The primitive ideal space of twisted covariant systems

83

Theorem 6. Let Nbea closed normal subgroup of the locally compact group G such that {G,N) IS E-n-regular Assume further that W is a locally closed G-mvariant subset of Prim(C*(N)) such that the stabilizer map from W into Jf{G) is continuous Let

^^ = {(5j,^)eC*(^(Gf ), Jg Ж kQTQ\N = J}

and

Prim ( C * ( G ) ) „ ^= { ker 71,716(5 such that n\N lives on W}

Then the map Ind ^^^-►Frim(C*(G))^, Ö-^ker(indf^ö) is constant on quasiorbits of б^цг and defines a homeomorphism between ^oi^w) ^^d Frim(C*(G))p^

Proof Let /i =кегЖ and /2 = кег(Ж\Ж) Then W can be canomcally identified with Prim(/2//i) Now let f be the twisting map of iV into the umtanes of ^(/2//i) defined by f(n)(a + /i) = T^^(n)a + /i, neN, ael2 Then, using Corollary 3 and Proposition 9, we see that we can identify Prim(C*(G))p^ with Prim(G, /2//1, т) and ^^ with the set ^ as usual defined for the system (G, IJh, ^ Moreover it is not hard to see that the inducing map from ^ to Prim(G, IJIu ^ is compatible with the inducing map from ^^ to Prim(C*(G))p^ Hence the theorem follows immedeatly from Theorem 3

A similar result has been proved by Fell in the separable case under the additional assumptions that Wis Hausdorff, N is of type I and the action of G on N IS smooth (see [1, Theorem 6 4-B])

Acknowledgements This paper constitutes a part of the author's doctoral dissertation which was written under the supervision of Professor E Kaniuth at the University of Paderborn The author would hke to take this opportunity to thank Professor Kaniuth for various discussions on the subject and for his encouragements He would also like to thank the referee for his detailed report, which led to the ehmination of some very long and technical arguments in the original proof of Theorem 1 and some other obscurities in the original text

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