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 continue our study of derivations, multipliers, weak amenability and Arens regularity of Segal algebras on locally compact groups. We also answer two questions on Arens regularity of the Lebesgue-Fourier algebra left open in our earlier work.

We study the Arens regularity of module actions of Banach left or right modules over Banach algebras. We prove that if has a brai (blai), then the right (left) module action of on * is Arens regular if and only if is reflexive. We find that Arens regularity is implied by the factorization of * or ** when is a left or a right ideal in **. The Arens regularity and strong irregularity of are related to those of the module actions of on the nth dual ${}^{\left(n\right)}$ of . Banach algebras for which Z( **) = but $⊊Z^t(**)$ are...

Let ${\tau }_X$ and ${\tau }_Y$ be representations of a topological group G on Banach spaces X and Y, respectively. We investigate the continuity of the linear operators Φ: X → Y with the property that $\Phi \circ {\tau }_X\left(t\right)={\tau }_Y\left(t\right)\circ \Phi$ for each t ∈ G in terms of the invariant vectors in Y and the automatic continuity of the invariant linear functionals on X.

We find representations for the automorphisms, derivations and multipliers of the Fréchet algebra $L{¹}_{loc}$ of locally integrable functions on the half-line ${ℝ}^{+}$. We show, among other things, that every automorphism θ of $L{¹}_{loc}$ is of the form $\theta ={\phi }_a{e}^{\lambda X}{e}^D$, where D is a derivation, X is the operator of multiplication by coordinate, λ is a complex number, a > 0, and ${\phi }_a$ is the dilation operator $\left({\phi }_{a}f\right)\left(x\right)=af\left(ax\right)$ ($f\in L{¹}_{loc}$, $x\in {ℝ}^{+}$). It is also shown that the automorphism group is a topological group with the topology of uniform convergence on bounded...

For any connected Lie group G and any Laplacian Λ = X²₁ + ⋯ + X²ₙ ∈ 𝔘𝔤 (X₁,...,Xₙ being a basis of 𝔤) one can define the commutant 𝔅 = 𝔅(Λ) of Λ in the convolution algebra ℒ¹(G) as well as the commutant ℭ(Λ) in the group C*-algebra C*(G). Both are involutive Banach algebras. We study these algebras in the case of a "distinguished Laplacian" on the "Iwasawa part AN" of a semisimple Lie group. One obtains a fairly good description of these algebras by objects derived from the semisimple group....

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...

Given a locally compact abelian group G with a measurable weight ω, it is shown that the Beurling algebra L¹(G,ω) admits either exactly one uniform norm or infinitely many uniform norms, and that L¹(G,ω) admits exactly one uniform norm iff it admits a minimum uniform norm.

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...

Let G be a locally compact group with left Haar measure μ, and let L1(G) be the convolution Banach algebra of integrable functions on G with respect to μ. In this paper we are concerned with the investigation of the structure of G in terms of analytic semigroups in L1(G).

The existence of a projection onto an ideal I of a commutative group algebra $L^1\left(G\right)$ depends on its hull Z(I) ⊆ Ĝ. Existing methods for constructing a projection onto I rely on a decomposition of Z(I) into simpler hulls, which are then reassembled one at a time, resulting in a chain of projections which can be composed to give a projection onto I. These methods are refined and examples are constructed to show that this approach does not work in general. Some answers are also given to previously asked...