Vol. 23, No. 1, 1967

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An L1 algebra for certain locally compact topological semigroups

Neal Jules Rothman

Vol. 23 (1967), No. 1, 143–151

This paper may be considered as another chapter in the theory of convolution algebras inaugurated by Hewitt and Zuckerman. The interest here is in finding an L1 theory for locally compact commutative topological semigroups which extends the known l1 theory for discrete commutative semigroups and L1 theory for locally compact topological groups. Let S be a locally compact commutative semigroup and m a nonnegative regular Borel measure on S such that if x ∕S and E S with m(E) = 0 then m(Ex1) = 0(Ex1 = [y : yx E]). When L1(S,m) is defined as the Banach space of all bounded complex measures μ M(S) which are absolutely continuous with respect to m, then L1(S,m) is a convolution algebra as a subalgebra of M(S). It is shown that there is a one to one correspondence between the measurable semicharacters on S and the multiplicative linear funclionals on L1(S,m) analogus to the group situation. Extensions of the above results to those S with a measure m satisfying the above condition in a local sense are also obtained.

Mathematical Subject Classification
Primary: 46.80
Received: 25 January 1966
Published: 1 October 1967
Neal Jules Rothman