Weighted Fourier and Fourier-Stieltjes Algebras

Let $G$ be a locally compact group and $omega$ be a symmetric weight function on $G$. We define a co-product $Gamma_omega$ on the weighted algebra $L^infty(G, omega^{-1})$ of essentially $omega$-bounded Borel measurable functions on $G$ and show that $L^infty(G, omega^{-1})$ becomes a Kac algebra with natural co-inverse $kappa_omega$ and Haar weight $phi_omega$. We use the machinery of Kac algebras to introduce the weighted Fourier and Fourier-Stieltjes algebra $ A(G,omega^{-1})$ and $ B(G,omega^{-1})$ of $G$.