About an isomorphism between the Beurling algebra with a weight dependent convolution and the $L^1(G)$ group algebra

TL;DR

The paper shows that the Beurling algebra with a weight-dependent convolution, $\mathscr{L}^1(G,\omega)$, is isomorphic to the classical group algebra $L^1(G)$ when the weight $\omega$ is symmetric. Using this isomorphism, it transfers the structure and proves an explicit correspondence between representations: unitary representations of the locally compact group $G$ correspond to non-degenerate $*$-representations of $\mathscr{L}^1(G,\omega)$, with explicit intertwining relations given by the weight-adjusted translations $\Gamma_\omega^s$ and $\Theta_\omega^s$. The main contributions include the construction of $\pi(f)$ from a unitary representation and the reconstruction of a unitary representation from a non-degenerate $*$-representation, thereby extending representation-theoretic results to the Beurling setting. This provides a unifying framework for transferring results between $L^1(G)$ and $\mathscr{L}^1(G,\omega)$ and clarifies the role of weight-dependent convolution in abstract harmonic analysis.

Abstract

We show that the Beurling algebra with a weight-dependent convolution and the group algebra $L^1(G)$ are isomorphic. In particular, using this isomorphism, we extend some results of the algebra $\mathscr{L}^1(G,ω)$ presented in recent articles. As a main result, we explicitly construct the equivalence between unitary representations of the group and non-degenerate $\ast$-representations of this algebra.

Theorems & Definitions (36)

• Definition 1.1
• Theorem 1.2
• Definition 1.3
• Proposition 1.4
• Definition 1.5
• Theorem 1.6
• Lemma 1.7
• proof
• Definition 1.8
• Definition 1.9