T. n Since T E G I ( A 4 ) , we have {G~}, {r/~} c 79, so t h a t X T = ~ _ t n X ~ n and T X = E t~ | Xtrl~ for each X E A4. 9 Trace functionals on O*-algebras for each X E M .

L be a canonical representation of T. By the closed graph t h e o r e m we have tM = tot(v). t. s proof. 6. T h e following s t a t e m e n t s are equivalent: (i) A4 is self-adjoint. (ii) I~I(Ad ) = 1 G ( M ) . P r o o f . 3, (2). (ii) =* (i) Take an a r b i t r a r y ~ E 79"(M). T h e n we have ~ @ ~ E I ~ ( M ) G I ( M ) , which implies ~ E 79. Hence M is self-adjoint. = For any T E 1 G ( A J ) we define two linear functionals on Ad by T f ( X ) = tr T X , f T ( X ) = tr Xt*T, X E M. 7. (1) fT = T f for each T E ~ l ( d ~ ) .

T . M . n=l P r o o f . (1) It follows from Kgthe [1], w T ~ s 5 that (~ | ~ (5 C i l ( ~ ) and tr T = ~ ( ~ l r / ~ ) . 4 and x (5 7-/. 9 Trace functionals on O*-algebras 33 oo which implies by the closedness of Ad that T T / c D and X T = E n X~ | 1 Similarly, we have T * ~ C D. Hence it. 2) that X T E ~1(~) oo and tr X T = E ( X ~ , d r b J , which means that T E ~ l ( 3 A ) . n 1 (2) Since X t X T , YtYT* E G I ( ~ ) for all X , Y E A/l, it. lI 2) 1/2 n n This completes the proof. 5. (1) Every element T of GI(Ad) can be written as T = (T1 - 1 3 ) + i ( T 3 T4) with 7~ E GI(M)+(i = 1 , .

### Advanced methods in applied mathematics, lecture course by Courant R.

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