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Fourier and related algebras occur naturally in the [[Harmonic analysis|harmonic analysis]] of locally compact groups (cf. also [[Harmonic analysis, abstract|Harmonic analysis, abstract]]). They play an important role in the duality theories of these groups.

==Fourier–Stieltjes algebra.==
The Fourier–Stieltjes algebra $B ( G )$ and the Fourier algebra $A ( G )$ of a locally compact group $G$ were introduced by P. Eymard in 1964 in [[#References|[a2]]] as respective replacements, in the case when $G$ is not Abelian, of the measure algebra $M ( \hat { G } )$ of finite measures on $\hat { C }$ and of the convolution algebra $L _ { 1 } ( \hat { G } )$ of integrable functions on $\hat { C }$, where $\hat { C }$ is the character group of the Abelian group $G$ (cf. also [[Character of a group|Character of a group]]). Indeed, if $G$ is a locally compact Abelian group, the Fourier–Stieltjes transform of a finite measure $\mu$ on $\hat { C }$ is the function $\hat{\mu}$ on $G$ defined by

\begin{equation*} \hat { \mu } ( x ) = \int _ { \hat{G} } \overline { \chi ( x ) } d \mu ( \chi ) , x \in G, \end{equation*}

and the space $B ( G )$ of these functions is an algebra under [[pointwise multiplication]], which is isomorphic to the measure algebra $M ( \hat { G } )$ (cf. also [[Algebra of measures|Algebra of measures]]). Restricted to $L _ { 1 } ( \hat { G } )$, viewed as a subspace of $M ( \hat { G } )$, the Fourier–Stieltjes transform is the [[Fourier transform|Fourier transform]] on $L _ { 1 } ( \hat { G } )$ and its image is, by definition, the Fourier algebra $A ( G )$. The generalized Bochner theorem states that a [[Measurable function|measurable function]] on $G$ is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on $\hat { C }$ if and only if it is positive definite. Thus, $B ( G )$ can be defined as the linear span of the set $P ( G )$ of continuous positive-definite functions on $G$. This definition is still valid when $G$ is not Abelian.

Let $G$ be a locally compact group. The elements of $B ( G )$ are exactly the matrix elements of the unitary representations of $G$: $\varphi \in B ( G )$ if and only if there exist a [[Unitary representation|unitary representation]] $\pi$ of $G$ in a [[Hilbert space|Hilbert space]] $H$ and vectors $\xi , \eta \in H$ such that

\begin{equation*} \varphi ( g ) = ( \xi , \eta ) ( g ) : = ( \pi ( g ) \xi , \eta ). \end{equation*}

The elements of $P ( G )$ are the matrix elements $( \xi , \xi )$. Because of the existence of the tensor product of unitary representations, $B ( G )$ is an algebra under pointwise multiplication. The norm defined as $\| \varphi \| = \operatorname { inf } \| \xi \| \| \eta \|$, where the infimum runs over all the representations $\varphi = ( \xi , \eta )$, makes it into a [[Banach algebra|Banach algebra]]. The Fourier algebra $A ( G )$ can be defined as the norm closure of the set of elements of $B ( G )$ with compact support. It consists exactly of the matrix elements of the [[Regular representation|regular representation]] on $L _ { 2 } ( G )$; equivalently, its elements are the functions of the form $\xi ^ { * } \widetilde { \eta }$, where $\xi , \eta \in L _ { 2 } ( G )$ and $\widetilde{\eta} ( x ) = \eta ( x ^ { - 1 } )$. It is a closed ideal in $B ( G )$.

The most visible role of $B ( G )$ and $A ( G )$ with respect to duality is that $B ( G )$ is the dual of the [[C*-algebra|$C ^ { * }$-algebra]] $C ^ { * } ( G )$ of the group $G$ and $A ( G )$ is the pre-dual of the [[Von Neumann algebra|von Neumann algebra]] $W ^ { * } ( G )$ of its regular representation. The pairing is given by $\langle \varphi , T \rangle = ( \pi ( T ) \xi , \eta )$, where $\varphi = ( \xi , \eta ) \in B ( G )$ and $T \in C ^ { * } ( G )$. The comparison with a similar result for $M ( G )$ and $L _ { 1 } ( G )$, namely $M ( G )$ is the dual of the Banach space $C _ { 0 } ( G )$ of continuous functions on $G$ vanishing at infinity and $L _ { 1 } ( G )$ is the pre-dual of $L _ { \infty } ( G )$, leads to the theory of Kac algebras and a generalized Pontryagin theorem (see below). Two complementary results suggest to view $A ( G )$ as a dual object of the group $G$; namely, Eymard's theorem states that the [[Topological space|topological space]] underlying $G$ can be recovered as the spectrum of the Fourier algebra $A ( G )$ and Walter's theorem states that a locally compact group $G$ is determined, up to topological isomorphism, by the normed algebra $A ( G )$, or by $B ( G )$; the second result should be compared with theorems of J.G. Wendel and of B.E. Johnson, which establish the same property for the normed algebras $L _ { 1 } ( G )$ and $M ( G )$, respectively; see [[#References|[a5]]] for a survey of these results.

==Multipliers.==
The multipliers of the Fourier algebra $A ( G )$ reflect interesting properties of the group $G$ (cf. also [[Multiplier theory|Multiplier theory]]). First, the unit $1$ (i.e., the constant function $1$) belongs to $A ( G )$ if and only if the group $G$ is compact. Leptin's theorem (see [[#References|[a3]]]) asserts that $A ( G )$ has a bounded approximate unit if and only if the group $G$ is amenable. A multiplier of the Fourier algebra $A ( G )$ is a function $\varphi$ on $G$ such that the operator $M _ { \varphi }$ of multiplication by $\varphi$ maps $A ( G )$ into itself. These multipliers form a Banach algebra under pointwise multiplication and the norm $\| \varphi \|_{ MA(G)} = \| M_\varphi \|$, denoted by $M A ( G )$. The transposed operator $\square ^ { t } M _ { \varphi }$ is a bounded linear mapping from $W ^ { * } ( G )$ into itself. One says that the multiplier $\varphi$ is completely bounded if the mapping $\square ^ { t } M _ { \varphi }$ is completely bounded, meaning that $\| \square ^ { t } M _ { \varphi } \| _ { \text{cb} } : = \operatorname { sup } \| \square ^ { t } M _ { \varphi } \otimes 1 _ { n } \|$ is finite, where the supremum runs over all integers $n \geq 1$ and $1_n$ is the identity operator from the $C ^ { * }$-algebra $M _ { n } ( \mathbf C )$ of complex $( n \times n )$-matrices into itself. For example, the matrix elements of uniformly bounded representations of $G$ are such multipliers. The completely bounded multipliers form also a Banach algebra under pointwise multiplication and the norm $\| \varphi \| _ {M_{0} A(G)} = \| M\|_{cb}$, denoted by $M _ { 0 } A ( G )$. There is an alternative description of completely bounded multipliers as Schur multipliers, initiated by M.G. Krein [[#References|[a1]]] (cf. also [[Schur multiplicator|Schur multiplicator]]) and related to the metric theory of Grothendieck's topological tensor products. Given a measure space $X$, a measurable function $\varphi$ on $X \times X$ is called a Schur multiplier if pointwise, or Schur, multiplication of kernels by $\varphi$ defines a bounded linear mapping $M _ { \varphi }$ from the space of bounded operators on $L _ { 2 } ( X )$ into itself; its Schur norm is then $\| \varphi \| _ { \text{S} } : = \| M_{ \varphi }\|$. The Schur multipliers form a Banach algebra $B ( X , X )$ under pointwise multiplication. According to the Bożekjko–Fendler theorem, a continuous function $\varphi$ on $G$ is a completely bounded multiplier of $A ( G )$ if and only if the function $\Gamma \varphi$ on $G \times G$ defined by $\Gamma \varphi ( x , y ) = \varphi ( x y ^ { - 1 } )$ is a Schur multiplier; moreover, the Schur norm and the completely bounded norms are equal. The continuous right-invariant Schur multipliers on $G \times G$ are called Herz–Schur multipliers; they form a subalgebra of $B ( G , G )$, denoted by $B _ { 2 } ( G )$, which is isometrically isomorphic to $M _ { 0 } A ( G )$. The following norm-decreasing inclusions hold:

\begin{equation*} B ( G ) \subset M _ { 0 } A ( G ) \subset M A ( G ). \end{equation*}

When $G$ is amenable, these inclusions are equalities; on the other hand, according to Losert's theorem, if $B ( G ) = M A ( G )$, then $G$ is amenable; the equality $B ( G ) = M _ { 0 } A ( G )$ gives the same conclusion, at least when $G$ is discrete (M. Bożekjko and J. Wysoczanski). A locally compact group $G$ is called weakly amenable if there exists an approximate unit in $A ( G )$ which is bounded in the norm of $M _ { 0 } A ( G )$. The Haagerup constant $\Lambda _ { G }$ is defined as the infimum of these bounds over all $M _ { 0 } A ( G )$-bounded approximate units. Free groups and, more generally simple Lie groups with finite centre and real rank one and their lattices, are weakly amenable and their Haagerup constants have been computed in [[#References|[a4]]]. For example, $\Lambda _ { G } = 1$ for $G = SO ( 1 , n )$ or $F _ { n }$ and $\Lambda _ { G } = 2 n - 1$ for $G = \operatorname { Sp } ( 1 , n )$ ($n \geq 2$). Groups of real rank greater than one are not weakly amenable. See also [[#References|[a4]]] for references to completely bounded multipliers.

==$L _ { p }$-Fourier algebras.==
An $L _ { p }$-version of the Fourier algebra has been developed for $1 &lt; p &lt; \infty$ (see [[#References|[a3]]] for a detailed account and references). Let $q$ be given by $1 / p + 1 / q = 1$. The Herz–Figa–Talamanca algebra $A _ { p } ( G )$ is the space of functions $\varphi$ on $G$ of the form

\begin{equation*} \varphi = \sum _ { k = 1 } ^ { \infty } f _ { k } * \widetilde{g} _ { k }, \end{equation*}

where

\begin{equation*} f _ { k } \in L _ { p } ( G ) , g _ { k } \in L _ { q } ( G ) , \sum _ { k = 1 } ^ { \infty } \| f _ { k } \| \| g _ { k } \| &lt; \infty, \end{equation*}

with pointwise multiplication. It is the quotient of the projective tensor product $L _ { p } ( G ) \otimes \widehat{} L _ { q } ( G )$ with respect to the mapping $P$ defined by $P ( f \otimes g ) = f  *  g$. Again, the amenability of $G$ is equivalent to the existence of a bounded approximate unit in $A _ { p } ( G )$. Just as above, one defines for a measure space $X$ the Schur multiplier algebra $B _ { p } ( X , X )$ as the space of functions $\varphi$ on $X \times X$ such that the Schur multiplication $M _ { \varphi }$ sends the space $\mathcal{L} ( L _ { q } ( X ) )$ of bounded operators on $L _ { q } ( X )$ (or, equivalently, its pre-dual $L _ { p } ( G ) \otimes \widehat{} L _ { q } ( G )$) into itself, and the Herz–Schur multiplier algebra $B _ { p } ( G )$ as the space of continuous functions $\varphi$ on $G$ such that $\Gamma \varphi$ belongs to $B _ { p } ( G , G )$; the product is pointwise multiplication. Since the mapping $P$ from $L _ { p } ( G ) \otimes \widehat{} L _ { q } ( G )$ onto $A _ { p } ( G )$ intertwines $M _ { \varphi }$ and $M_ { \Gamma \varphi}$, a Herz–Schur multiplier $\varphi \in B _ { p } ( G )$ is a multiplier of $A _ { p } ( G )$ and the inclusion $B _ { p } ( G ) \subset M A _ { p } ( G )$ decreases the norm. It is an equality if $G$ is amenable. These algebras are also related to convolution operators. In particular, the dual of $A _ { p } ( G )$ is the weak closure $P M _ { q } ( G )$ of $L _ { 1 } ( G )$ in $\mathcal{L} ( L _ { q } ( X ) )$, where $L _ { 1 } ( G )$ acts by left convolution. Banach algebra properties of the Fourier algebras $A ( G )$ and $A _ { p } ( G )$ have been much studied; see [[#References|[a3]]] for a bibliography up to 1984.

==Kac algebras.==
Fourier algebras are natural objects in the $C ^ { * }$-algebraic theory of [[Quantum groups|quantum groups]] and groupoids. In particular, Kac algebras (see [[#References|[a5]]]) provide a symmetric framework for duality, which extends the classical [[Pontryagin duality|Pontryagin duality]] theory for locally compact Abelian groups. Each Kac algebra $K$ has a dual Kac algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120080/f120080187.png"/> and the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120080/f120080188.png"/> is isomorphic to $K$. The Fourier algebra $A ( K ) \subset K$ is the pre-dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f120/f120080/f120080191.png"/> and the Fourier–Stieltjes algebra $B ( K )$ is the dual of the enveloping $C ^ { * }$-algebra of $A ( \widehat{K} )$. If $K$ is the Kac algebra $L _ { \infty } ( G )$ of a locally compact group $G$, then the dual Kac algebra is $\hat { K } = W ^ { * } ( G )$ and the corresponding Fourier and Fourier–Stieltjes algebras are: $A ( K ) = A ( G )$, $B ( K ) = B ( G )$, $A ( \hat { K } ) = L _ { 1 } ( G )$ and $B ( \hat { K } ) = M ( G )$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Krein,   "Hermitian-positive kernels on homogeneous spaces I–II"  ''Amer. Math. Soc. Transl. (2)'' , '''34'''  (1963)  pp. 69–164  ''Ukrain. Mat. Z.'' , '''1–2'''  (1949/50)  pp. 64–98; 10–59</td></tr><tr><td valign="top">[a2]</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">[a3]</td> <td valign="top">  J.-P. Pier,   "Amenable locally compact groups" , Wiley–Interscience  (1984)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  M. Cowling,   U. Haagerup,   "Completely bounded multipliers of the Fourier algebra of a simple Lie groups of real rank one"  ''Invent. Math.'' , '''96''' :  3  (1989)  pp. 507–549</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  M. Enock,   J.-M. Schwartz,   "Kac algebras and duality of locally compact groups" , Springer  (1992)</td></tr></table>