Vol. 47, No. 1, 1973

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Approximate identities for convolution measure algebras

Charles Dwight Lahr

Vol. 47 (1973), No. 1, 147–159

Let (A,) be a commutative semisimple convolution measure algebra with structure semigroup Γ. It is proved that A has a weak bounded approximate identity if and only if Γ has a finite set of relative units; moreover, Γ has an identity if and only if some weak bounded approximate identity is of norm one. Considering now a commutative semigroup S, the existence of a bounded (norm) approximate identity in A = l1(S) is equivalent to the existence in S of a finite number of nets {up(i)}p(i)∈ℱi,i = 1,2,,n with the property that for every x S there exist j and ρ(j)x such that ρ(j) ρ(j)x implies xuρ(j) = x.

Mathematical Subject Classification 2000
Primary: 46J05
Secondary: 43A10
Received: 27 March 1972
Published: 1 July 1973
Charles Dwight Lahr