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A topological algebra with [[Involution|involution]] formed by certain functions on the group with multiplication in it defined as convolution. Let the Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452301.png" /> be constructed using a left-invariant [[Haar measure|Haar measure]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452302.png" /> on a locally compact topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452303.png" /> and let the multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452304.png" /> be defined as the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452305.png" />; also, let the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452306.png" /> be given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452307.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452308.png" /> is the modular function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g0452309.png" />. The resulting [[Banach algebra|Banach algebra]] with involution is said to be the group algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523010.png" /> and is also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523011.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523012.png" /> is a finite group, then the definition of the group algebra coincides with the ordinary algebraic definition of the [[group algebra]] over the field of complex numbers.
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The concept of a group algebra makes it possible to use the general methods of the theory of Banach algebras in problems of group theory and, in particular, in abstract harmonic analysis. The properties of a group algebra, as a Banach algebra, reflect the properties of topological groups; thus, a group algebra contains a unit element if and only if the group is discrete; the group algebra is the direct (topological) sum of its finite-dimensional minimal two-sided ideals if and only if the group is compact. Of special importance is the concept of a group algebra in the theory of unitary representations (cf. [[Unitary representation]]) of groups: Between the continuous unitary representations of a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523013.png" /> and the non-degenerate symmetric representations (cf. [[Involution representation]]) of the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523014.png" /> there exists a one-to-one correspondence. This correspondence puts a continuous unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523016.png" /> in a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523017.png" /> into correspondence with the representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523018.png" /> defined by
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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/g/g045/g045230/g04523019.png" /></td> </tr></table>
+
A topological algebra with [[Involution|involution]] formed by certain functions on the group with multiplication in it defined as convolution. Let the Banach space  $  L _ {1} ( G) $
 +
be constructed using a left-invariant [[Haar measure|Haar measure]]  $  d g $
 +
on a locally compact topological group  $  G $
 +
and let the multiplication in  $  L _ {1} ( G) $
 +
be defined as the convolution  $  ( f _ {1} , f _ {2} ) \rightarrow f _ {1} \star f _ {2} $;  
 +
also, let the involution  $  f \rightarrow f ^ { * } $
 +
be given by the formula  $  f ^ { * } ( g) = \overline{ {f ( g  ^ {-} 1 ) }}\; \Delta ( g) $,
 +
where  $  \Delta $
 +
is the modular function of  $  G $.  
 +
The resulting [[Banach algebra|Banach algebra]] with involution is said to be the group algebra of  $  G $
 +
and is also denoted by  $  L _ {1} ( G) $.  
 +
If  $  G $
 +
is a finite group, then the definition of the group algebra coincides with the ordinary algebraic definition of the [[group algebra]] over the field of complex numbers.
  
Group algebras of locally compact groups have a number of properties in common. In fact, any group algebra contains an approximate unit element (cf. [[Banach algebra]]), formed by the family of characteristic functions on a neighbourhood of the unit element ordered by inclusion (in decreasing order). For this reason it is possible to establish for a group algebra a correspondence between the positive functionals on the group algebra and its symmetric representations. Any group algebra is a semi-simple algebra, and has a symmetric [[faithful representation]]. In particular, the representation of a group algebra determined by the [[regular representation]] of the group is faithful. The closed left ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523020.png" /> are the closed vector subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523021.png" /> that are invariant with respect to left translation.
+
The concept of a group algebra makes it possible to use the general methods of the theory of Banach algebras in problems of group theory and, in particular, in abstract harmonic analysis. The properties of a group algebra, as a Banach algebra, reflect the properties of topological groups; thus, a group algebra contains a unit element if and only if the group is discrete; the group algebra is the direct (topological) sum of its finite-dimensional minimal two-sided ideals if and only if the group is compact. Of special importance is the concept of a group algebra in the theory of unitary representations (cf. [[Unitary representation]]) of groups: Between the continuous unitary representations of a topological group $  G $
 +
and the non-degenerate symmetric representations (cf. [[Involution representation]]) of the group algebra  $  L _ {1} ( G) $
 +
there exists a one-to-one correspondence. This correspondence puts a continuous unitary representation  $  \pi $
 +
of $  G $
 +
in a Hilbert space  $  H $
 +
into correspondence with the representation of $  L _ {1} ( G) $
 +
defined by
  
The name group algebra is also sometimes given to the Banach algebra with involution obtained from the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523022.png" /> by the adjunction of a unit. There exist several other algebras with involutions, which are sometimes referred to as group algebras. These include, in particular: the algebra of measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523023.png" /> with respect to convolution, algebras with respect to ordinary multiplication, such as the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523024.png" /> of essentially-bounded functions that are measurable by the Haar measure, and the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523025.png" /> spanned by the set of complex positive-definite functions. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523026.png" /> and the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523027.png" /> of continuous functions with compact support form algebras both with respect to convolution and with respect to ordinary multiplication. One obtains the following table, in which arrows denote inclusions:
+
$$
 +
f  \rightarrow  \int\limits _ { G } f ( g) \pi ( g)  dg,\ \
 +
f \in L _ {1} ( G).
 +
$$
  
<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/g/g045/g045230/g04523028.png" /></td> </tr></table>
+
Group algebras of locally compact groups have a number of properties in common. In fact, any group algebra contains an approximate unit element (cf. [[Banach algebra]]), formed by the family of characteristic functions on a neighbourhood of the unit element ordered by inclusion (in decreasing order). For this reason it is possible to establish for a group algebra a correspondence between the positive functionals on the group algebra and its symmetric representations. Any group algebra is a semi-simple algebra, and has a symmetric [[faithful representation]]. In particular, the representation of a group algebra determined by the [[regular representation]] of the group is faithful. The closed left ideals of  $  L _ {1} ( G) $
 +
are the closed vector subspaces of  $  L _ {1} ( G) $
 +
that are invariant with respect to left translation.
 +
 
 +
The name group algebra is also sometimes given to the Banach algebra with involution obtained from the group algebra  $  L _ {1} ( G) $
 +
by the adjunction of a unit. There exist several other algebras with involutions, which are sometimes referred to as group algebras. These include, in particular: the algebra of measures  $  M ( G) $
 +
with respect to convolution, algebras with respect to ordinary multiplication, such as the algebra  $  L _  \infty  ( G) $
 +
of essentially-bounded functions that are measurable by the Haar measure, and the algebra  $  P ( G) $
 +
spanned by the set of complex positive-definite functions. The set  $  P _ {1} ( G) = P ( G) \cap L _ {1} ( G) $
 +
and the set  $  K ( G) $
 +
of continuous functions with compact support form algebras both with respect to convolution and with respect to ordinary multiplication. One obtains the following table, in which arrows denote inclusions:
 +
 
 +
$$
 +
 
 +
\begin{array}{rcl}
 +
{}  &P _ {1}  & \rightarrow  P  \rightarrow  L _  \infty  \\
 +
{}  &\downarrow  &{}  \\
 +
K  \rightarrow  &L _ {1}  & \rightarrow  M  \\
 +
\end{array}
 +
.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Guichardet,  "Analyse harmonique commutative" , Dunod  (1968)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Guichardet,  "Analyse harmonique commutative" , Dunod  (1968)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The set of all complex continuous positive-definite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523029.png" /> is not an algebra since 1 is in it but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523030.png" /> is not. However, the set of all complex linear combinations of its members, which is usually denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523031.png" />, is an algebra under the usual [[pointwise multiplication]] (see [[#References|[a1]]], (32.10)). For all non-discrete locally compact Abelian groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523032.png" /> there exist continuous compactly-supported functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523033.png" /> that are not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523034.png" /> (see [[#References|[a1]]], (33.3) and (41.19)). Thus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523035.png" /> is not a subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523036.png" />.
+
The set of all complex continuous positive-definite functions on $  G $
 +
is not an algebra since 1 is in it but $  - 1 $
 +
is not. However, the set of all complex linear combinations of its members, which is usually denoted $  B ( G) $,  
 +
is an algebra under the usual [[pointwise multiplication]] (see [[#References|[a1]]], (32.10)). For all non-discrete locally compact Abelian groups $  G $
 +
there exist continuous compactly-supported functions on $  G $
 +
that are not in $  B ( G) $(
 +
see [[#References|[a1]]], (33.3) and (41.19)). Thus $  K ( G) $
 +
is not a subset of $  B ( G) $.
  
For the notion of the modular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523038.png" />, cf. [[Haar measure|Haar measure]]. A symmetric representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523039.png" /> of an algebra with involution in a Hilbert space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523040.png" /> is one which satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523041.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523042.png" />. Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523043.png" /> on the right denotes taking adjoints and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523044.png" /> on the left refers to the involution on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523045.png" />.
+
For the notion of the modular function $  \Delta $
 +
of $  G $,  
 +
cf. [[Haar measure|Haar measure]]. A symmetric representation $  \pi $
 +
of an algebra with involution in a Hilbert space over $  \mathbf C $
 +
is one which satisfies $  \pi ( a  ^ {*} ) = \pi ( a)  ^ {*} $
 +
for all $  a \in A $.  
 +
Here the $  * $
 +
on the right denotes taking adjoints and the $  * $
 +
on the left refers to the involution on $  A $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''1–2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523046.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''1–2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g045/g045230/g04523046.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR></table>

Revision as of 19:42, 5 June 2020


A topological algebra with involution formed by certain functions on the group with multiplication in it defined as convolution. Let the Banach space $ L _ {1} ( G) $ be constructed using a left-invariant Haar measure $ d g $ on a locally compact topological group $ G $ and let the multiplication in $ L _ {1} ( G) $ be defined as the convolution $ ( f _ {1} , f _ {2} ) \rightarrow f _ {1} \star f _ {2} $; also, let the involution $ f \rightarrow f ^ { * } $ be given by the formula $ f ^ { * } ( g) = \overline{ {f ( g ^ {-} 1 ) }}\; \Delta ( g) $, where $ \Delta $ is the modular function of $ G $. The resulting Banach algebra with involution is said to be the group algebra of $ G $ and is also denoted by $ L _ {1} ( G) $. If $ G $ is a finite group, then the definition of the group algebra coincides with the ordinary algebraic definition of the group algebra over the field of complex numbers.

The concept of a group algebra makes it possible to use the general methods of the theory of Banach algebras in problems of group theory and, in particular, in abstract harmonic analysis. The properties of a group algebra, as a Banach algebra, reflect the properties of topological groups; thus, a group algebra contains a unit element if and only if the group is discrete; the group algebra is the direct (topological) sum of its finite-dimensional minimal two-sided ideals if and only if the group is compact. Of special importance is the concept of a group algebra in the theory of unitary representations (cf. Unitary representation) of groups: Between the continuous unitary representations of a topological group $ G $ and the non-degenerate symmetric representations (cf. Involution representation) of the group algebra $ L _ {1} ( G) $ there exists a one-to-one correspondence. This correspondence puts a continuous unitary representation $ \pi $ of $ G $ in a Hilbert space $ H $ into correspondence with the representation of $ L _ {1} ( G) $ defined by

$$ f \rightarrow \int\limits _ { G } f ( g) \pi ( g) dg,\ \ f \in L _ {1} ( G). $$

Group algebras of locally compact groups have a number of properties in common. In fact, any group algebra contains an approximate unit element (cf. Banach algebra), formed by the family of characteristic functions on a neighbourhood of the unit element ordered by inclusion (in decreasing order). For this reason it is possible to establish for a group algebra a correspondence between the positive functionals on the group algebra and its symmetric representations. Any group algebra is a semi-simple algebra, and has a symmetric faithful representation. In particular, the representation of a group algebra determined by the regular representation of the group is faithful. The closed left ideals of $ L _ {1} ( G) $ are the closed vector subspaces of $ L _ {1} ( G) $ that are invariant with respect to left translation.

The name group algebra is also sometimes given to the Banach algebra with involution obtained from the group algebra $ L _ {1} ( G) $ by the adjunction of a unit. There exist several other algebras with involutions, which are sometimes referred to as group algebras. These include, in particular: the algebra of measures $ M ( G) $ with respect to convolution, algebras with respect to ordinary multiplication, such as the algebra $ L _ \infty ( G) $ of essentially-bounded functions that are measurable by the Haar measure, and the algebra $ P ( G) $ spanned by the set of complex positive-definite functions. The set $ P _ {1} ( G) = P ( G) \cap L _ {1} ( G) $ and the set $ K ( G) $ of continuous functions with compact support form algebras both with respect to convolution and with respect to ordinary multiplication. One obtains the following table, in which arrows denote inclusions:

$$ \begin{array}{rcl} {} &P _ {1} & \rightarrow P \rightarrow L _ \infty \\ {} &\downarrow &{} \\ K \rightarrow &L _ {1} & \rightarrow M \\ \end{array} . $$

References

[1] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)
[2] A. Guichardet, "Analyse harmonique commutative" , Dunod (1968)

Comments

The set of all complex continuous positive-definite functions on $ G $ is not an algebra since 1 is in it but $ - 1 $ is not. However, the set of all complex linear combinations of its members, which is usually denoted $ B ( G) $, is an algebra under the usual pointwise multiplication (see [a1], (32.10)). For all non-discrete locally compact Abelian groups $ G $ there exist continuous compactly-supported functions on $ G $ that are not in $ B ( G) $( see [a1], (33.3) and (41.19)). Thus $ K ( G) $ is not a subset of $ B ( G) $.

For the notion of the modular function $ \Delta $ of $ G $, cf. Haar measure. A symmetric representation $ \pi $ of an algebra with involution in a Hilbert space over $ \mathbf C $ is one which satisfies $ \pi ( a ^ {*} ) = \pi ( a) ^ {*} $ for all $ a \in A $. Here the $ * $ on the right denotes taking adjoints and the $ * $ on the left refers to the involution on $ A $.

References

[a1] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1–2 , Springer (1979)
[a2] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
How to Cite This Entry:
Group algebra of a locally compact group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Group_algebra_of_a_locally_compact_group&oldid=47141
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article