On discrete twisted \(C^*\)-dynamical systems, Hilbert \(C^*\)-modules and regularity (Q2869232)

Let \(A\) be a unital \(C^*\)-algebra, \(G\) a discrete group with identity \(e\), \((\alpha,\sigma)\) a twisted action of \(G\) on \(A\), i.e., the maps \(\alpha:G\to\text{Aut}(A)\), \(\sigma:G\times G\to U(A)\) satisfy \(\alpha_g\circ\alpha_h=\text{Ad}(\sigma(g,h))\circ\alpha_{gh}\); \(\sigma(g,h)\sigma(gh,k)=\alpha_g(\sigma(h,k))\sigma(g,hk)\); and \(\sigma(g,e)=\sigma(e,g)=1\) for any \(g,h,k\in G\). Then the quadruple \(\Sigma=(A,G,\alpha,\sigma)\) is called a twisted \(C^*\)-dynamical system. To each twisted \(C^*\)-dynamical system \(\Sigma\) one may associate its full (resp., reduced) twisted crossed product \(C^*(\Sigma)\) (resp., \(C^*_r(\Sigma))\).NEWLINENEWLINEThe authors introduce the notion of equivariant representations of twisted \(C^*\)-dynamical systems and their product with covariant representations and prove a version of the Fell absorption principle: the product of an induced regular equivariant representation with a covariant faithful representation is weakly equivalent to an induced regular covariant representation. This principle allows the authors to prove that a certain property, formally weaker than Exel's approximation property, ensures that a twisted \(C^*\)-dynamical system is regular, i.e., that \(C^*(\Sigma)=C^*_r(\Sigma)\).