Essential spectrum Γ(β) of a dual action on a von Neumann algebra

Abstract

For a dual action β of a locally compact group G on a von Neumann algebra N we define the essential spectrum Γ(β) as the intersection of all spectrum sp βp of the restriction βp of β to Np when p runs over all nonzero projections in Nβ. T(β) is then an algebraic invariant for a covariant dual system {N, β}. Γ(β) is a closed subgroup of G (Theorem 3.7). We introduce three kinds of concept for β such as integrable, regular and dominant (§§4, 5). The former concepts are weaker than the dominance, ϊf β is regular, Γ(/3) coincides with the kerne! of the action β on the center of the crossed dual product N ζξ)dβG (Theorem 6.1). If β is regular, Γ(β) is normal and Γ(β) — Γ(β). If β is ergodic on the center Z(N) and Γ(β) = G, then N(£)dβG is a factor and vice versa (Theorem 6.4). If β is regular, Γ(β) = G is equivalent to Z(N β )CZ(N) (Proposition 6.3). If β is integrable on a factor N and if Γ(β)= G, then there is a lattice isomorphism between the closed subgroups of G and the von Neumann subalgebras of N containing Nβ (Theorem 8.4). Moreover, by N§ξ)dβ(H\G) we mean the von Neumann algebra generated by β(N)and 1 (g)(Lx(G)Π λ'(H)'), where H is a closed subgroup of G and Λ' is the right regular representation of G. N§ξ>dβ(H\G) coincides with the set of x Ξ N(g)^G such that βt(x) = x for all t E H (Theorem 7.2). 0. Introduction. In our previous paper [17, 16, 21] we have generalized Takesaki's duality to a general locally compact group in terms of a dual action and a crossed dual product as the following: In this paper we continue our study on dual actions and Takesaki's duality obtained in the above from the view point of covariant systems {M, a} and covariant dual systems {JV, β}. Then we naturally raise some questions: a. What is an invariant of equivalent covariant dual systems? b. When does Takesaki's duality hold as a covariant (dual) system? Using the spectrum of β given in [17, §5], we can define the essential spectrum Γ(j8) of β in §3 by the same manner as S set. Then Γ(β) is a closed subgroup of G and an algebraic invariant of dual actions on a