Sr~ozR, I. M., and J. Wz~B Math. Annalen, Bd. 129, S. 260--264 (1955). - PDF document

Sr~ozR, I. M., and J. Wz~B Math. Annalen, Bd. 129, S. 260--264 (1955). Derivations on Commutative Normed Algebras. By I. M. SINGER and J. WERMER in New York City and Providence, Rhode Island. A derivation D on an algebra is a transformation on the algebra such (al) (a)), a in 9.1. Then q~ is a linear functional and we claim ~ is multi- plieative. For by (i) and (iii), _ D ~(a) D i(b) Dn(ab) ~ i! j' n! i+j=n / (D n (ab)) ~ / (D i (a)). / (D i ~ • n=O i+j~n (i~_O A~/(D~(a)) i i i ()~ flail = ~ function of of the the die Unbesehr~nktheit der Operatoren der Quantenmechanik", Math. Ann. 121, 21 (1949---50)] that if a, b are bounded operators on a normed vector space, then a b b a # a b b a a b b a = c b IID(c)ll .2 2 -~ = 0 = a b b a of Math. C a t o n = L n L n (to) = ~to the functional ~b *-a~ CO D / = 1, D = ax, a 2 = D a 2 + B ? We need = d~(ax) • q~(az) + there exists a non.zero (bounded) then there a commutative extension B a non-zero (bounded) deri- vation D I t g B a non.zero (bounded) derivation into B but the radical then there a non.zero (bounded) 2 a (a2, ~s) = (as multiplicative functional functional be the set of all (a, ~) with 2 = in B, ) (a2, ) + To prove the partial = ~ ) = B, where as follows. follows. ~(a)= 0}, where ~ is a multiplicative linear functional. Then M~ is M~ be M~ be a = • 1, = l((a~ 1) (a~ • 1)) + q~(ax) + ~(a~) multiplicative functional ~ 0 have the: M~ for ] E =/+--/- where/+,/- f 6 "On the Amer. Math. are bounded A C C A B C to be Brown Univ. Dezember 1954.)