A derivation D on an algebra is a transformation on the algebra such that (i) D(a -4-b) ---D(a) -4-D(b) (ii) D(~t a) = ~ D(a), ~ any scalar (iii) D(a b) = D(a) b + a D(b).We are concerned with derivations on commutative Banach algebras over the complex field, where by a Banach algebra we mean a normed algebra 92 which is complete in its norm. The radical of 9A is the intersection of all maximal ideals M in 91 which are such that 9A/M has a unit. If the radical reduces to the zero element, 91 is called semi.simple.A derivation on 91 is said to be bounded if (iv) sup IID(a)ll = IIDfl < c¢ ~[atr=l Theorem 11): Let ~l be a commutative Banach algebra and D a bounded derivation on 91. Then D maps 91 into its radical. In particular, if 9.1 is semisimple, D = O. Proof of Theorem 1: A non-zero linear functional / on 91 is called multiplicative if/(a b) =/(a) f(b) for all a, b in 91. We need the following result, due to GELFAND : (1} If ~ is multiplicative, 1/(all ~ IIan for each a. Since D is bounded, --n-~---< ~ if t < ~, and so for any complex number ,~ the series --n-i 4 converges to a bounded operator on 91 which we call e ~D. For finite-zt~O dimensional algebras it is well-known 2) that e ~/) (a b) = e ~D (a) e ~D (b) for a, b in 91.Guided by the formal process, we proceed as follows:1) Slimy showed in his paper "On a property of rings of functions", Doklady Akad. Nauk SSSR. (N.S.) 58, 985--988 (1947), that the algebra of all infinitely differentiable functions on an interval cannot be normed so as to be a Banach algebra. Prof. I. Ka-PL~CSttl~ conjectured that the "reason" for this was that non-zero derivations could not exist on a commutative semisimple Banach algebra. Theorem 1 proves this conjecture for bounded derivations. It seems probable that hypothesis (iv) is superfluous.
In this paper we first prove certain formulas concerning rational inner functions in one and two complex variables, and then use these formulas to give operator theoretic consequences, in particular Ando's inequality for two commuting contractions on Hilbert space.The work in this paper is closely related to earlier work by Jim Agler inNote that f ν , g ν are rational functions, analytic on D 2 .
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