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Is an element z such that z = Б x Iim S ^x . Since (e - x)* N n=0 N, N+1 (e + x + . . + X ) = e - x e - 0 = e, we obtain (4. 2) by letting N tend to infinity. Definition 4. 2. A continuous linear functional I(x), Banach algebra ^ , is called multiplicative if I(x. y) = I(x)I(y) for all . X, у€^ defined in a 44 CONVOLUTION UNITS AND THE GROUP A LG E B R A Definition 4, 2a. /. 3) such that I(z) / 0. z) = I(e)I(z) and hence 1(e) = I. Let be the unital algebra associated with the algebra For each multiplicative functional I in ^ character I^ of whose restriction to there is a unique ^ is I.

With F « L^) then ||fjj - f|| Lebesgue dominated convergence theorem. 0 by the THE SPACES Given p, C AND L AND TH E IR DUALS I < P< 00 13 , p' will always stand for its conjugate index such that l / p + 1/p' = I (we use the convention l/oo = 0). There exist f, g € L^ for which fg ^ l \ L then fg € L I but if f € and IX always (in particular, if g(x) = e ). More generally, Hôlder's inequality asserts that, if f € L^ and g € L^ , I< P < , then fg € L and 00 I /f(x)g(x)dx| < ||f IlP Ilg lip, We write I^(g) = = f f(x)g(x)dx and I^(Ç) = f ç(x)dii, SOthatif f € and g e = .

4) it follows that v is в. measure (this is immediate for and positive measures, and is obtained in the general case by linear combinations). 1) is satisfied for ф = x-g* Hence it holds for ф any simple function and, by the standard monotone limit process, it holds for any continuous Ф of compact support. (ÿ € Cq Thus I (¢) = 1 (¢) for all у M * and hence also for all Ф€ C^,as wewanted toprove. Proposition 3. 2. If f e L^, I < P < 00 and \i € then V h=f | jl is a function such that (a) h(x) = f f(x - y)dfi(y), x e IR^, (3,5) (b) the integral in (a) exists in the Lebesgue sense a.