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 β^{ρ} of the restriction β^{p} of β to
N_{p} when p runs over all nonzero projections in N^{β}. 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 ⊗_{β}^{d}G (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 ⊗_{β}^{d}G is a factor and vice versa (Theorem 6.4). If β is regular, Γ(β) = G
is equivalent to Z(N^{β}) ⊂ 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 N^{β}
(Theorem 8.4). Moreover, by N ⊗_{β}^{d}(H∖G) we mean the von Neumann algebra
generated by β(N) and 1 ⊗ (L^{∞}(G) ∩ λ^{f}(H)′), where H is a closed subgroup of
G and λ′ is the right regular representation of G. N ⊗_{β}^{d}(H∖G) coincides
with the set of x ∈ N ⊗_{β}^{d}G such that β_{t}(x) = x for all t ∈ H (Theorem
7.2).
