View source for Fourier-Stieltjes algebra

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Let $G$ be an arbitrary locally [[Compact group|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|Unitary representation]]). The completion of $L _ { \mathbf{C} } ^ { 1 } ( G )$ with respect to this norm is a [[Banach algebra|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 [[#References|[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 $G$ ([[#References|[a3]]]; see also [[#References|[a9]]], [[#References|[a6]]]).

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

a) The [[Fourier-algebra(2)|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 [[#References|[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 [[#References|[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 ([[#References|[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 [[#References|[a12]]] proved the converse assertion: if (a1) holds, then $G$ must be amenable!

In a difficult paper [[#References|[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 [[#References|[a2]]], [[#References|[a5]]].

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

See also [[Figà-Talamanca algebra|Figà-Talamanca algebra]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Cowling,   "The Fourier–Stieltjes algebra of a semisimple Lie group"  ''Colloq. Math.'' , '''41'''  (1979)  pp. 89–94</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  M. Cowling,   G. Fendler,   "On representations in Banach spaces"  ''Math. Ann.'' , '''266'''  (1984)  pp. 307–315</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Derighetti,   "Some results on the Fourier–Stieltjes algebra of a locally compact group"  ''Comment. Math. Helv.'' , '''45'''  (1970)  pp. 219–228</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  P. Eymard,   "L'algèbre de Fourier d'un groupe localement compact"  ''Bull. Soc. Math. France'' , '''92'''  (1964)  pp. 181–236</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  G. Fendler,   "An $L ^ { p }$-version of a theorem of D.A. Raikov"  ''Ann. Inst. Fourier (Grenoble)'' , '''35''' :  1  (1985)  pp. 125–135</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  E.E. Granirer,   M. Leinert,   "On some topologies which coincide on the unit sphere of the Fourier–Stieltjes algebra $B ( G )$ and of the measure algebra $M ( G )$"  ''Rocky Mount. J. Math.'' , '''11'''  (1981)  pp. 459–472</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  C. Herz,   "Une généralisation de la notion de transformée de Fourier–Stieltjes"  ''Ann. Inst. Fourier (Grenoble)'' , '''24''' :  3  (1974)  pp. 145–157</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  B. Host,   "Le théorème des idempotents dans $B ( G )$"  ''Bull. Soc. Math. France'' , '''114'''  (1986)  pp. 215–223</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  K. McKennon,   "Multipliers, positive functionals, positive-definite functions, and Fourier–Stieltjes transforms"  ''Memoirs Amer. Math. Soc.'' , '''111'''  (1971)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  M. Lefranc,   "Sur certaines algèbres sur un groupe"  ''C.R. Acad. Sci. Paris Sér. A'' , '''274'''  (1972)  pp. 1882–1883</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  M. Lefranc,   "Sur certaines algèbres sur un groupe"  ''Thèse de Doctorat d'État, Univ. Sci. et Techn. du Languedoc''  (1972)</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  V. Losert,   "Properties of the Fourier algebra that are equivalent to amenability"  ''Proc. Amer. Math. Soc.'' , '''92'''  (1984)  pp. 347–354</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  J.-P. Pier,   "Amenable locally compact groups" , Wiley  (1984)</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  M.E. Walter,   "$W ^ { * }$-algebras and nonabelian harmonic analysis"  ''J. Funct. Anal.'' , '''11'''  (1972)  pp. 17–38</td></tr></table>