Fourier transform as the Gelfand transform of an LCA group algebra

For $b=(z,f)$ and $c=(v,g)$, the norm estimate in [F1] gives $\mathrm{\parallel }bc\mathrm{\parallel }\le \mathrm{\mid }zv\mathrm{\mid }+\mathrm{\mid }z\mathrm{\mid }\mathrm{\parallel }g{\mathrm{\parallel }}_1+\mathrm{\mid }v\mathrm{\mid }\mathrm{\parallel }f{\mathrm{\parallel }}_1+\mathrm{\parallel }f{\mathrm{\parallel }}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1=(\mathrm{\mid }z\mathrm{\mid }+\mathrm{\parallel }f{\mathrm{\parallel }}_1)(\mathrm{\mid }v\mathrm{\mid }+\mathrm{\parallel }g{\mathrm{\parallel }}_1)$. A Cauchy sequence in $B$ has Cauchy scalar and $A$ coordinates; completeness of $\mathbb{C}$ from [F3] and of $A$ from [F1] makes it converge in the sum norm.

Bilinearity and commutativity follow from [F1], and $(1,0)$ is the identity. For $b=(z,f)$, $c=(v,g)$ and $d=(u,k)$, either bracketing of $bcd$ has scalar part $zvu$ and $A$ part $zvk+zug+vuf+z(g\ast k)+v(f\ast k)+u(f\ast g)+(f\ast g)\ast k$, by convolution associativity. Conjugate-linearity, involutivity and isometry of the star follow coordinatewise from [F1]; expanding the product and applying $(f\ast g{)}^{\ast }=g^{\ast }\ast f^{\ast }$ gives $(bc{)}^{\ast }=c^{\ast }{b}^{\ast }$.

In the conjugate-phase notation of [F1], set $\gamma =\bar{w}$. Then $h_{\gamma }^{+}(z,f)=z+{\int }_{G}f(t)\overline{\gamma (t)}dm(t)=h_w(z,f)$. The map $w\mapsto \bar{w}$ is its own inverse and preserves uniform convergence on each compact set, hence is a homeomorphism of the compact-open dual. Therefore [F1] makes every character of $B$ one of the displayed $h_w$ or $q$, with the asserted topology. Applying [F2] gives ${\mathrm{\Gamma }}_B(z,f)(h_w)=h_w(z,f)=z+\widehat{f}(w)$ and ${\mathrm{\Gamma }}_B(z,f)(q)=q(z,f)=z$. Taking $z=0$ and restricting to the dual gives the Fourier/Gelfand identity.