The Topology on the Primitive Ideal Space of Transformation Group C ∗ - Algebras and C.C.R. Transformation Group C ∗ -Algebras

Unitary representations of diamond groups

“…This is a special case of a much more general theorem, see [19], mentioned in the introduction. On the other hand the space riv C Ã Q is homeomorphic to S Â M a $ see [17] or [22]. Thus the above lemma says among others that indeed riv C Ã Q is homeomorphic to the Q-quasi-orbit space aQ.…”

“…The homeomorphy of GaZN with U Â NaZN a $ follows from [22] and [17, Theorem 3.3 and the remarks in front of Theorem 3.7]. The homeomorphy of the open subset G n GaZN with U Â R Â is equally easy.…”

“…As ZN is central in our algebra or, more rigorously, as point measures at elements in ZN yield central elements in the corresponding adjoint (or multiplier) algebra, again we distinguish between the representations which are trivial on ZN and those which are equal to a non-trivial unitary character 0Y t U 3 e i !t for 0Y t P ZN & N C n  R. Representations of the first type can be considered as representations of C à SY C à NaZN bY T . As C à NaZN b is commutative the results of [17,22] apply. First we observe that S acts on the structure space b of b by sb b s .…”

On transformation group 𝐶*-algebras with continuous trace

“…It is a classical result that the induced representation indfw(7T, co) of (G, Cn(Q)) is always irreducible if n is irreducible. Moreover, if n and p are unitary representations of Su , then the approximation arguments given in the proof of [18,Proposition 4.2] show that any intertwining operator of indf[o(n, co) and indfw(/>, co) also intertwines the representations (indfm n, M) and (indfw p, M) of the imprimitivity algebra C*(G, G/Sa) (in the setting of [18, Lemma 4.1]), from which follows by the imprimitivity theorem that ivtdSiij(n, co) is equivalent to indsm(p, co) if and only if n is equivalent to p. Since a C* -algebra A with continuous trace always has a Hausdorff spectrum Â, the answer to Ql follows from Proposition 1. Let (G,¿1) be a transformation group such that C*(G, Q)~ is Hausdorff and every stability group is amenable.…”

ON THE TOPOLOGY OF THE DUAL SPACE OF CROSSED PRODUCT C*-ALGEBRAS WITH FINITE GROUPS

“…Describing this structure in a general setting is a difficult task. To gain any meaningful insight about A ⋊ σ G one has had to impose various conditions on A and G [1,2,4,5,7,8]. Recently Echterhoff and Williams gave a concrete description of the dual space in the case of a strictly proper action on a continuous trace C * -algebra [3].…”