Vol. 82, No. 2, 1979

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Convolution cut-down in some radical convolution algebras

Lee Albert Rubel

Vol. 82 (1979), No. 2, 523–525

Let 𝒜 = Lloc1(R+) be the algebra of locally integrable functions on the positive real axis, with convolution as multiplication, given by

(f ∗g)(x) = 0 f(x − t)g(t)dt.

Sometimes 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 𝒜 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 cut-down”; namely whether given a radical convolution algebra A and an f A, there must exist an h A (h0) such that f h L1(R+). We show, at least, that one cannot always choose h L1. As a corollary, we show that simultaneous convolution cut-down is not always possible.

Mathematical Subject Classification 2000
Primary: 43A15
Received: 17 August 1978
Published: 1 June 1979
Lee Albert Rubel