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On induced covariant systems. (English) Zbl 0692.46054
Let H be a closed subgroup of a locally compact group G such that H acts strongly continuously by *-automorphism on a \(C^*\)-algebra D. The symbol Ind D denotes the induced \(C^*\)-algebra.
It is known that there is a continuous G-invariant map \(\phi\) : prim(Ind D)\(\to G/H\), which is defined by \(\phi (J)=xH\) if J contains the ideal \(I(x)=\{f\in Ind D:\) \(f(x)=0\}.\)
In this note the author shows the converse of this proposition namely that if (G,A) is a covariant system and if \(I=\cap \{J:\quad J\in \phi^{-1}(\{eH\})\}\) and \(D=A/I\), then (G,A) is isomorphic to (G,Ind D). Here the G-equivariant isomorphism \(\phi\) : \(A\to ind D\) is specified. In addition, there are two interesting corollaries.
Reviewer: J.C.Rho

MSC:
46L55 Noncommutative dynamical systems
46L05 General theory of \(C^*\)-algebras
22D30 Induced representations for locally compact groups
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