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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302101.png" /> be an arbitrary locally [[Compact group|compact group]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302102.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302104.png" /> is the set of all equivalence classes of unitary continuous representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302105.png" /> (cf. also [[Unitary representation|Unitary representation]]). The completion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302106.png" /> with respect to this norm is a [[Banach algebra|Banach algebra]], denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302107.png" /> and called the full <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f1302109.png" />-algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021010.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021011.png" /> is Abelian and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021012.png" /> its dual group, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021013.png" /> is isometrically isomorphic to the Banach algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021014.png" /> of all complex-valued continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021015.png" /> vanishing at infinity.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021016.png" /> be the complex linear span of the set of all continuous positive-definite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021017.png" />.
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1) The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021018.png" />-vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021019.png" /> is isomorphic to the dual space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021020.png" />. With the dual norm and the pointwise product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021022.png" /> is a commutative Banach algebra [[#References|[a4]]].
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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.
  
This Banach algebra is called the Fourier–Stieltjes algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021024.png" /> is Abelian, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021025.png" /> is isometrically isomorphic to the Banach algebra of all bounded Radon measures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021026.png" />.
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Let $B ( G )$ be the complex linear span of the set of all continuous positive-definite functions on $G$.
  
2) On the boundary of the unit ball of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021027.png" /> (i.e. on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021028.png" />) the weak topology <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021029.png" /> coincides with the compact-open topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021030.png" /> ([[#References|[a3]]]; see also [[#References|[a9]]], [[#References|[a6]]]).
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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]]].
 +
 
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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 $\hat { C }$.
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 +
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]]]):
 
3) The following properties are satisfied ([[#References|[a4]]]):
  
a) The [[Fourier-algebra(2)|Fourier algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021031.png" /> is a closed ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021032.png" />;
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a) The [[Fourier-algebra(2)|Fourier algebra]] $A ( G )$ is a closed ideal of $B ( G )$;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021033.png" />;
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b) $B ( G ) \cap C _ { 00 } ( G ; \mathbf{C} ) \subset A ( G )$;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021034.png" /> coincides with the closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021035.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021036.png" />;
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c) $A ( G )$ coincides with the closure in $B ( G )$ of $B ( G ) \cap C _ { 00 } ( G ; {\bf C} )$;
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021037.png" />, with equality of the corresponding norms. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021038.png" /> is the algebra of functions of compact support on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021039.png" />. In [[#References|[a14]]], M.E. Walter showed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021040.png" /> (and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021041.png" />) completely characterizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021042.png" />. More precisely, assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021044.png" /> are locally compact groups; then the following assertions are equivalent:
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021046.png" /> are topologically isomorphic;
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the locally compact groups $G_1$ and $G_2$ are topologically isomorphic;
  
the Banach algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021047.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021048.png" /> are isometrically isomorphic;
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the Banach algebras $B ( G _ { 1 } )$ and $B ( G_{2} )$ are isometrically isomorphic;
  
the Banach algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021050.png" /> are isometrically isomorphic.
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the Banach algebras $A ( G _ { 1 } )$ and $A ( G _ { 2 } )$ are isometrically isomorphic.
  
He also gave a description of the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021051.png" />.
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He also gave a description of the dual of $B ( G )$.
  
For a connected semi-simple Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021052.png" />, M. Cowling [[#References|[a1]]] has given a description of the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021053.png" />; surprisingly, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021054.png" /> is Abelian, then the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021055.png" /> seems to be much more complicated than in the non-Abelian case!
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021056.png" /> is amenable, then ([[#References|[a3]]])
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If $G$ is amenable, then ([[#References|[a3]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021057.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021058.png" /> must be amenable!
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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]]].
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021060.png" /> for arbitrary locally compact groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021061.png" /> ([[#References|[a10]]], [[#References|[a8]]]; see also [[#References|[a11]]] for detailed proofs).
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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]].
 
See also [[Figà-Talamanca algebra|Figà-Talamanca algebra]].
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021062.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021063.png" /> and of the measure algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021064.png" />"  ''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021065.png" />"  ''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,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f130/f130210/f13021066.png" />-algebras and nonabelian harmonic analysis"  ''J. Funct. Anal.'' , '''11'''  (1972)  pp. 17–38</TD></TR></table>
+
<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>

Latest revision as of 15:30, 1 July 2020

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])

\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 [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).

See also Figà-Talamanca algebra.

References

[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
How to Cite This Entry:
Fourier-Stieltjes algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier-Stieltjes_algebra&oldid=13242
This article was adapted from an original article by Antoine Derighetti (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article