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And M. Riesz theorem, v is absolutely continuous with respect to m 0 ; let g = dv/dm0. By Proposition 5 (and the remark following its proof), B is weak-star dense in L°°. Since ffg dm0 = ffdv = 0 for all / in B> it follows that g = 0, and hence that v = 0, as desired. We could have defined the algebra A as the uniform closure in C of the polynomials. The uniform density of the polynomials in A follows, for example, from a classical theorem of Fejer [21, p. 17] which states that the Cesaro means of the Fourier series of a function in C converge uniformly to the function; if the function is in A, these Cesaro means are polynomials.

Math. S o c , Providence, R. , 1957. MR 19, 664. 21. K. Hoffman, Banach spaces of analytic functions, Prentice-Hall Ser. in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. , 1962. MR 24 #A2844. 22. K. Hoffman and I. M. Singer, Maximal algebras of continuous functions, Acta Math. 103 (1960), 2 1 7 - 2 4 1 . MR 22 #8318. 23. G. K. Kalisch, A functional analysis proof of Titchmarsh's theorem on convolution, J. Math. Anal. Appl. 5 (1962), 176-183 . MR 25 #4307. 24. P. Lax, Translation invariant spaces, Acta Math.

Hence, the composite map / = FJ0 transforms the group {Wt} into the group of translations on 7,2(— °°, °°). A subspace of L2(~ °°, °°) is invariant under all right translations if and only if it is the image under J of an invariant subspace of U0. This in conjunction with Theorem 5 describes the right-translation invariant subspaces of Z, 2 (- °°, °°). One right-translation invariant subspace of L2(— °°, °°) is the subspace of functions that vanish almost everywhere on (— °°, 0); we may identify this subspace in the obvious way with L2(0y °°).

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