Convolution cut-down in some radical convolution algebras
Lee A. Rubel · Pacific Journal of Mathematics · 1979
Let ^=Lϊ oc tR + ) be the algebra of locally integrable functions on the positive real axis, with convolution as multiplication, given by (f*g)(x)=\"f(x-t)g(t)dt. JoSometimes it is convenient to think of our functions as being defined on all of R, but vanishing for negative x.We are interested in subalgebras of ^/ that are Banach algebras in some norm, and that are radical in the sense that there exist no (nontrivial) complex homomorphisms.We call these algebras radical convolution algebras.Such algebras A present a challenge because there is no Fourier transform for them.We are concerned with the problem of "convolution cutdown"; namely whether given a radical convolution algebra A and an/eA, there must exist an heA(hΦ0 such that fahε&iR*).We show, at least, that one cannot always choose h e ZΛ As a corollary, we show that simultaneous convolution cut-down is not always possible.1 ?This problem arose in discussion with Jamil A.