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You can view its source [e]" accesskey="e"><span>View source</span></a></li> <li id="ca-history" class="collapsible"><a href="/index.php?title=Positive-definite_function&action=history" title="Past revisions of this page [h]" accesskey="h"><span>History</span></a></li> </ul> </div>   <div id="p-cactions" class="springereomMenu emptyPortlet"> <h5><span>Actions</span><a href="#"></a></h5> <div class="menu"> <ul> </ul> </div> </div>  </div> </div> <div id="InnerContent"> <div id="mw-js-message" style="display:none;"></div>  <h1 id="firstHeading" class="firstHeading">Positive-definite function</h1>   <div id="bodyContent">  <div id="siteSub">From Encyclopedia of Mathematics</div>   <div id="contentSub"></div>   <div id="jump-to-nav"> Jump to: <a href="#mw-head">navigation</a>, <a href="#p-search">search</a> </div>   <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr"><div class="mw-parser-output"><p><br /> A complex-valued function $ \phi $ on a group $ G $ satisfying </p><p>$$ \sum _ {i,j= 1 } ^ { m } \alpha _ {i} \overline \alpha \; _ {j} \phi ( x _ {j} ^ {-1} x _ {i} ) \geq 0 $$ </p><p>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. </p><p>The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. <a href="/wiki/Positive_functional" title="Positive functional">Positive functional</a>) on the group algebra $ \mathbf C G $ and unitary representations of the group $ G $( cf. <a href="/wiki/Unitary_representation" title="Unitary representation">Unitary representation</a>). 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 </p><p>$$ l _ \phi \left ( \sum _ {g \in G } \alpha _ {g} g \right ) = \ \sum _ {g \in G } \phi ( g) \alpha _ {g} ; $$ </p><p>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. </p><p>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) $. </p><p>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. </p><p>$$ \int\limits _ { G } \phi ( g) ( \sigma ( g) \xi , \xi ) dg \geq 0 $$ </p><p>for any representation $ \sigma $ and any vector $ \xi \in H _ \sigma $, where $ H _ \sigma $ is the space of $ \sigma $. </p> <h4><span class="mw-headline" id="References">References</span></h4> <table><tbody><tr><td valign="top">[1]</td> <td valign="top"> E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , <b>2</b> , 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></tbody></table> <h4><span class="mw-headline" id="Comments">Comments</span></h4> <p>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 <a href="/wiki/Cyclic_module" title="Cyclic module">Cyclic module</a> for an analogous concept in ring and module theory. </p><p>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 </p><p>$$ \langle A, B \rangle = l ( A ^ {*} B ) , $$ </p><p>and define a left ideal of $ {\mathcal A} $ by </p><p>$$ {\mathcal I} = \{ {A \in {\mathcal A} } : {l( A ^ {*} A ) = 0 } \} . $$ </p><p>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: </p><p>$$ \pi _ {l} ( A) ([ B ]) \simeq [ AB], $$ </p><p>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} $. </p><p>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 <a href="/wiki/Net_(directed_set)" title="Net (directed set)">net (directed set)</a> $ \{ 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. <a href="#References">[1]</a>, vol. 1, p. 321 and <a href="#References">[a5]</a>, Sects. 2.2.3, 2.3.1 and 2.3.3 for more details on all this. </p><p>A positive functional on a $ C ^ {*} $- algebra of norm 1 is often called a state, especially in the theoretical physics literature. </p> <h4><span class="mw-headline" id="References_2">References</span></h4> <table><tbody><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 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" , <b>1</b> , Springer (1979)</td></tr></tbody></table>    </div><div id="citeRevision" class="springerCitation"><strong>How to Cite This Entry:</strong><br /> Positive-definite function. <i>Encyclopedia of Mathematics.</i> URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=55150</div><div id="originalIsbn" class="springerCitation">This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. <a href="http://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=19269">See original article</a></div></div>   <div class="printfooter"> Retrieved from "<a dir="ltr" href="https://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=55150">https://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=55150</a>" </div>   <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:Categories" title="Special:Categories">Categories</a>: <ul><li><a href="/wiki/Category:TeX_auto" title="Category:TeX auto">TeX auto</a></li><li><a href="/wiki/Category:TeX_done" title="Category:TeX done">TeX done</a></li></ul></div></div>  <div class="visualClear"></div>   </div>  </div> </div> </div> <div id="footer"> <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 16 January 2024, at 20:29.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="/wiki/Encyclopedia_of_Mathematics:Privacy_policy" title="Encyclopedia of Mathematics:Privacy policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Encyclopedia_of_Mathematics:About" class="mw-redirect" title="Encyclopedia of Mathematics:About">About Encyclopedia of Mathematics</a></li> <li id="footer-places-disclaimer"><a href="/wiki/Encyclopedia_of_Mathematics:General_disclaimer" title="Encyclopedia of Mathematics:General disclaimer">Disclaimers</a></li> </ul> <ul id="footer-icons"> <li><a href="/index.php/Encyclopedia_of_Mathematics:Copyrights">Copyrights</a> </li> <li> <a href="/index.php/Encyclopedia_of_Mathematics:Legal">Impressum-Legal</a></li> </ul> <div style="clear:both"></div> <span class="manage-cookies"><a class="optanon-show-settings">Manage Cookies</a></span> </div>  <script type="<br /> <b>Warning</b>: Undefined array key "jsmimetype" in <b>/var/www/html/includes/skins/QuickTemplate.php</b> on line <b>99</b><br /> "> if ( window.isMSIE55 ) fixalpha(); </script>   <script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgPageParseReport":{"limitreport":{"cputime":"0.009","walltime":"0.013","ppvisitednodes":{"value":60,"limit":1000000},"postexpandincludesize":{"value":126,"limit":2097152},"templateargumentsize":{"value":8,"limit":2097152},"expansiondepth":{"value":5,"limit":40},"expensivefunctioncount":{"value":0,"limit":100},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":0,"limit":5000000},"timingprofile":["100.00% 1.666 1 -total"," 94.23% 1.570 2 Template:TEX"]},"cachereport":{"timestamp":"20241125233036","ttl":86400,"transientcontent":false}}});mw.config.set({"wgBackendResponseTime":71});});</script></div> </div> </body> </html>