On a Certain C ∗ -Crossed Product inside a W ∗ -Crossed Product

Abstract

To each W*-dynamical system (T, G, a) corresponds canonically a C*-dynamical system (9VRc, G, aI'D1t). We show that the C*-crossed product G Xa 9C can be identified with a certain C*-subalgebra of the W*-crossed product G x a DThe major part of the theory of noncommutative dynamical systems and their crossed products is Takesaki's work; see e.g. [7] and [8]. An important contribution, however, was made by Landstad who in [1] characterized those operator algebras that are crossed products with a given locally compact group G. Landstad's theory of G-products for abelian groups was exploited in [4] and [2] and we shall use it again to solve a problem arising from the difference between C*and W*-crossed products. A general exposition of noncommutative dynamical systems can be found in Chapters 7 and 8 of [5], but only the elementary parts of the theory will be needed here. Recall that a triple (6(, G, a) is called a C*-dynamical system if &6 is a C*-algebra and a is a representation of the locally compact abelian group G as automorphisms on &, such that each function t >a,(x), x E &, is norm continuous. If 6IT is a von Neumann algebra we define analogously a W*-dynamical system (6OR, G, a), but now only with the requirement that each function t -a(x), x E 'DX, is a-weakly continuous. Given a W*-dynamical system ('1., G, a) define DUZC to be the set of elements x in 6R for which the function t -* at(x) is norm continuous, see [5, 7.5.1]. Clearly DU is a G-invariant C*-subalgebra of 6Th containing all elements of the form a/f(y) = fat(y)f(t) dt, y E 6Th,f E L1(G) (since translation is continuous on L1(G)). Using an approximate unit in L1(G) we see that DIU is in fact generated by elements af(y), and therefore a-weakly dense in 69R. Thus we obtain from (6Th, G, a) a canonically defined C*-dynamical system (91Xc, G, a1lTc). We shall study the relation between the W*-crossed product G x a 6Th and the C*-crossed product G x a 9ZD. Recall from [8, ?4] (cf. [5, 7.10.3]) that to each W*-dynamical system (9T, G, a) we can construct the dual system (G xa 9, G, a). We may identify 9Th with the von Neumann subalgebra of G xa 9Th consisting of the fixed points for G under Received by the editors April 5, 1979 and, in revised form, August 17, 1979. AMS (MOS) subject classifications (1970). Primary 46L05; Secondary 46L10.