Let (_??_,
G, α) be a
C*-dynamical system, and
C*r(_??_; α) the reduced
C*-crossed product of _??_ by α. We construct a “dual”
C*-crossed product
C*d(
C*r(_??_; α); β) of
C*r(_??_; α) by an isomorphism β from
C*r(_??_; α) into the full operator algebra _??_(_??_) on a Hilbert space _??_. Then, it is isomorphic to the
C*-tensor product _??_ _??_
*C(
L2(
G)) of _??_ and the
C*-algebra
C(
L2(
G)) of all compact operators on
L2(
G).
In the abelian case, there exists a continuous action α of the dual group
G of
G on the
C*-crossed product
C*(_??_; α) of _??_ by α such that the
C*-crossed
C*(
C*(_??_; α); α) of
C*(_??_; α) by α is isomorphic to
C*d (
C*r(_??_; α); β).
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