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])

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

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).

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