PROOF
Print ISSN: 2944-9162, E-ISSN: 2732-9941 An Open Access International Journal of Applied Science and Engineering
Volume 3, 2023
Twisted Dynamical Systems, Schrodinger Representations
Author:
Abstract: Assume $$G$$ is a locally compact Hausdorff group, $$A$$is a $$C*$$ -algebra, and (A,G, ω) is a dynamical system, we consider a Takai theorem that states the isomorphism $$Φ:(Ax_{ω}G){x_{\hat{ω}}}\hat{G}\rightarrow A \otimes LK (L{^2}(G))$$ is equivariant for $$\hat{\hat{ω}}: G \rightarrow (Ax_{ω}G)x_{\hat{ω}}\hat{G}$$ and for $$\hat{\hat{ω}}\otimes Ad (ρ): G \rightarrow A \otimes LK (L^{2}(G)) $$. Also, we show that *-surjective mapping $$Υ: Cc (G,A) \rightarrow Cc (G,A,τ) $$ can be extended to quotient mapping $$\widetilde{Υ}: Ax_{ω}G\rightarrow Ax^{τ}_{ω} G=Ax_{ω} G / (I \bigcap Ax_{ω} G)$$ for the twisted dynamical system $$(A, G, ω, τ)$$. We establish that there exists an isomorphism of the Schrodinger $$C*$$ -algebra $$Sch^{τ}_{Λ}( Ax^{τ}_{ω}G)$$ to the reduced crossed product $$[ Ax^{τ}_{ω}G]_{red};$$ and show the representation $$Β^{τ}(A)\mapsto Sch^{τ}_{Λ}( Ax^{τ}_{ω}G)\subset LB(L^{2}(G))$$ is faithful for each amenable group $$G$$.
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Keywords: Takai Duality, $$γ -duality$$, Wigner function, $$C^{*} -algebra$$, Pontryagin duality, induced representation, cross product
Pages: 14-20
DOI: 10.37394/232020.2023.3.3