Vol. 54, No. 1, 1974

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Symmetric maximal ideals in M(G)

Sadahiro Saeki

Vol. 54 (1974), No. 1, 229–243
Abstract

Let G be a nondiscrete locally compact abelian group, and M(G) the convolution algebra of bounded regular measures on G. In this paper, the followin g is proved: Let {λk}k=0 be a countable subset of Mc+(G),0λ0 M0(G), and {Ck}k=0 a countable family of σ-compact subsets of G such that λk(x + Ck) = 0 for all x G and all k = 0,1,2, . Then there exists a nonzero measure σ M0+(suppλ0) with compact support such that λk[x + Ck + Gp(suppσ)] = 0 for all x G and all k = 0,1,2, . A consequence of this result is the following: Let Y be the closed ideal in M(G) which is generated by ∪{L1(λk) : k = 0,1,2,} for some countable subset {λk}λ=0 of Mc(G). Then there exist “fairly many” symmetric maximal ideals in M(G) which contain ∪{L1(μ) : μ Y }∪ Ma(G) but not M0(G). Here L1(μ) denotes the set of the measures in M(G) which are absolutely continuous with respect to |μ|.

Mathematical Subject Classification 2000
Primary: 43A10
Milestones
Received: 15 March 1973
Revised: 22 June 1973
Published: 1 September 1974
Authors
Sadahiro Saeki