Vol. 70, No. 2, 1977

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

Yoshiomi Nakagami

Vol. 70 (1977), No. 2, 437–479

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 βρ of the restriction βp of β to Np when p runs over all nonzero projections in Nβ. Γ(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. If β is regular, Γ(β) coincides with the kernel of the action β on the center of the crossed dual product N βdG (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 βdG is a factor and vice versa (Theorem 6.4). If β is regular, Γ(β) = G is equivalent to Z(Nβ) Z(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 βd(HG) we mean the von Neumann algebra generated by β(N) and 1 (L(G) λf(H)), where H is a closed subgroup of G and λis the right regular representation of G. N βd(HG) coincides with the set of x N βdG such that βt(x) = x for all t H (Theorem 7.2).

Mathematical Subject Classification 2000
Primary: 46L10
Secondary: 22D35
Received: 12 November 1976
Revised: 17 March 1977
Published: 1 June 1977
Yoshiomi Nakagami