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To each rf"*-dynamical system (1)11, G, a) corresponds canonically a C*-dynamical system (D1LC, G, a|9Hc). We show that the C*-crossed product G Xa ÇH*7 can be identified with a certain C*-subalgebra of the If*-crossed product GX, 91. The 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 (â, G, a) is called a C*-dynamical system if 6E 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 G &, is norm continuous. If 9H is a von Neumann algebra we define analogously a W*-dynamical system CDU, G, a), but now only with the requirement that each function / -* a,(x), x G 911, is a-weakly continuous. Given a If^-dynamical system (9lt, G, a) define 91te to be the set of elements x in 9lt for which the function t -» a,(x) is norm continuous, see [5, 7.5.1]. Clearly 91tc is a G-invariant C*-subalgebra of 911 containing all elements of the form aA[y) = fa,(y)f(t) dt, y E 9H,/ G LX(G) (since translation is continuous on LX(G)). Using an approximate unit in LX(G) we see that 91tc is in fact generated by elements af(y), and therefore a-weakly dense in 911. Thus we obtain from (9It, G, a) a canonically defined C*-dynamical system (91tc, G, a|9He). We shall study the relation between the W*-crossed product G xa 911 and the C*-crossed product G Xa 91tc. Recall from [8, §4] (cf. [5, 7.10.3]) that to each ^-dynamical system <91t, G, a) we can construct the dual system (G Xa 91L, G, a). We may identify 91L with the von Neumann subalgebra of G X 0 911 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.

On a certain 𝐶*-crossed product inside a 𝑊*-crossed product

On a certain 𝐶-crossed product inside a 𝑊-crossed product