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''measure algebra''
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The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113901.png" /> of complex-valued regular Borel measures on a locally compact Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113902.png" /> that have bounded variation, with the ordinary linear operations and with convolution as multiplication (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]). The convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113903.png" /> of the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113904.png" /> is completely defined by the condition that, for any continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113905.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113906.png" /> with compact support,
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{{TEX|done}}
  
<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/a/a011/a011390/a0113907.png" /></td> </tr></table>
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''measure algebra''
 
 
If the total variation of a measure is taken as norm, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113908.png" /> becomes a commutative [[Banach algebra|Banach algebra]] over the field of complex numbers. The algebra of measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a0113909.png" /> has a unit which is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139010.png" />-measure located at the zero of the group. The set of discrete measures contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139011.png" /> forms a closed subalgebra.
 
 
 
To each function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139012.png" /> which belongs to the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139013.png" /> may be assigned a corresponding measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139014.png" /> in accordance with the rule
 
  
<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/a/a011/a011390/a01139015.png" /></td> </tr></table>
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The algebra  $  M(G) $
 +
of complex-valued regular Borel measures on a locally compact Abelian group  $  G $
 +
that have bounded variation, with the ordinary linear operations and with convolution as multiplication (cf. [[Harmonic analysis, abstract|Harmonic analysis, abstract]]). The convolution  $  \lambda \star \mu $
 +
of the measures  $  \lambda , \mu \in M (G) $
 +
is completely defined by the condition that, for any continuous function  $  f $
 +
on  $  G $
 +
with compact support,
  
(integral with respect to the Haar measure). The result is an isometric isomorphic imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139016.png" />. Under this imbedding the image is a closed ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139017.png" />.
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$$
 +
\int\limits _ { G } f  d ( \lambda \star \mu ) = \int\limits _ { G } \int\limits _ { G }
 +
f ( x + y )  d \lambda ( x )  d \mu ( y ) .
 +
$$
  
The Fourier–Stieltjes transform of a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139018.png" /> is the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139019.png" /> on the dual group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139020.png" /> defined by the formula
+
If the total variation of a measure is taken as norm,  $  M(G) $
 +
becomes a commutative [[Banach algebra|Banach algebra]] over the field of complex numbers. The algebra of measures  $  M(G) $
 +
has a unit which is the $  \delta $-
 +
measure located at the zero of the group. The set of discrete measures contained in  $  M(G) $
 +
forms a closed subalgebra.
  
<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/a/a011/a011390/a01139021.png" /></td> </tr></table>
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To each function  $  f $
 +
which belongs to the [[Group algebra|group algebra]]  $  L _ {1} (G) $
 +
may be assigned a corresponding measure  $  \mu _ {f} \in M(G) $
 +
in accordance with the rule
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139023.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139024.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139025.png" /> is an algebra without a radical.
+
$$
 +
\mu _ {f} ( E )  = \int\limits _ { E } f  d x
 +
$$
  
If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139026.png" /> is not discrete, then the structure of the measure algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139027.png" /> is extremely complicated: It is not symmetric and its space of maximal ideals has a number of pathological properties. For instance, this space contains infinite-dimensional analytic sets, and the naturally imbedded group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139028.png" /> in it is not dense even in the [[Shilov boundary]]. Nevertheless, the idempotent measures, i.e. measures for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139029.png" />, are known. Each idempotent measure is a finite integer combination <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139031.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139032.png" /> is the Haar measure of a compact subgroup, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139033.png" /> is a character. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139034.png" /> this means that a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139035.png" /> of zeros and ones is the Fourier–Stieltjes transform of some measure on the circle if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139036.png" /> differs from a periodic sequence by not more than a finite number of terms.
+
(integral with respect to the Haar measure). The result is an isometric isomorphic imbedding  $  L _ {1} (G) \rightarrow M(G) $.  
 +
Under this imbedding the image is a closed ideal in $  M(G) $.
  
In the general case the theorem on idempotent measures can be naturally interpreted in terms of the cohomology spaces of dimension zero of the space of maximal ideals. A satisfactory description is also known for other cohomology groups of the space of maximal ideals of the measure algebra. This makes it possible to tell, in particular, if a logarithm of an invertible measure from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011390/a01139037.png" /> can be taken (one-dimensional integral cohomology).
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The Fourier–Stieltjes transform of a measure $  \mu \in M(G) $
 
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is the function  $ \widehat \mu   $
====References====
+
on the dual group $ \widehat{G} $
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> W. Rudin,   "Fourier analysis on groups" , Interscience  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Taylor,  "The cohomology of the spectrum of a measure algebra" ''Acta Math.'' , '''126''' (1971) pp. 195–225</TD></TR></table>
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defined by the formula
  
 +
$$
 +
\widehat \mu  ( \chi )  =  \int\limits _ { G } \overline \chi \;  d \mu .
 +
$$
  
 +
Then  $  \widehat{ {\lambda \star \mu }}  = \widehat \lambda  \cdot \widehat \mu  $
 +
and  $  \| \mu \| = 0 $
 +
if  $  \widehat \mu  \equiv 0 $.
 +
In particular,  $  M(G) $
 +
is an algebra without a radical.
  
====Comments====
+
If the group  $  G $
 +
is not discrete, then the structure of the measure algebra  $  M(G) $
 +
is extremely complicated: It is not symmetric and its space of maximal ideals has a number of pathological properties. For instance, this space contains infinite-dimensional analytic sets, and the naturally imbedded group  $  \widehat{G}  $
 +
in it is not dense even in the [[Shilov boundary]]. Nevertheless, the idempotent measures, i.e. measures for which  $  \mu \star \mu = \mu $,
 +
are known. Each idempotent measure is a finite integer combination  $  n _ {1} \mu _ {1} + \dots + n _ {k} \mu _ {k} $,
 +
where  $  \mu _ {i} = \chi _ {i} \nu _ {i} $,
 +
and where  $  \nu _ {i} $
 +
is the Haar measure of a compact subgroup, and  $  \chi _ {i} $
 +
is a character. In the case  $  G = \mathbf Z $
 +
this means that a sequence  $  (c _ {m} ) $
 +
of zeros and ones is the Fourier–Stieltjes transform of some measure on the circle if and only if  $  (c _ {m} ) $
 +
differs from a periodic sequence by not more than a finite number of terms.
  
 +
In the general case the theorem on idempotent measures can be naturally interpreted in terms of the cohomology spaces of dimension zero of the space of maximal ideals. A satisfactory description is also known for other cohomology groups of the space of maximal ideals of the measure algebra. This makes it possible to tell, in particular, if a logarithm of an invertible measure from  $  M(G) $
 +
can be taken (one-dimensional integral cohomology).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Graham,  O.C. McGehee,  "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Taylor,  "Measure algebras" , Amer. Math. Soc.  (1972)</TD></TR></table>
+
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Rudin,  "Fourier analysis on groups" , Interscience  (1962)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.L. Taylor,  "The cohomology of the spectrum of a measure algebra"  ''Acta Math.'' , '''126'''  (1971)  pp. 195–225</TD></TR>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.C. Graham,  O.C. McGehee,  "Essays in commutative harmonic analysis" , Springer  (1979)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.L. Taylor,  "Measure algebras" , Amer. Math. Soc.  (1972)</TD></TR></table>

Latest revision as of 13:07, 7 April 2023


measure algebra

The algebra $ M(G) $ of complex-valued regular Borel measures on a locally compact Abelian group $ G $ that have bounded variation, with the ordinary linear operations and with convolution as multiplication (cf. Harmonic analysis, abstract). The convolution $ \lambda \star \mu $ of the measures $ \lambda , \mu \in M (G) $ is completely defined by the condition that, for any continuous function $ f $ on $ G $ with compact support,

$$ \int\limits _ { G } f d ( \lambda \star \mu ) = \int\limits _ { G } \int\limits _ { G } f ( x + y ) d \lambda ( x ) d \mu ( y ) . $$

If the total variation of a measure is taken as norm, $ M(G) $ becomes a commutative Banach algebra over the field of complex numbers. The algebra of measures $ M(G) $ has a unit which is the $ \delta $- measure located at the zero of the group. The set of discrete measures contained in $ M(G) $ forms a closed subalgebra.

To each function $ f $ which belongs to the group algebra $ L _ {1} (G) $ may be assigned a corresponding measure $ \mu _ {f} \in M(G) $ in accordance with the rule

$$ \mu _ {f} ( E ) = \int\limits _ { E } f d x $$

(integral with respect to the Haar measure). The result is an isometric isomorphic imbedding $ L _ {1} (G) \rightarrow M(G) $. Under this imbedding the image is a closed ideal in $ M(G) $.

The Fourier–Stieltjes transform of a measure $ \mu \in M(G) $ is the function $ \widehat \mu $ on the dual group $ \widehat{G} $ defined by the formula

$$ \widehat \mu ( \chi ) = \int\limits _ { G } \overline \chi \; d \mu . $$

Then $ \widehat{ {\lambda \star \mu }} = \widehat \lambda \cdot \widehat \mu $ and $ \| \mu \| = 0 $ if $ \widehat \mu \equiv 0 $. In particular, $ M(G) $ is an algebra without a radical.

If the group $ G $ is not discrete, then the structure of the measure algebra $ M(G) $ is extremely complicated: It is not symmetric and its space of maximal ideals has a number of pathological properties. For instance, this space contains infinite-dimensional analytic sets, and the naturally imbedded group $ \widehat{G} $ in it is not dense even in the Shilov boundary. Nevertheless, the idempotent measures, i.e. measures for which $ \mu \star \mu = \mu $, are known. Each idempotent measure is a finite integer combination $ n _ {1} \mu _ {1} + \dots + n _ {k} \mu _ {k} $, where $ \mu _ {i} = \chi _ {i} \nu _ {i} $, and where $ \nu _ {i} $ is the Haar measure of a compact subgroup, and $ \chi _ {i} $ is a character. In the case $ G = \mathbf Z $ this means that a sequence $ (c _ {m} ) $ of zeros and ones is the Fourier–Stieltjes transform of some measure on the circle if and only if $ (c _ {m} ) $ differs from a periodic sequence by not more than a finite number of terms.

In the general case the theorem on idempotent measures can be naturally interpreted in terms of the cohomology spaces of dimension zero of the space of maximal ideals. A satisfactory description is also known for other cohomology groups of the space of maximal ideals of the measure algebra. This makes it possible to tell, in particular, if a logarithm of an invertible measure from $ M(G) $ can be taken (one-dimensional integral cohomology).

References

[1] W. Rudin, "Fourier analysis on groups" , Interscience (1962)
[2] J.L. Taylor, "The cohomology of the spectrum of a measure algebra" Acta Math. , 126 (1971) pp. 195–225
[a1] C.C. Graham, O.C. McGehee, "Essays in commutative harmonic analysis" , Springer (1979) pp. Chapt. 5
[a2] J.L. Taylor, "Measure algebras" , Amer. Math. Soc. (1972)
How to Cite This Entry:
Algebra of measures. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebra_of_measures&oldid=42922
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article