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We study the relationship between the classical invariance properties of amenable locally compact groups G and the approximate diagonals possessed by their associated group algebras L¹(G). From the existence of a weak form of approximate diagonal for L¹(G) we provide a direct proof that G is amenable. Conversely, we give a formula for constructing a strong form of approximate diagonal for any amenable locally compact group. In particular we have a new proof of Johnson's Theorem: A locally compact...
We show that the dual version of our factorization [J. Funct. Anal. 261 (2011)] of contractive homomorphisms φ: L¹(F) → M(G) between group/measure algebras fails to hold in the dual, Fourier/Fourier-Stieltjes algebra, setting. We characterize the contractive w*-w* continuous homomorphisms between measure algebras and (reduced) Fourier-Stieltjes algebras. We consider the problem of describing all contractive homomorphisms φ: L¹(F) → L¹(G).
We show how the measure theory of regular compacted-Borel measures defined on the -ring of compacted-Borel subsets of a weighted locally compact group provides a compatible framework for defining the corresponding Beurling measure algebra , thus filling a gap in the literature.
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