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C*-algebra <i>C*</i>(<i>G</i>)</span> </div> </a> <ul id="toc-The_group_C*-algebra_C*(G)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-The_reduced_group_C*-algebra_Cr*(G)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#The_reduced_group_C*-algebra_Cr*(G)"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>The reduced group C*-algebra <i>C<sub>r</sub>*</i>(<i>G</i>)</span> </div> </a> <ul id="toc-The_reduced_group_C*-algebra_Cr*(G)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-von_Neumann_algebras_associated_to_groups" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#von_Neumann_algebras_associated_to_groups"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>von Neumann algebras associated to groups</span> </div> </a> <ul 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</div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about topological algebras associated to topological groups. For the purely algebraic case (without any topology), see <a href="/wiki/Group_ring" title="Group ring">group ring</a>.</div><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Topological algebra associated to continuous groups</div> <p>In <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> and related areas of <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, the <b>group algebra</b> is any of various constructions to assign to a <a href="/wiki/Locally_compact_group" title="Locally compact group">locally compact group</a> an <a href="/wiki/Operator_algebra" title="Operator algebra">operator algebra</a> (or more generally a <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebra</a>), such that representations of the algebra are related to representations of the group. As such, they are similar to the <a href="/wiki/Group_ring" title="Group ring">group ring</a> associated to a discrete group. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="The_algebra_Cc(G)_of_continuous_functions_with_compact_support"><span id="The_algebra_Cc.28G.29_of_continuous_functions_with_compact_support"></span>The algebra <i>C<sub>c</sub></i>(<i>G</i>) of continuous functions with compact support</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=1" title="Edit section: The algebra Cc(G) of continuous functions with compact support"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>G</i> is a <a href="/wiki/Locally_compact_group" title="Locally compact group">locally compact Hausdorff group</a>, <i>G</i> carries an essentially unique left-invariant countably additive <a href="/wiki/Borel_measure" title="Borel measure">Borel measure</a> <i>μ</i> called a <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a>. Using the Haar measure, one can define a <a href="/wiki/Convolution" title="Convolution">convolution</a> operation on the space <i>C<sub>c</sub></i>(<i>G</i>) of complex-valued continuous functions on <i>G</i> with <a href="/wiki/Compact_support" class="mw-redirect" title="Compact support">compact support</a>; <i>C<sub>c</sub></i>(<i>G</i>) can then be given any of various <a href="/wiki/Norm_(mathematics)" title="Norm (mathematics)">norms</a> and the <a href="/wiki/Completeness_(order_theory)" title="Completeness (order theory)">completion</a> will be a group algebra. </p><p>To define the convolution operation, let <i>f</i> and <i>g</i> be two functions in <i>C<sub>c</sub></i>(<i>G</i>). For <i>t</i> in <i>G</i>, define </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [f*g](t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [f*g](t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fec7f312d8e5a03ede4d2733df315ef76e06a5e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.846ex; height:5.676ex;" alt="{\displaystyle [f*g](t)=\int _{G}f(s)g\left(s^{-1}t\right)\,d\mu (s).}"></span></dd></dl> <p>The fact that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de088e4a3777d3b5d2787fdec81acd91e78a719e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}"></span> is continuous is immediate from the <a href="/wiki/Dominated_convergence_theorem" title="Dominated convergence theorem">dominated convergence theorem</a>. Also </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Support} (f*g)\subseteq \operatorname {Support} (f)\cdot \operatorname {Support} (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Support</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>Support</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>Support</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Support} (f*g)\subseteq \operatorname {Support} (f)\cdot \operatorname {Support} (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f84ee93a8af6c5ad917e5c9f3b5eacc2caaf18f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.636ex; height:2.843ex;" alt="{\displaystyle \operatorname {Support} (f*g)\subseteq \operatorname {Support} (f)\cdot \operatorname {Support} (g)}"></span></dd></dl> <p>where the dot stands for the product in <i>G</i>. <i>C<sub>c</sub></i>(<i>G</i>) also has a natural <a href="/wiki/Involution_(mathematics)" title="Involution (mathematics)">involution</a> defined by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}(s)={\overline {f(s^{-1})}}\,\Delta (s^{-1})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{*}(s)={\overline {f(s^{-1})}}\,\Delta (s^{-1})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47774a87f9474760290a066cc7839d2cfcc5a28f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.555ex; height:3.843ex;" alt="{\displaystyle f^{*}(s)={\overline {f(s^{-1})}}\,\Delta (s^{-1})}"></span></dd></dl> <p>where Δ is the <a href="/wiki/Haar_measure#The_modular_function" title="Haar measure">modular function</a> on <i>G</i>. With this involution, it is a <a href="/wiki/*-algebra" title="*-algebra">*-algebra</a>. </p> <blockquote><p><b>Theorem.</b> With the norm: </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{1}:=\int _{G}|f(s)|\,d\mu (s),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>:=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{1}:=\int _{G}|f(s)|\,d\mu (s),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da6a52b6a360fdb49a43f67ba5bc668a023e7f6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.63ex; height:5.676ex;" alt="{\displaystyle \|f\|_{1}:=\int _{G}|f(s)|\,d\mu (s),}"></span></dd></dl> <p><i>C<sub>c</sub></i>(<i>G</i>) becomes an involutive <a href="/wiki/Normed_algebra" title="Normed algebra">normed algebra</a> with an <a href="/wiki/Approximate_identity" title="Approximate identity">approximate identity</a>.</p></blockquote> <p>The approximate identity can be indexed on a neighborhood basis of the identity consisting of compact sets. Indeed, if <i>V</i> is a compact neighborhood of the identity, let <i>f<sub>V</sub></i> be a non-negative continuous function supported in <i>V</i> such that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{V}f_{V}(g)\,d\mu (g)=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{V}f_{V}(g)\,d\mu (g)=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c4402b8a1b25b730b490fb63942ea33148b2bcb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:19.574ex; height:5.676ex;" alt="{\displaystyle \int _{V}f_{V}(g)\,d\mu (g)=1.}"></span></dd></dl> <p>Then {<i>f<sub>V</sub></i>}<sub><i>V</i></sub> is an approximate identity. A group algebra has an identity, as opposed to just an approximate identity, if and only if the topology on the group is the <a href="/wiki/Discrete_topology" class="mw-redirect" title="Discrete topology">discrete topology</a>. </p><p>Note that for discrete groups, <i>C<sub>c</sub></i>(<i>G</i>) is the same thing as the complex group ring <b>C</b>[<i>G</i>]. </p><p>The importance of the group algebra is that it captures the <a href="/wiki/Unitary_representation" title="Unitary representation">unitary representation</a> theory of <i>G</i> as shown in the following </p> <blockquote><p><b>Theorem.</b> Let <i>G</i> be a locally compact group. If <i>U</i> is a strongly continuous unitary representation of <i>G</i> on a Hilbert space <i>H</i>, then </p><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi _{U}(f)=\int _{G}f(g)U(g)\,d\mu (g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pi _{U}(f)=\int _{G}f(g)U(g)\,d\mu (g)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5eb9c4a76d9ce91be30e3bfdb57c1fe88d52f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.049ex; height:5.676ex;" alt="{\displaystyle \pi _{U}(f)=\int _{G}f(g)U(g)\,d\mu (g)}"></span></dd></dl> <p>is a non-degenerate bounded *-representation of the normed algebra <i>C<sub>c</sub></i>(<i>G</i>). The map </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\mapsto \pi _{U}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">↦</mo> <msub> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mapsto \pi _{U}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c13be819df2768d41367fc238c8bdb9d12b1d81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.214ex; height:2.509ex;" alt="{\displaystyle U\mapsto \pi _{U}}"></span></dd></dl><p> is a bijection between the set of strongly continuous unitary representations of <i>G</i> and non-degenerate bounded *-representations of <i>C<sub>c</sub></i>(<i>G</i>). This bijection respects unitary equivalence and <a href="/w/index.php?title=Strong_containment&action=edit&redlink=1" class="new" title="Strong containment (page does not exist)">strong containment</a>. In particular, <span class="texhtml mvar" style="font-style:italic;">π</span><sub><i>U</i></sub> is irreducible if and only if <i>U</i> is irreducible.</p></blockquote> <p>Non-degeneracy of a representation <span class="texhtml mvar" style="font-style:italic;">π</span> of <i>C<sub>c</sub></i>(<i>G</i>) on a Hilbert space <i>H</i><sub><span class="texhtml mvar" style="font-style:italic;">π</span></sub> means that </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{\pi (f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mi>π</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mi>ξ</mi> <mo>:</mo> <mi>f</mi> <mo>∈</mo> <msub> <mi mathvariant="normal">C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>ξ</mi> <mo>∈</mo> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\pi (f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6420e4e51dfe8c60543e2bf008e4ffdefc4a5ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.1ex; height:2.843ex;" alt="{\displaystyle \left\{\pi (f)\xi :f\in \operatorname {C} _{c}(G),\xi \in H_{\pi }\right\}}"></span></dd></dl> <p>is dense in <i>H</i><sub><span class="texhtml mvar" style="font-style:italic;">π</span></sub>. </p> <div class="mw-heading mw-heading2"><h2 id="The_convolution_algebra_L1(G)"><span id="The_convolution_algebra_L1.28G.29"></span>The convolution algebra <i>L</i><sup>1</sup>(<i>G</i>)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=2" title="Edit section: The convolution algebra L1(G)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is a standard theorem of <a href="/wiki/Measure_theory" class="mw-redirect" title="Measure theory">measure theory</a> that the completion of <i>C<sub>c</sub></i>(<i>G</i>) in the <i>L</i><sup>1</sup>(<i>G</i>) norm is isomorphic to the space <a href="/wiki/Lp_space" title="Lp space"><i>L</i><sup>1</sup>(<i>G</i>)</a> of equivalence classes of functions which are integrable with respect to the <a href="/wiki/Haar_measure" title="Haar measure">Haar measure</a>, where, as usual, two functions are regarded as equivalent if and only if they differ only on a set of Haar measure zero. </p> <blockquote><p><b>Theorem.</b> <i>L</i><sup>1</sup>(<i>G</i>) is a <a href="/wiki/Banach_*-algebra" class="mw-redirect" title="Banach *-algebra">Banach *-algebra</a> with the convolution product and involution defined above and with the <i>L</i><sup>1</sup> norm. <i>L</i><sup>1</sup>(<i>G</i>) also has a bounded approximate identity.</p></blockquote> <div class="mw-heading mw-heading3"><h3 id="The_group_C*-algebra_C*(G)"><span id="The_group_C.2A-algebra_C.2A.28G.29"></span>The group C*-algebra <i>C*</i>(<i>G</i>)</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=3" title="Edit section: The group C*-algebra C*(G)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <b>C</b>[<i>G</i>] be the <a href="/wiki/Group_ring" title="Group ring">group ring</a> of a <a href="/wiki/Discrete_group" title="Discrete group">discrete group</a> <i>G</i>. </p><p>For a locally compact group <i>G</i>, the group <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebra</a> <i>C*</i>(<i>G</i>) of <i>G</i> is defined to be the C*-enveloping algebra of <i>L</i><sup>1</sup>(<i>G</i>), i.e. the completion of <i>C<sub>c</sub></i>(<i>G</i>) with respect to the largest C*-norm: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{C^{*}}:=\sup _{\pi }\|\pi (f)\|,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mo>:=</mo> <munder> <mo movablelimits="true" form="prefix">sup</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>π</mi> </mrow> </munder> <mo fence="false" stretchy="false">‖</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{C^{*}}:=\sup _{\pi }\|\pi (f)\|,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57f601387c43047b9af54eea8ffa062cb51e9d49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.964ex; height:4.343ex;" alt="{\displaystyle \|f\|_{C^{*}}:=\sup _{\pi }\|\pi (f)\|,}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">π</span> ranges over all non-degenerate *-representations of <i>C<sub>c</sub></i>(<i>G</i>) on Hilbert spaces. When <i>G</i> is discrete, it follows from the triangle inequality that, for any such <span class="texhtml mvar" style="font-style:italic;">π</span>, one has: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|\pi (f)\|\leq \|f\|_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|\pi (f)\|\leq \|f\|_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fba0e0ea79caff88a865f9b44301d829b36a22a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.148ex; height:2.843ex;" alt="{\displaystyle \|\pi (f)\|\leq \|f\|_{1},}"></span></dd></dl> <p>hence the norm is well-defined. </p><p>It follows from the definition that, when G is a discrete group, <i>C*</i>(<i>G</i>) has the following <a href="/wiki/Universal_property" title="Universal property">universal property</a>: any *-homomorphism from <b>C</b>[<i>G</i>] to some <b>B</b>(<i>H</i>) (the C*-algebra of <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operators</a> on some <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> <i>H</i>) factors through the <a href="/wiki/Inclusion_map" title="Inclusion map">inclusion map</a>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {C} [G]\hookrightarrow C_{\max }^{*}(G).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>G</mi> <mo stretchy="false">]</mo> <mo stretchy="false">↪</mo> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {C} [G]\hookrightarrow C_{\max }^{*}(G).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17be3a281cad53339c54dfb159766f47309b9340" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.194ex; height:2.843ex;" alt="{\displaystyle \mathbf {C} [G]\hookrightarrow C_{\max }^{*}(G).}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="The_reduced_group_C*-algebra_Cr*(G)"><span id="The_reduced_group_C.2A-algebra_Cr.2A.28G.29"></span>The reduced group C*-algebra <i>C<sub>r</sub>*</i>(<i>G</i>)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=4" title="Edit section: The reduced group C*-algebra Cr*(G)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The reduced group C*-algebra <i>C<sub>r</sub>*</i>(<i>G</i>) is the completion of <i>C<sub>c</sub></i>(<i>G</i>) with respect to the norm </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{C_{r}^{*}}:=\sup \left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> </mrow> </msub> <mo>:=</mo> <mo movablelimits="true" form="prefix">sup</mo> <mrow> <mo>{</mo> <mrow> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>:</mo> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{C_{r}^{*}}:=\sup \left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/512a908108b352d77c7dce5086f7d52412fe03c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:35.593ex; height:3.176ex;" alt="{\displaystyle \|f\|_{C_{r}^{*}}:=\sup \left\{\|f*g\|_{2}:\|g\|_{2}=1\right\},}"></span></dd></dl> <p>where </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|f\|_{2}={\sqrt {\int _{G}|f|^{2}\,d\mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>f</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>μ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|f\|_{2}={\sqrt {\int _{G}|f|^{2}\,d\mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f00674546f777c9924c6d86080b5b991463670c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.915ex; height:7.676ex;" alt="{\displaystyle \|f\|_{2}={\sqrt {\int _{G}|f|^{2}\,d\mu }}}"></span></dd></dl> <p>is the <i>L</i><sup>2</sup> norm. Since the completion of <i>C<sub>c</sub></i>(<i>G</i>) with regard to the <i>L</i><sup>2</sup> norm is a Hilbert space, the <i>C<sub>r</sub>*</i> norm is the norm of the bounded operator acting on <i>L</i><sup>2</sup>(<i>G</i>) by convolution with <i>f</i> and thus a C*-norm. </p><p>Equivalently, <i>C<sub>r</sub>*</i>(<i>G</i>) is the C*-algebra generated by the image of the left regular representation on <i>ℓ</i><sup>2</sup>(<i>G</i>). </p><p>In general, <i>C<sub>r</sub>*</i>(<i>G</i>) is a quotient of <i>C*</i>(<i>G</i>). The reduced group C*-algebra is isomorphic to the non-reduced group C*-algebra defined above if and only if <i>G</i> is <a href="/wiki/Amenable_group" title="Amenable group">amenable</a>. </p> <div class="mw-heading mw-heading2"><h2 id="von_Neumann_algebras_associated_to_groups">von Neumann algebras associated to groups</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=5" title="Edit section: von Neumann algebras associated to groups"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The group von Neumann algebra <i>W*</i>(<i>G</i>) of <i>G</i> is the enveloping von Neumann algebra of <i>C*</i>(<i>G</i>). </p><p>For a discrete group <i>G</i>, we can consider the <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> ℓ<sup>2</sup>(<i>G</i>) for which <i>G</i> is an <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a>. Since <i>G</i> operates on ℓ<sup>2</sup>(<i>G</i>) by permuting the basis vectors, we can identify the complex group ring <b>C</b>[<i>G</i>] with a subalgebra of the algebra of <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operators</a> on ℓ<sup>2</sup>(<i>G</i>). The weak closure of this subalgebra, <i>NG</i>, is a <a href="/wiki/Von_Neumann_algebra" title="Von Neumann algebra">von Neumann algebra</a>. </p><p>The center of <i>NG</i> can be described in terms of those elements of <i>G</i> whose <a href="/wiki/Conjugacy_class" title="Conjugacy class">conjugacy class</a> is finite. In particular, if the identity element of <i>G</i> is the only group element with that property (that is, <i>G</i> has the <a href="/wiki/Infinite_conjugacy_class_property" title="Infinite conjugacy class property">infinite conjugacy class property</a>), the center of <i>NG</i> consists only of complex multiples of the identity. </p><p><i>NG</i> is isomorphic to the <a href="/wiki/Hyperfinite_type_II-1_factor" class="mw-redirect" title="Hyperfinite type II-1 factor">hyperfinite type II<sub>1</sub> factor</a> if and only if <i>G</i> is <a href="/wiki/Countable" class="mw-redirect" title="Countable">countable</a>, <a href="/wiki/Amenable_group" title="Amenable group">amenable</a>, and has the infinite conjugacy class property. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Graph_algebra" title="Graph algebra">Graph algebra</a></li> <li><a href="/wiki/Incidence_algebra" title="Incidence algebra">Incidence algebra</a></li> <li><a href="/wiki/Hecke_algebra_of_a_locally_compact_group" class="mw-redirect" title="Hecke algebra of a locally compact group">Hecke algebra of a locally compact group</a></li> <li><a href="/wiki/Path_algebra" class="mw-redirect" title="Path algebra">Path algebra</a></li> <li><a href="/wiki/Groupoid_algebra" title="Groupoid algebra">Groupoid algebra</a></li> <li><a href="/wiki/Stereotype_algebra" class="mw-redirect" title="Stereotype algebra">Stereotype algebra</a></li> <li><a href="/wiki/Stereotype_group_algebra" class="mw-redirect" title="Stereotype group algebra">Stereotype group algebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Group_algebra_of_a_locally_compact_group&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFLang2002" class="citation book cs1">Lang, S. (2002). <a rel="nofollow" class="external text" href="https://www.springer.com/gp/book/9780387953854"><i>Algebra</i></a>. Graduate Texts in Mathematics. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4613-0041-0" title="Special:BookSources/978-1-4613-0041-0"><bdi>978-1-4613-0041-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=2002&rft.isbn=978-1-4613-0041-0&rft.aulast=Lang&rft.aufirst=S.&rft_id=https%3A%2F%2Fwww.springer.com%2Fgp%2Fbook%2F9780387953854&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+algebra+of+a+locally+compact+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVinberg2003" class="citation book cs1">Vinberg, E. (10 April 2003). <a rel="nofollow" class="external text" href="https://www.ams.org/books/gsm/056/"><i>A Course in Algebra</i></a>. Graduate Studies in Mathematics. Vol. 56. American Mathematical Society. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fgsm%2F056">10.1090/gsm/056</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3413-8" title="Special:BookSources/978-0-8218-3413-8"><bdi>978-0-8218-3413-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Course+in+Algebra&rft.series=Graduate+Studies+in+Mathematics&rft.pub=American+Mathematical+Society&rft.date=2003-04-10&rft_id=info%3Adoi%2F10.1090%2Fgsm%2F056&rft.isbn=978-0-8218-3413-8&rft.aulast=Vinberg&rft.aufirst=E.&rft_id=https%3A%2F%2Fwww.ams.org%2Fbooks%2Fgsm%2F056%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+algebra+of+a+locally+compact+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDixmier1982" class="citation book cs1">Dixmier, Jacques (1982). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=P34ZAQAAIAAJ&q=C*-Algebras%20Jacques%20Dixmier"><i>C*-algebras</i></a>. North-Holland. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-444-86391-1" title="Special:BookSources/978-0-444-86391-1"><bdi>978-0-444-86391-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=C%2A-algebras&rft.pub=North-Holland&rft.date=1982&rft.isbn=978-0-444-86391-1&rft.aulast=Dixmier&rft.aufirst=Jacques&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DP34ZAQAAIAAJ%26q%3DC%2A-Algebras%2520Jacques%2520Dixmier&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+algebra+of+a+locally+compact+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirillov1976" class="citation book cs1">Kirillov, Aleksandr A. (1976). <a rel="nofollow" class="external text" href="https://link.springer.com/book/10.1007/978-3-642-66243-0"><i>Elements of the Theory of Representations</i></a>. Grundlehren der mathematischen Wissenschaften. Vol. 220. Springer-Verlag. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-66243-0">10.1007/978-3-642-66243-0</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-642-66245-4" title="Special:BookSources/978-3-642-66245-4"><bdi>978-3-642-66245-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Elements+of+the+Theory+of+Representations&rft.series=Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer-Verlag&rft.date=1976&rft_id=info%3Adoi%2F10.1007%2F978-3-642-66243-0&rft.isbn=978-3-642-66245-4&rft.aulast=Kirillov&rft.aufirst=Aleksandr+A.&rft_id=https%3A%2F%2Flink.springer.com%2Fbook%2F10.1007%2F978-3-642-66243-0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+algebra+of+a+locally+compact+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLoomis2011" class="citation book cs1">Loomis, Lynn H. (19 July 2011). <i>Introduction to Abstract Harmonic Analysis (Dover Books on Mathematics) by Lynn H. Loomis (2011) Paperback</i>. Dover Publications. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-48123-4" title="Special:BookSources/978-0-486-48123-4"><bdi>978-0-486-48123-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Abstract+Harmonic+Analysis+%28Dover+Books+on+Mathematics%29+by+Lynn+H.+Loomis+%282011%29+Paperback&rft.pub=Dover+Publications&rft.date=2011-07-19&rft.isbn=978-0-486-48123-4&rft.aulast=Loomis&rft.aufirst=Lynn+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+algebra+of+a+locally+compact+group" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFA.I._Shtern2001" class="citation cs2">A.I. Shtern (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Group_algebra_of_a_locally_compact_group">"Group algebra of a locally compact group"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Group+algebra+of+a+locally+compact+group&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft.au=A.I.+Shtern&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DGroup_algebra_of_a_locally_compact_group&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGroup+algebra+of+a+locally+compact+group" class="Z3988"></span> <i>This article incorporates material from Group $C^*$-algebra on <a href="/wiki/PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the <a href="/wiki/Wikipedia:CC-BY-SA" class="mw-redirect" title="Wikipedia:CC-BY-SA">Creative Commons Attribution/Share-Alike License</a>.</i></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Group_algebra_of_a_locally_compact_group&oldid=1242476828">https://en.wikipedia.org/w/index.php?title=Group_algebra_of_a_locally_compact_group&oldid=1242476828</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Algebras" title="Category:Algebras">Algebras</a></li><li><a href="/wiki/Category:C*-algebras" title="Category:C*-algebras">C*-algebras</a></li><li><a href="/wiki/Category:Von_Neumann_algebras" title="Category:Von Neumann algebras">Von Neumann algebras</a></li><li><a href="/wiki/Category:Unitary_representation_theory" title="Category:Unitary representation theory">Unitary representation theory</a></li><li><a href="/wiki/Category:Harmonic_analysis" title="Category:Harmonic analysis">Harmonic analysis</a></li><li><a href="/wiki/Category:Lie_groups" title="Category:Lie groups">Lie groups</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li><li><a href="/wiki/Category:Wikipedia_articles_incorporating_text_from_PlanetMath" title="Category:Wikipedia articles incorporating text from PlanetMath">Wikipedia articles incorporating text from PlanetMath</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 27 August 2024, at 00:36<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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