Editing Gerbe (section)

=== C*-algebras ===
There are natural examples of Gerbes that arise from studying the algebra of compactly supported complex valued functions on a paracompact space <math>X</math><ref>{{cite arXiv |last1=Block |first1=Jonathan |last2=Daenzer |first2=Calder |date=2009-01-09 |title=Mukai duality for gerbes with connection |class=math.QA |eprint=0803.1529 }}</ref><sup>pg 3</sup>. Given a cover <math>\mathcal{U} = \{U_i\}</math> of <math>X</math> there is the Cech groupoid defined as<blockquote><math>\mathcal{G} = \left\{ \coprod_{i,j}U_{ij} \rightrightarrows \coprod U_i \right\} </math></blockquote>with source and target maps given by the inclusions<blockquote><math>\begin{align}
s: U_{ij} \hookrightarrow U_j \\
t: U_{ij} \hookrightarrow U_i
\end{align}</math></blockquote>and the space of composable arrows is just<blockquote><math>\coprod_{i,j,k}U_{ijk}</math></blockquote>Then a degree 2 cohomology class <math>\sigma \in H^2(X;U(1))</math> is just a map<blockquote><math>\sigma: \coprod U_{ijk} \to U(1)</math></blockquote>We can then form a non-commutative [[C*-algebra]] <math>C_c(\mathcal{G}(\sigma))</math>, which is associated to the set of compact supported complex valued functions of the space<blockquote><math>\mathcal{G}_1 = \coprod_{i,j}U_{ij}</math></blockquote>It has a non-commutative product given by<blockquote><math>a* b(x,i,k) := \sum_j a(x,i,j)b(x,j,k)\sigma(x,i,j,k)</math></blockquote>where the cohomology class <math>\sigma</math> twists the multiplication of the standard <math>C^*</math>-algebra product.