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Tomasz Kania
Tomasz Kania
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The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.

Let $G$ be a locally compact Abelian group and let $bG$ denote its Bohr compactification (the Pontryagin dual of $G$$\widehat{G}$ with the discrete topology). Denote by $\mathfrak{A}$ the space $L_1(G)^{**}$ furnished with either Arens product.

Is there a canonical action of $M(bG)$ (the measure algebra on $bG$) on $L_\infty(G)$ that would give rise to an isometric homomorphism $M(bG)\to \mathfrak{A}$?

The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.

Let $G$ be a locally compact Abelian group and let $bG$ denote its Bohr compactification (the Pontryagin dual of $G$ with the discrete topology). Denote by $\mathfrak{A}$ the space $L_1(G)^{**}$ furnished with either Arens product.

Is there a canonical action of $M(bG)$ (the measure algebra on $bG$) on $L_\infty(G)$ that would give rise to an isometric homomorphism $M(bG)\to \mathfrak{A}$?

The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.

Let $G$ be a locally compact Abelian group and let $bG$ denote its Bohr compactification (the Pontryagin dual of $\widehat{G}$ with the discrete topology). Denote by $\mathfrak{A}$ the space $L_1(G)^{**}$ furnished with either Arens product.

Is there a canonical action of $M(bG)$ (the measure algebra on $bG$) on $L_\infty(G)$ that would give rise to an isometric homomorphism $M(bG)\to \mathfrak{A}$?

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Tomasz Kania
Tomasz Kania
  • 11.3k
  • 2
  • 39
  • 75

Measure algebra on the Bohr compactification vs the bidual algebras

The following question probably reduces to some standard abstract harmonic analysis Twister play, but I'd still welcome some comments on it.

Let $G$ be a locally compact Abelian group and let $bG$ denote its Bohr compactification (the Pontryagin dual of $G$ with the discrete topology). Denote by $\mathfrak{A}$ the space $L_1(G)^{**}$ furnished with either Arens product.

Is there a canonical action of $M(bG)$ (the measure algebra on $bG$) on $L_\infty(G)$ that would give rise to an isometric homomorphism $M(bG)\to \mathfrak{A}$?