View source for Figà-Talamanca algebra

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Let $G$ be a locally compact group, $1 < p < \infty$ and $p ^ { \prime } = p / p - 1$. Consider the set $\mathcal{A} _ { p } ( G )$ of all pairs $( ( k _ { n } ) _ { n = 1 } ^ { \infty } , ( l _ { n } ) _ { n = 1 } ^ { \infty } )$, with $( k _ { n } ) _ { n = 1 } ^ { \infty }$ a sequence in ${\cal L} _ {\bf C } ^ { p } ( G )$ and $( l _ { n } ) _ { n = 1 } ^ { \infty } $ a sequence in $\mathcal{L} _ { \text{C} } ^ { p ^ { \prime } } ( G )$ such that $\sum _ { n = 1 } ^ { \infty } N _ { p } ( k _ { n } ) N _ { p^{\prime} }  ( l _ { n } ) < \infty$. Here, $N _ { p } ( f )$ is defined by $N _ { p } ( f ) = ( \int _ { G } | f ( x ) | ^ { p } d m ( x ) ) ^ { 1 / p }$, where $m$ is some left-invariant [[Haar measure|Haar measure]] on $G$. Let $A _ { p } ( G )$ denote the set of all $u \in {\bf C} ^ { G }$ for which there is a pair $( ( k _ { n } ) _ { n = 1 } ^ { \infty } , ( l _ { n } ) _ { n = 1 } ^ { \infty } ) \in \mathcal{A} _ { p } ( G )$ such that $u ( x ) = \sum _ { n = 1 } ^ { \infty } \overline { k _ { n } } * \check{l} _ { n } ( x )$, where $\check{\varphi} { P } ( x ) = \varphi ( x ^ { - 1 } )$. The set $A _ { p } ( G )$ is a linear subspace of the $\mathbf{C}$-vector space of all continuous complex-valued functions on $G$ vanishing at infinity. For $u \in A _ { p } ( G )$ one sets

<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130100/f13010024.png"/></td> </tr></table>

\begin{equation*} \left\{ \sum _ { n = 1 } ^ { \infty } N _ { p } ( k _ { n } ) N _ { p^{\prime} }  ( l _ { n } ) : \begin{array} { c c } {((k_n)^{\infty}_{n=1} , (l_n)^{\infty}_{n=1}) \in \mathcal{A}_{p} (G)    } \\ {\text { with } u = \sum _ { n = 1 } ^ { \infty } \overline { k _ { n } } * \check{l}_ { n }} \end{array} \right\}. \end{equation*}

1) For the pointwise product on $G$, $A _ { p } ( G )$ is a [[Banach algebra|Banach algebra]]. This algebra is called the Figà-Talamanca algebra of $G$. If $G$ is Abelian, $A _ { 2 } ( G )$ is isometrically isomorphic to $L _ { \text{C} } ^ { 1 } ( \hat { G } )$, where $\hat { C }$ is the dual group of $G$. For $G$ not necessarily Abelian, $A _ { 2 } ( G )$ is precisely the [[Fourier-algebra(2)|Fourier algebra]] of $G$.

2) If $G$ is amenable, then $A _ { 2 } ( G ) \subset A _ { p } ( G )$. The algebra $A _ { p } ( G )$ is a useful tool for studying the $p$-convolution operators of $G$ (see [[#References|[a2]]], [[#References|[a7]]], [[#References|[a8]]]). For a function $\varphi$ on $G$ and $a , x \in G$ one sets $\square_{a} \varphi ( x ) = \varphi ( a x )$. A continuous [[Linear operator|linear operator]] $T$ on $L _ { \text{C} } ^ { p } ( G )$ is said to be a $p$-convolution operator of $G$ if $T ( \square _ { \alpha } \varphi ) = \square _ { \alpha } ( T ( \varphi ) )$ for every $a \in G$ and every $\varphi \in L _ { \text{C} } ^ { p } ( G )$. Let $C V _ { p } ( G )$ be the set of all $p$-convolution operators of $G$. It is a closed subalgebra of the Banach algebra $\mathcal{L} ( L _ { \text{C} } ^ { p } ( G ) )$ of all continuous linear operators on $L _ { \text{C} } ^ { p } ( G )$. For a complex bounded [[Measure|measure]] $\mu$ on $G$ (i.e. $\mu \in M _ { \text{C} } ^ {1} ( G )$) and a continuous complex-valued function $\varphi$ with compact support on $G$ ($\varphi \in C _ { 00 } ( G ; \mathbf C )$), the rule $\lambda ^ { p } ( \mu ) [ \varphi ] = [ \varphi  *  \Delta _ { G } ^ { 1 / p ^ { \prime } } \check{\mu} ]$ defines a $p$-convolution operator $\lambda ^ { p } ( \mu )$. Of course, for $f \in \mathbf{C} ^ { G }$, $[ f ]$ denotes the set of all $g \in \mathbf{C} ^ { G }$ with $g ( x ) = f ( x )$ $m$-almost everywhere.

Even for $G = \mathbf{R}$ one has $C V _ { p } ( G ) \neq \lambda ^ { p } ( M ^ { 1 } ( G ) )$. Let $P M _ { p } ( G )$ be the closure in $C V _ { p } ( G )$ of $\lambda ^ { p } ( M ^ { 1 } ( G ) )$ with respect to the ultraweak operator topology on $\mathcal{L} ( L _ { \text{C} } ^ { p } ( G ) )$.

3) The dual $A _ { p } ( G ) ^ { \prime }$ of the Banach space $A _ { p } ( G )$ is canonically isometrically isomorphic to $P M _ { p } ( G )$. Also, $A _ { p } ( G ) ^ { \prime }$ with the topology $\sigma ( A _ { p } ( G ) ^ { \prime } , A _ { p } ( G ) )$ is homeomorphic to $P M _ { p } ( G )$ with the ultraweak operator topology on $\mathcal{L} ( L _ { \text{C} } ^ { p } ( G ) )$. As a consequence, for $G$ amenable or for $G$ arbitrary but with $p = 2$, $P M _ { p } ( G ) = C V _ { p } ( G )$.

This duality between $A _ { p } ( G )$ and $P M _ { p } ( G )$ also permits one to develop (see [[#References|[a1]]]) a kind of  "non-commutative harmonic analysis on G" , where (for $G$ Abelian) $A _ { p } ( G )$ replaces $L _ { \text{C} } ^ { 1 } ( \hat { G } )$ and $C V _ { p } ( G )$ replaces $L _ { C } ^ { \infty } ( \hat { G } )$. (Cf. also [[Harmonic analysis, abstract|Harmonic analysis, abstract]].)

Let $T \in C V _ { p } ( G )$. Then the support of $T$, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130100/f13010099.png"/>, is the set of all $x \in G$ for which for all open subsets $U$, $V$, of $G$ with $e \in U$ and $x \in \mathbf{V}$ there are $\phi , \psi \in C _ { 00 } ( G ; \mathbf{C} )$ with $\text{supp}\,  \phi \subset U$, $\operatorname{supp} \psi \subset V$ and

\begin{equation*} \langle T [ \phi ] , [ \psi ] \rangle _ { L _ { \text{C} } ^ { p } ( G ) , L _ { \text{C} } ^ { p^{\prime} } ( G ) } \neq 0. \end{equation*}

If $\mu \in M _ { \text{C} } ^ {1} ( G )$, then $\operatorname {supp}\lambda _ { G } ^ { p } ( \mu ) = ( \operatorname { supp } \mu ) ^ { - 1 }$. For $G$ Abelian, 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 $A _ { 2 } ( G )$. Let $u \in L _ { \text{C} } ^ { \infty } ( \hat { G } )$ and $x \in G$. Then $x$  "belongs to the spectrum of u"  (written as $x \in \operatorname { sp } u$) if $[ \epsilon ( x ) ]$ lies in the [[linear closure]] of $\left\{ \square _ { \chi } u : \chi \in \widehat { G } \right\}$ in $L _ { C } ^ { \infty } ( \hat { G } )$, for the [[Weak topology|weak topology]] $\sigma ( L _ {\bf C } ^ { \infty } ( \hat { G } ) , L _ {\bf C } ^ { 1 } ( \hat { G } ) )$. Let $T \in C V _ { p } ( G )$; then $\operatorname {sp} \hat { T } = ( \operatorname { supp } T ) ^ { - 1 }$. For $G$ not necessarily amenable and $T \in C V _ { p } ( G )$, $T = 0$ if and only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130100/f130100133.png"/> is empty. This assertion is a non-commutative version of the Wiener theorem! Similarly, there is also a version of the Carleman–Kaplansky theorem: for $T \in C V _ { p } ( G )$, $\operatorname { supp } T = \{ x _ { 1 } , \dots , x _ { n } \}$ if and only there exist $c_1 , \ldots , c_n \in \mathbf{C}$ such that $T = c _ { 1 } \lambda ^ { p } ( \delta _ { x _ { 1 } } ) + \ldots + c _ { n } \lambda ^ { p } ( \delta _ { x _ { n } } )$, where $\delta _ { x }$ denotes the Dirac measure in $x$ (cf. also [[Dirac distribution|Dirac distribution]]). In fact, even for $G = \mathbf{T}$ or for $G = \mathbf{R}$ (but $p \neq 2$) the situation is not classical!

The Banach space $A _ { p } ( G )$ has been first introduced by A. Figà-Talamanca in [[#References|[a3]]] for $G$ Abelian or $G$ non-Abelian but compact. For these classes of groups he obtained assertion 3) above. The statement for a general locally compact group is due to C.S. Herz [[#References|[a5]]]. Assertion 1) is also due to Herz [[#References|[a4]]].

The Banach algebra $A _ { p }$ also satisfies the following properties:

a) Let $H$ be a closed subgroup of $G$. Then $\operatorname { Res } _ { H } A _ { p } ( G ) = A _ { p } ( H )$. More precisely, for every $u \in A _ { p } ( H )$ and for every $\epsilon > 0$ there is a $v \in A _ { p } ( G )$ with $\operatorname { Res } _ { H } v = u$ and $\|v\|_{A_p (G)} \leq \| u \| _ { A_p(H) } + \epsilon$ (see [[#References|[a5]]]).

b) The Banach algebra $A _ { p } ( G )$ has bounded approximate units (i.e. there is a $C > 0$ such that for every $u \in A _ { p } ( G )$ and for every $\epsilon > 0$ there is a $v \in A _ { p } ( G )$ with $\|v \| _ { A _ { p } ( G ) } \leq C$ and $\| u - u v \| _ { A _ { p } ( G ) } < \epsilon$) if and only if the locally compact group $G$ is amenable (see [[#References|[a5]]] and [[#References|[a6]]] for $p = 2$). This algebra is often called the Figà-Talamanca–Herz algebra.

See also [[Fourier-algebra(2)|Fourier algebra]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  A. Derighetti,   "Quelques observations concernant les ensembles de Ditkin d'un groupe localement compact"  ''Monatsh. Math.'' , '''101'''  (1986)  pp. 95–113</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P. Eymard,   "Algèbres $A _ { p }$ et convoluteurs de $L ^ { p } ( G )$" , ''Sem. Bourbaki 1969/70, Exp. 367'' , ''Lecture Notes in Math.'' , '''180''' , Springer  (1971)  pp. 364–381</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  A. Figà-Talamanca,   "Translation invariant operators in $L ^ { p }$"  ''Duke Math. J.'' , '''32'''  (1965)  pp. 495–501</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  C. Herz,   "The theory of $p$-spaces"  ''Trans. Amer. Math. Soc.'' , '''154'''  (1971)  pp. 69–82</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C. Herz,   "Harmonic synthesis for subgroups"  ''Ann. Inst. Fourier (Grenoble)'' , '''23''' :  3  (1973)  pp. 91–123</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  H. Leptin,   "Sur l'algèbre de Fourier d'un groupe localement compact"  ''C.R. Acad. Sci. Paris Sér. A'' , '''266'''  (1968)  pp. 1180–1182</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  N. Lohoué,   "Algèbres $A _ { p }$ et convoluteurs de $L ^ { p } ( G )$"  ''Thèse, Univ. Paris-Sud''  (1971)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  N. Lohoué,   "Estimations $L ^ { p }$ des coefficients de représentations et opérateurs de convolution"  ''Adv. Math.'' , '''38'''  (1980)  pp. 178–221</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  J.R. McMullen,   "Extensions of positive-definite functions"  ''Memoirs Amer. Math. Soc.'' , '''117'''  (1972)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  J.-P. Pier,   "Amenable locally compact groups" , Wiley  (1984)</td></tr></table>