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Let G be a finite group of order n and let A be a (real or complex) Banach algebra. Rudin and Schneider [31 ask whether a mapping f: G -A satisfying IIf(x)II = 1 and f(x) = (l/n)YyEGf(xy ')f(y) is necessarily a homomorphism (Question 1, p. 602). They give an affirmative answer if A is either commutative and semisimple or strictly convex. Here, we prove this result for general Banach algebras, and at the same time prove the natural generalization to compact groups. This allows us to characterize norm one idempotents in generalized group algebras. Suppose that G is a compact group with identity e and that X is a normed space. A representation of G on X is a homomorphism T: G -> B (X). T is an isometric representation if in addition each Tx is an isometry on X; in this case Te = I and each Tx is invertible. T is semi-isometric if TX II X is Bochner measurable if it is the almost everywhere limit of a sequence of simple functions, and Bochner integrable if in addition f J11(x)II dx B (X) is strongly measurable if for each t E X, the map x -> Tx is Bochner measurable (see Hille-Phillips [2, pp. 72-74]). If for each c E X, Txt is Bochner integrable and Tt = JGTx dx, then we shall write T = JGTx dx. LEMMA 1. If x -> Tx is a strongly measurable mapping from G into B (X), then so is the map yi-> Txy -Ty We omit the proof of Lemma 1, which follows by approximation by simple functions. LEMMA 2. If 4: G -> X is Bochner integrable then lim 11-0(ux 4)(u)) I du= 0. PROOF. See [2, Theorem 3.8.3] for the case G = Rn; the same proof applies here. The lemma is proved first for simple functions and follows in general by approximation. Again we omit the details. Received by the editors August 6, 1975. AMS (MOS) subject classifications (1970). Primary 43A20; Secondary 43A65. X) American Mathematical Society 1976

Idempotents of norm one and Banach algebra representations of compact groups