### A characterization of algebraic measures

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Let G be a locally compact abelian group and M(G) its measure algebra. Two measures μ and λ are said to be equivalent if there exists an invertible measure ϖ such that ϖ*μ = λ. The main result of this note is the following: A measure μ is invertible iff |μ̂| ≥ ε on Ĝ for some ε > 0 and μ is equivalent to a measure λ of the form λ = a + θ, where a ∈ L¹(G) and θ ∈ M(G) is an idempotent measure.

In the current work, a new notion of $n$-weak amenability of Banach algebras using homomorphisms, namely $(\varphi ,\psi )$-$n$-weak amenability is introduced. Among many other things, some relations between $(\varphi ,\psi )$-$n$-weak amenability of a Banach algebra $\mathcal{A}$ and ${M}_{m}\left(\mathcal{A}\right)$, the Banach algebra of $m\times m$ matrices with entries from $\mathcal{A}$, are studied. Also, the relation of this new concept of amenability of a Banach algebra and its unitization is investigated. As an example, it is shown that the group algebra ${L}^{1}\left(G\right)$ is ($\varphi ,\psi $)-$n$-weakly amenable for any...

We define a Banach algebra 𝔄 to be dual if 𝔄 = (𝔄⁎)* for a closed submodule 𝔄⁎ of 𝔄*. The class of dual Banach algebras includes all W*-algebras, but also all algebras M(G) for locally compact groups G, all algebras ℒ(E) for reflexive Banach spaces E, as well as all biduals of Arens regular Banach algebras. The general impression is that amenable, dual Banach algebras are rather the exception than the rule. We confirm this impression. We first show that under certain conditions an amenable...

We observe that the classical theorem of Hardy on Fourier transform pairs can be reformulated in terms of the heat kernel associated with the Laplacian on the Euclidean space. This leads to an interesting version of Hardy's theorem for the sublaplacian on the Heisenberg group. We also consider certain Rockland operators on the Heisenberg group and Schrödinger operators on ℝⁿ related to them.

We give an elementary proof for the case of the circle group of the theorem of O. Hatori and E. Sato, which states that every measure on a compact abelian group G can be decomposed into a sum of two measures with a natural spectrum and a discrete measure.

Semisimple commutative Banach algebras 𝓐 admitting exactly one uniform norm (not necessarily complete) are investigated. 𝓐 has this Unique Uniform Norm Property iff the completion U(𝓐) of 𝓐 in the spectral radius r(·) has UUNP and, for any non-zero spectral synthesis ideal ℐ of U(𝓐), ℐ ∩ 𝓐 is non-zero. 𝓐 is regular iff U(𝓐) is regular and, for any spectral synthesis ideal ℐ of 𝓐, 𝓐/ℐ has UUNP iff U(𝓐) is regular and for any spectral synthesis ideal ℐ of U(𝓐), ℐ = k(h(𝓐 ∩ ℐ)) (hulls...

Let G be a locally compact group, and consider the weakly almost periodic functionals on M(G), the measure algebra of G, denoted by WAP(M(G)). This is a C*-subalgebra of the commutative C*-algebra M(G)*, and so has character space, say ${K}_{WAP}$. In this paper, we investigate properties of ${K}_{WAP}$. We present a short proof that ${K}_{WAP}$ can naturally be turned into a semigroup whose product is separately continuous; at the Banach algebra level, this product is simply the natural one induced by the Arens products. This...