Fourier algebra

Eymard algebra

Let $G$ be a locally compact group. The algebra $A _ { 2 } ( G )$ (see also Figà-Talamanca algebra for notations) is called the Fourier algebra of $G$. In fact,

\begin{equation*} A _ { 2 } ( G ) = \left\{ \overline { k } * \breve{ l } : k , l \in \mathcal{L} _ { C } ^ { 2 } ( G ) \right\} \end{equation*}

and for $u \in A _ { 2 } ( G )$,

\begin{equation*} \| u \| A _ { 2 ^{ ( G )}} = \operatorname { inf } \{ N _ { 2 } ( k ) N _ { 2 } ( l ) : k , l \in \mathcal{L} _ { \text{C} } ^ { 2 } ( G ) , u = \overline { k } * \check{l} \}. \end{equation*}

Let $G$ be Abelian and let $\varepsilon$ be the canonical mapping from $G$ onto $\widehat {\widehat {G} }$. Then $f \mapsto ( \widehat { f } \circ \varepsilon )$, where $ \check{\varphi} ( \chi ) = \varphi ( \chi ^ { - 1 } )$, is an isometric isomorphism of the Banach algebra $L ^ { 1 } ( \hat { G } )$ onto $A _ { 2 } ( G )$. Therefore $A _ { 2 } ( G )$ can be considered as a substitute of $L _ { \text{C} } ^ { 1 } ( \hat { G } )$ if $G$ is non-Abelian.

One always has $P M _ { 2 } ( G ) = C V _ { 2 } ( G )$: $A _ { 2 } ( G )$ is precisely the pre-dual of the von Neumann algebra $C V _ { 2 } ( G )$. For the definition of $C V _ { 2 } ( G )$ and $P M _ { 2 } ( G )$, see Figà-Talamanca algebra. Consequently, in analogy with the Abelian case, $A _ { 2 } ( G )$ is weakly $\sigma ( A _ { 2 } ( G ) , C V _ { 2 } ( G ) )$ sequentially complete.

The existence, for $G$ not amenable, of approximate units in $A _ { 2 } ( G )$ is still in doubt (as of 2000). However, such exist for all closed subgroups of $SO ( n , 1 )$ and $ \operatorname{SU} ( n , 1 )$. Such approximate units in $A _ { 2 }$ can be used in the study of lattices of non-compact simple Lie groups of real rank one ([a1]).

For a study of certain ideals of $A _ { 2 } ( G )$, see [a2], [a4].

The unimodular case was first investigated by W.F. Stinespring, using a very interesting non-commutative integration theory [a6]. The case of a general locally compact group $G$ was initiated by P. Eymard [a3] on the basis of an extensive use of the theories of $C ^ { * }$-algebras and von Neumann algebras.

If $G$ is non-Abelian, $A _ { 2 } ( G )$ is also called the Eymard algebra of $G$ and is denoted by $A ( G )$.

See also Fourier–Stieltjes algebra.

References

[a1]	M. Cowling, U. Haagerup, "Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one." Invent. Math. , 96 (1989) pp. 507–549
[a2]	J. Delaporte, A. Derighetti, "Best bounds for the approximate units of certain ideals of $L ^ { 1 } ( G )$ and of $A _ { p } ( G )$" Proc. Amer. Math. Soc. , 124 (1996) pp. 1159–1169
[a3]	P. Eymard, "L'algèbre de Fourier d'un groupe localement compact" Bull. Soc. Math. France , 92 (1964) pp. 181–236
[a4]	E. Kaniuth, A.T. Lau, "A separation property of positive definite functions on locally compact groups and applications to Fourier algebras" J. Funct. Anal. , 175 (2000) pp. 89–110
[a5]	J.-P. Pier, "Amenable locally compact groups" , Wiley (1984)
[a6]	W.F. Stinespring, "Integration theorems for gages and duality for unimodular groups" Trans. Amer. Math. Soc. , 90 (1959) pp. 15–56