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Characterization, structure and analysis on Abelian ${ℒ}_{\infty }$ groups. (English) Zbl 0563.43005
An abelian topological group is an ${ℒ}_{\infty }$ group if and only if it is a locally $\sigma$-compact k-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian ${ℒ}_{\infty }$ group G is topologically isomorphic to ${ℝ}^{\alpha }\oplus {G}_{0}$ where $\alpha \le {\aleph }_{0}$ and ${G}_{0}$ is an abelian ${ℒ}_{\infty }$ group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra, ${L}_{1}$-algebra, Fourier transforms of abelian ${ℒ}_{\infty }$ groups are given and their properties are studied.

##### MSC:
 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 43A20 ${L}^{1}$-algebras on groups, semigroups, etc.
##### References:
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