# Fourier-Stieltjes algebra

Let $G$ be an arbitrary locally compact group. For $f \in L _ { \mathbf{C} } ^ { 1 } ( G )$, let $\| f \| = \operatorname { sup } \{ \| \pi ( f ) \| : \pi \in \Sigma \}$, where $\Sigma$ is the set of all equivalence classes of unitary continuous representations of $G$ (cf. also Unitary representation). The completion of $L _ { \mathbf{C} } ^ { 1 } ( G )$ with respect to this norm is a Banach algebra, denoted by $C ^ { * } ( G )$ and called the full $C ^ { * }$-algebra of $G$. If $G$ is Abelian and $\hat { C }$ its dual group, then $C ^ { * } ( G )$ is isometrically isomorphic to the Banach algebra $C _ { 0 } ( \hat { G } ; \mathbf{C} )$ of all complex-valued continuous functions on $\hat { C }$ vanishing at infinity.

Let $B ( G )$ be the complex linear span of the set of all continuous positive-definite functions on $G$.

1) The $\mathbf{C}$-vector space $B ( G )$ is isomorphic to the dual space of $C ^ { * } ( G )$. With the dual norm and the pointwise product on $G$, $B ( G )$ is a commutative Banach algebra [a4].

This Banach algebra is called the Fourier–Stieltjes algebra of $G$. If $G$ is Abelian, then $B ( G )$ is isometrically isomorphic to the Banach algebra of all bounded Radon measures on $\hat { C }$.

2) On the boundary of the unit ball of $B ( G )$ (i.e. on $\{ u \in B ( G ) : \| u \| _ { B ( G ) } = 1 \}$) the weak topology $\sigma ( \mathcal{L} _ { \mathbf{C} } ^ { \infty } ( G ) , \mathcal{L} _ { \mathbf{C} } ^ { 1 } ( G ) )$ coincides with the compact-open topology on $G$ ([a3]; see also [a9], [a6]).

3) The following properties are satisfied ([a4]):

a) The Fourier algebra $A ( G )$ is a closed ideal of $B ( G )$;

b) $B ( G ) \cap C _ { 00 } ( G ; \mathbf{C} ) \subset A ( G )$;

c) $A ( G )$ coincides with the closure in $B ( G )$ of $B ( G ) \cap C _ { 00 } ( G ; {\bf C} )$;

d) $B ( G ) = B ( G _ { d } ) \cap C ( G ; \mathbf{C} )$, with equality of the corresponding norms. Here, $C_{00} ( G ; \mathbf{C} )$ is the algebra of functions of compact support on $G$. In [a14], M.E. Walter showed that $B ( G )$ (and also $A ( G )$) completely characterizes $G$. More precisely, assume that $G_1$ and $G_2$ are locally compact groups; then the following assertions are equivalent:

the locally compact groups $G_1$ and $G_2$ are topologically isomorphic;

the Banach algebras $B ( G _ { 1 } )$ and $B ( G_{2} )$ are isometrically isomorphic;

the Banach algebras $A ( G _ { 1 } )$ and $A ( G _ { 2 } )$ are isometrically isomorphic.

He also gave a description of the dual of $B ( G )$.

For a connected semi-simple Lie group $G$, M. Cowling [a1] has given a description of the spectrum of $B ( G )$; surprisingly, if $G$ is Abelian, then the spectrum of $B ( G )$ seems to be much more complicated than in the non-Abelian case!

If $G$ is amenable, then ([a3])

\begin{equation} \tag{a1} B ( G ) = \{ u \in \mathbf{C} ^ { G } : u v \in A ( G ) \text { for every } \ v \in A ( G ) \}. \end{equation}

V. Losert [a12] proved the converse assertion: if (a1) holds, then $G$ must be amenable!

In a difficult paper [a7], C.S. Herz tried to extend the preceding results, replacing unitary representations by representations in Banach spaces. He partially succeeded in the amenable case. See also [a2], [a5].

M. Lefranc generalized Paul Cohen's idempotent theorem to $B ( G )$ for arbitrary locally compact groups $G$ ([a10], [a8]; see also [a11] for detailed proofs).