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A complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738901.png" /> on a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738902.png" /> satisfying
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<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/p/p073/p073890/p0738903.png" /></td> </tr></table>
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 +
{{TEX|done}}
  
for all choices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738905.png" />. The set of positive-definite functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738906.png" /> forms a cone in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738907.png" /> of all bounded functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738908.png" /> which is closed with respect to the operations of multiplication and complex conjugation.
+
A complex-valued function  $  \phi $
 +
on a group  $  G $
 +
satisfying
  
The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. [[Positive functional|Positive functional]]) on the group algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p0738909.png" /> and unitary representations of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389010.png" /> (cf. [[Unitary representation|Unitary representation]]). More precisely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389011.png" /> be any function and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389012.png" /> be the functional given by
+
$$
 +
\sum _ {i,j= 1 } ^ { m }  \alpha _ {i} \overline \alpha \; _ {j} \phi ( x _ {j}  ^ {-} 1 x _ {i} ) \geq  0
 +
$$
  
<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/p/p073/p073890/p07389013.png" /></td> </tr></table>
+
for all choices  $  x _ {1} \dots x _ {m} \in G $,
 +
$  \alpha _ {1} \dots \alpha _ {m} \in \mathbf C $.
 +
The set of positive-definite functions on  $  G $
 +
forms a cone in the space  $  M( G) $
 +
of all bounded functions on  $  G $
 +
which is closed with respect to the operations of multiplication and complex conjugation.
  
then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389014.png" /> to be positive it is necessary and sufficient that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389015.png" /> be a positive-definite function. Further, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389016.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389017.png" />-representation of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389018.png" /> on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389019.png" />, and therefore a unitary representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389020.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389022.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389023.png" />. Conversely, for any representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389024.png" /> and any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389025.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389026.png" /> is a positive-definite function.
+
The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. [[Positive functional|Positive functional]]) on the group algebra $  \mathbf C G $
 +
and unitary representations of the group $  G $(
 +
cf. [[Unitary representation|Unitary representation]]). More precisely, let  $  \phi : G \rightarrow \mathbf C $
 +
be any function and let  $  l _  \phi  : \mathbf C G\rightarrow \mathbf C $
 +
be the functional given by
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389027.png" /> is a topological group, the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389028.png" /> is weakly continuous if and only if the positive-definite function is continuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389029.png" /> is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389030.png" />.
+
$$
 +
l _  \phi  \left ( \sum _ {g \in G } \alpha _ {g} g \right )  = \
 +
\sum _ {g \in G } \phi ( g) \alpha _ {g} ;
 +
$$
  
For commutative locally compact groups, the class of continuous positive-definite functions coincides with the class of Fourier transforms of finite positive regular Borel measures on the dual group. There is an analogue of this assertion for compact groups: A continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389031.png" /> on a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389032.png" /> is a positive-definite function if and only if its Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389033.png" /> takes positive (operator) values on each element of the dual object, i.e.
+
then for  $  l _  \phi  $
 +
to be positive it is necessary and sufficient that  $  \phi $
 +
be a positive-definite function. Further,  $  l _  \phi  $
 +
defines a  $  * $-
 +
representation of the algebra  $  \mathbf C G $
 +
on a Hilbert space  $  H _  \phi  $,
 +
and therefore a unitary representation  $  \pi _  \phi  $
 +
of the group $  G $,
 +
where  $  \phi ( g) = ( \pi _  \phi  ( g) \xi , \xi ) $
 +
for some  $  \xi \in H _  \phi  $.  
 +
Conversely, for any representation  $  \pi $
 +
and any vector  $  \xi \in H _  \phi  $,
 +
the function $  g \rightarrow ( \pi ( g) \xi , \xi ) $
 +
is a positive-definite function.
  
<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/p/p073/p073890/p07389034.png" /></td> </tr></table>
+
If  $  G $
 +
is a topological group, the representation  $  \pi _  \phi  $
 +
is weakly continuous if and only if the positive-definite function is continuous. If  $  G $
 +
is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on  $  L _ {1} ( G) $.
  
for any representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389035.png" /> and any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389037.png" /> is the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389038.png" />.
+
For commutative locally compact groups, the class of continuous positive-definite functions coincides with the class of Fourier transforms of finite positive regular Borel measures on the dual group. There is an analogue of this assertion for compact groups: A continuous function  $  \phi $
 +
on a compact group  $  G $
 +
is a positive-definite function if and only if its Fourier transform  $  \widehat \phi  ( b) $
 +
takes positive (operator) values on each element of the dual object, i.e.
 +
 
 +
$$
 +
\int\limits _ { G } \phi ( g) ( \sigma ( g) \xi , \xi )  dg  \geq  0
 +
$$
 +
 
 +
for any representation  $  \sigma $
 +
and any vector $  \xi \in H _  \sigma  $,  
 +
where $  H _  \sigma  $
 +
is the space of $  \sigma $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Hewitt,  K.A. Ross,  "Abstract harmonic analysis" , '''2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1984)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389039.png" /> associated to positive functionals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389040.png" /> mentioned above are cyclic representations. A cyclic representation of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389042.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389043.png" /> is a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389044.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389045.png" />-algebra of bounded operators on the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389046.png" />, such that there is a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389047.png" /> such that the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389048.png" /> is all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389049.png" />. These are the basic components of any representation. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389050.png" /> is non-degenerate, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389052.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389053.png" /> is a direct sum of cyclic representations. Cf. also [[Cyclic module|Cyclic module]] for an analogous concept in ring and module theory.
+
The representations of $  \mathbf C G $
 +
associated to positive functionals $  l $
 +
mentioned above are cyclic representations. A cyclic representation of a $  C  ^ {*} $-
 +
algebra $  {\mathcal A} $
 +
is a representation $  \rho : {\mathcal A} \rightarrow B( H) $,  
 +
the $  C  ^ {*} $-
 +
algebra of bounded operators on the Hilbert space $  H $,  
 +
such that there is a vector $  \xi \in H $
 +
such that the closure of $  \{ {A \xi } : {A \in {\mathcal A} } \} $
 +
is all of $  H $.  
 +
These are the basic components of any representation. Indeed, if $  \rho $
 +
is non-degenerate, i.e. $  \{ {\xi \in H } : {\rho ( A) ( \xi ) = 0  \textrm{ for  all  }  A \in {\mathcal A} } \} = 0 $,  
 +
then $  \rho $
 +
is a direct sum of cyclic representations. Cf. also [[Cyclic module|Cyclic module]] for an analogous concept in ring and module theory.
  
The cyclic representation associated to a positive functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389054.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389055.png" /> is a suitably completed quotient of the regular representation. More precisely, the construction is as follows. Define an inner product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389056.png" /> by
+
The cyclic representation associated to a positive functional $  l $
 +
on $  {\mathcal A} $
 +
is a suitably completed quotient of the regular representation. More precisely, the construction is as follows. Define an inner product on $  {\mathcal A} $
 +
by
  
<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/p/p073/p073890/p07389057.png" /></td> </tr></table>
+
$$
 +
\langle  A, B \rangle  = l ( A  ^ {*} B ) ,
 +
$$
  
and define a left ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389058.png" /> by
+
and define a left ideal of $  {\mathcal A} $
 +
by
  
<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/p/p073/p073890/p07389059.png" /></td> </tr></table>
+
$$
 +
{\mathcal I}  = \{ {A \in {\mathcal A} } : {l( A  ^ {*} A ) = 0 } \}
 +
.
 +
$$
  
The inner product just defined descends to define an inner product on the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389060.png" />. Now complete this space to obtain a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389061.png" />, and define the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389062.png" /> by:
+
The inner product just defined descends to define an inner product on the quotient space $  {\mathcal A} / {\mathcal I} $.  
 +
Now complete this space to obtain a Hilbert space $  H _ {l} $,  
 +
and define the representation $  \pi _ {l} $
 +
by:
  
<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/p/p073/p073890/p07389063.png" /></td> </tr></table>
+
$$
 +
\pi _ {l} ( A) ([ B ])  \simeq  [ AB],
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389064.png" /> denotes the class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389065.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389066.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389067.png" /> extends to a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389068.png" />.
+
where $  [ B] $
 +
denotes the class of $  B \in {\mathcal A} $
 +
in $  {\mathcal A} / {\mathcal I} \subset  H _ {l} $.  
 +
The operator $  \pi _ {l} ( A) $
 +
extends to a bounded operator on $  H _ {l} $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389069.png" /> contains an identity, then the class of that identity is a cyclic vector for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389071.png" /> does not contain an identity, such is first adjoined to obtain a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389072.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389073.png" /> and the construction is repeated for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389074.png" />. To prove that then the class of 1 is cyclic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389075.png" /> (not just <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389076.png" />) one uses an approximate identity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389077.png" />, i.e. a [[Net (directed set)|net (directed set)]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389078.png" /> of positive elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389079.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389081.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389083.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389084.png" />. Such approximate identities always exist. See e.g. [[#References|[1]]], vol. 1, p. 321 and [[#References|[a5]]], Sects. 2.2.3, 2.3.1 and 2.3.3 for more details on all this.
+
If $  {\mathcal A} $
 +
contains an identity, then the class of that identity is a cyclic vector for $  \pi _ {l} $.  
 +
If $  {\mathcal A} $
 +
does not contain an identity, such is first adjoined to obtain a $  C  ^ {*} $-
 +
algebra $  {\mathcal A}  tilde $
 +
and the construction is repeated for $  {\mathcal A}  tilde $.  
 +
To prove that then the class of 1 is cyclic for $  {\mathcal A} $(
 +
not just $  {\mathcal A}  tilde $)  
 +
one uses an approximate identity for $  {\mathcal I} $,  
 +
i.e. a [[Net (directed set)|net (directed set)]] $  \{ E _  \alpha  \} $
 +
of positive elements $  E _  \alpha  \in {\mathcal I} $
 +
such that $  \| E _  \alpha  \| \leq  1 $,  
 +
$  \alpha \leq  \beta $
 +
implies $  E _  \alpha  \leq  E _  \beta  $
 +
and $  \lim\limits _  \alpha  \| AE _  \alpha  - A \| = 0 $
 +
for all $  A \in {\mathcal I} $.  
 +
Such approximate identities always exist. See e.g. [[#References|[1]]], vol. 1, p. 321 and [[#References|[a5]]], Sects. 2.2.3, 2.3.1 and 2.3.3 for more details on all this.
  
A positive functional on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389085.png" />-algebra of norm 1 is often called a state, especially in the theoretical physics literature.
+
A positive functional on a $  C  ^ {*} $-
 +
algebra of norm 1 is often called a state, especially in the theoretical physics literature.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bochner,  "Lectures on Fourier integrals" , Princeton Univ. Press  (1959)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Fourier analysis on groups" , Wiley  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Reiter,  "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389087.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  O. Bratteli,  D.W. Robinson,  "Operator algebras and quantum statistical mechanics" , '''1''' , Springer  (1979)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Bochner,  "Lectures on Fourier integrals" , Princeton Univ. Press  (1959)  (Translated from German)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  W. Rudin,  "Fourier analysis on groups" , Wiley  (1962)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Reiter,  "Classical harmonic analysis and locally compact groups" , Clarendon Press  (1968)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J. Dixmier,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073890/p07389087.png" /> algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  O. Bratteli,  D.W. Robinson,  "Operator algebras and quantum statistical mechanics" , '''1''' , Springer  (1979)</TD></TR></table>

Revision as of 08:07, 6 June 2020


A complex-valued function $ \phi $ on a group $ G $ satisfying

$$ \sum _ {i,j= 1 } ^ { m } \alpha _ {i} \overline \alpha \; _ {j} \phi ( x _ {j} ^ {-} 1 x _ {i} ) \geq 0 $$

for all choices $ x _ {1} \dots x _ {m} \in G $, $ \alpha _ {1} \dots \alpha _ {m} \in \mathbf C $. The set of positive-definite functions on $ G $ forms a cone in the space $ M( G) $ of all bounded functions on $ G $ which is closed with respect to the operations of multiplication and complex conjugation.

The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. Positive functional) on the group algebra $ \mathbf C G $ and unitary representations of the group $ G $( cf. Unitary representation). More precisely, let $ \phi : G \rightarrow \mathbf C $ be any function and let $ l _ \phi : \mathbf C G\rightarrow \mathbf C $ be the functional given by

$$ l _ \phi \left ( \sum _ {g \in G } \alpha _ {g} g \right ) = \ \sum _ {g \in G } \phi ( g) \alpha _ {g} ; $$

then for $ l _ \phi $ to be positive it is necessary and sufficient that $ \phi $ be a positive-definite function. Further, $ l _ \phi $ defines a $ * $- representation of the algebra $ \mathbf C G $ on a Hilbert space $ H _ \phi $, and therefore a unitary representation $ \pi _ \phi $ of the group $ G $, where $ \phi ( g) = ( \pi _ \phi ( g) \xi , \xi ) $ for some $ \xi \in H _ \phi $. Conversely, for any representation $ \pi $ and any vector $ \xi \in H _ \phi $, the function $ g \rightarrow ( \pi ( g) \xi , \xi ) $ is a positive-definite function.

If $ G $ is a topological group, the representation $ \pi _ \phi $ is weakly continuous if and only if the positive-definite function is continuous. If $ G $ is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on $ L _ {1} ( G) $.

For commutative locally compact groups, the class of continuous positive-definite functions coincides with the class of Fourier transforms of finite positive regular Borel measures on the dual group. There is an analogue of this assertion for compact groups: A continuous function $ \phi $ on a compact group $ G $ is a positive-definite function if and only if its Fourier transform $ \widehat \phi ( b) $ takes positive (operator) values on each element of the dual object, i.e.

$$ \int\limits _ { G } \phi ( g) ( \sigma ( g) \xi , \xi ) dg \geq 0 $$

for any representation $ \sigma $ and any vector $ \xi \in H _ \sigma $, where $ H _ \sigma $ is the space of $ \sigma $.

References

[1] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1979)
[2] M.A. Naimark, "Normed rings" , Reidel (1984) (Translated from Russian)

Comments

The representations of $ \mathbf C G $ associated to positive functionals $ l $ mentioned above are cyclic representations. A cyclic representation of a $ C ^ {*} $- algebra $ {\mathcal A} $ is a representation $ \rho : {\mathcal A} \rightarrow B( H) $, the $ C ^ {*} $- algebra of bounded operators on the Hilbert space $ H $, such that there is a vector $ \xi \in H $ such that the closure of $ \{ {A \xi } : {A \in {\mathcal A} } \} $ is all of $ H $. These are the basic components of any representation. Indeed, if $ \rho $ is non-degenerate, i.e. $ \{ {\xi \in H } : {\rho ( A) ( \xi ) = 0 \textrm{ for all } A \in {\mathcal A} } \} = 0 $, then $ \rho $ is a direct sum of cyclic representations. Cf. also Cyclic module for an analogous concept in ring and module theory.

The cyclic representation associated to a positive functional $ l $ on $ {\mathcal A} $ is a suitably completed quotient of the regular representation. More precisely, the construction is as follows. Define an inner product on $ {\mathcal A} $ by

$$ \langle A, B \rangle = l ( A ^ {*} B ) , $$

and define a left ideal of $ {\mathcal A} $ by

$$ {\mathcal I} = \{ {A \in {\mathcal A} } : {l( A ^ {*} A ) = 0 } \} . $$

The inner product just defined descends to define an inner product on the quotient space $ {\mathcal A} / {\mathcal I} $. Now complete this space to obtain a Hilbert space $ H _ {l} $, and define the representation $ \pi _ {l} $ by:

$$ \pi _ {l} ( A) ([ B ]) \simeq [ AB], $$

where $ [ B] $ denotes the class of $ B \in {\mathcal A} $ in $ {\mathcal A} / {\mathcal I} \subset H _ {l} $. The operator $ \pi _ {l} ( A) $ extends to a bounded operator on $ H _ {l} $.

If $ {\mathcal A} $ contains an identity, then the class of that identity is a cyclic vector for $ \pi _ {l} $. If $ {\mathcal A} $ does not contain an identity, such is first adjoined to obtain a $ C ^ {*} $- algebra $ {\mathcal A} tilde $ and the construction is repeated for $ {\mathcal A} tilde $. To prove that then the class of 1 is cyclic for $ {\mathcal A} $( not just $ {\mathcal A} tilde $) one uses an approximate identity for $ {\mathcal I} $, i.e. a net (directed set) $ \{ E _ \alpha \} $ of positive elements $ E _ \alpha \in {\mathcal I} $ such that $ \| E _ \alpha \| \leq 1 $, $ \alpha \leq \beta $ implies $ E _ \alpha \leq E _ \beta $ and $ \lim\limits _ \alpha \| AE _ \alpha - A \| = 0 $ for all $ A \in {\mathcal I} $. Such approximate identities always exist. See e.g. [1], vol. 1, p. 321 and [a5], Sects. 2.2.3, 2.3.1 and 2.3.3 for more details on all this.

A positive functional on a $ C ^ {*} $- algebra of norm 1 is often called a state, especially in the theoretical physics literature.

References

[a1] S. Bochner, "Lectures on Fourier integrals" , Princeton Univ. Press (1959) (Translated from German)
[a2] W. Rudin, "Fourier analysis on groups" , Wiley (1962)
[a3] H. Reiter, "Classical harmonic analysis and locally compact groups" , Clarendon Press (1968)
[a4] J. Dixmier, " algebras" , North-Holland (1977) (Translated from French)
[a5] O. Bratteli, D.W. Robinson, "Operator algebras and quantum statistical mechanics" , 1 , Springer (1979)
How to Cite This Entry:
Positive-definite function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=19269
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article