❯❋❋❊ ❍❆❆●❊❘❯P❙ ❚❍■❘❉ ❚❆▲❑ ❆❚ ▼❆❙❚❊❘❈▲❆❙❙ ❖◆ ❱❖◆ ◆❊❯▼❆◆◆ ❆▲●❊❇❘❆❙ ❆◆❉ ●❘❖❯P ❆❈❚■❖◆❙ ✶✳ ❈r♦ss❡❞ ♣r♦❞✉❝ts ■❢ A ⊆ B(H) ✐s ❛ ✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛ ❛♥❞ G ✐s ❛ ❞✐s❝r❡t❡ ❣r♦✉♣ ❛♥❞ α : G → ❆✉t(A) ✐s ❛ ❛❝t✐♦♥ ♦❢ G ♦♥ A✱ ❧❡t ✭✶✳✶✮ (π(a)ξ)(g) = α−1
g (a)ξ(g),
∀ξ ∈ l2(G, H), a ∈ A ❛♥❞ ✭✶✳✷✮ (λ(g)ξ)(h) = ξ(h−1g), ∀ξ ∈ l2(G, H), g ∈ G ❉❡✜♥❡ ✭✶✳✸✮ M := A ⋊ G := (π(A) ∪ λ(G))′′ t❤❡♥ ✭✶✳✹✮ λ(g)π(a)λ(g)∗ = π(αg(a)) ❛♥❞ ✐❢ ✇❡ ✐❞❡♥t✐❢② a ✇✐t❤ π(a) ∈ M✱ ✇❡ ❣❡t ✭✶✳✺✮ M := (A ∪ λ(G))′′ ❛♥❞ ✭✶✳✻✮ λ(g)aλ(g)∗ = αg(a). ✷✳ ●r♦✉♣ ♠❡❛s✉r❡ s♣❛❝❡ ❝♦♥str✉❝t✐♦♥ ❆s ❛ s♣❡❝✐❛❧ ❝❛s❡ t❛❦❡ A = L∞(X, µ) ✇❤❡r❡ X ✐s ❛ st❛♥❞❛r❞ ❇♦r❡❧ s♣❛❝❡ ❛♥❞ µ ✐s ❛ σ✲✜♥✐t❡ ♠❡❛s✉r❡✱ ❛♥❞ ❧❡t αg(f) = f(σ−1
g x) ❢♦r ❛♥ ❛❝t✐♦♥ σ : g → ❆✉t(X, [µ])✱
t❤❡ ❇♦r❡❧ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ X ♣r❡s❡r✈✐♥❣ t❤❡ ♠❡❛s✉r❡ ❝❧❛ss✳ ❉❡✜♥✐t✐♦♥ ✶✳ σ ✐s ❛♥ ❡r❣♦❞✐❝ ❛❝t✐♦♥ ✐✛ ❢♦r ❡✈❡r② G✲✐♥✈❛r✐❛♥t ❇♦r❡❧s❡t B ⊆ X ❡✐t❤❡r µ(B) = 0 ♦r µ(X\B) = 0✳ ❉❡✜♥✐t✐♦♥ ✷✳ σ ✐s ❢r❡❡ ✐❢ ❢♦r µ✲❛❧♠♦st ❛❧❧ x ∈ X g → gx ✐s ❛ ✶✲t♦✲✶✲♠❛♣ ❢r♦♠ G t♦ X✳ ❚❤❡♦r❡♠ ✸ ✭▼✉rr❛② ✰ ✈♦♥ ◆❡✉♠❛♥♥✱ ≈ ✶✾✹✵✮✳ ■❢ σ ✐s ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝ t❤❡♥ M = L∞(X, µ) ⋊α G ✐s ❛ ❢❛❝t♦r ❛♥❞ A = L∞(X, µ) ✐s ❛ ▼❆❙❆ ✭♠❛①✐♠❛❧ ❛❜❡❧✐❛♥ s❡❧❢❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛✮ ✐♥ M✳ ❚❤❡♦r❡♠ ✹ ✭▼✉rr❛② ✰ ✈♦♥ ◆❡✉♠❛♥♥✱ ≈ ✶✾✹✵✮✳ ❆ss✉♠❡ ❛ ❢r❡❡ ❛♥❞ ❡r❣♦❞✐❝ ❛❝t✐♦♥
- M ✐s ❛ I∞✲❢❛❝t♦r ✐✛ Ω ✐s ✐♥✜♥✐t❡ ❜✉t ❝♦✉♥t❛❜❧❡
- M ✐s ❛ II1✲❢❛❝t♦r ✐✛ Ω ✐s ✉♥❝♦✉♥t❛❜❧❡ ❛♥❞ t❤❡r❡ ❡①✐st ❛ G✲✐♥✈❛r✐❛♥t ✜♥✐t❡
♠❡❛s✉r❡ ν ∈ [µ]
- M ✐s ❛ II∞✲❢❛❝t♦r ✐✛ Ω ✐s ✉♥❝♦✉♥t❛❜❧❡ ❛♥❞ t❤❡r❡ ❡①✐sts ❛ G✲✐♥✈❛r✐❛♥t σ✲
✜♥✐t❡✱ ❜✉t ♥♦t ✜♥✐t❡ ♠❡❛s✉r❡ ν ∈ [µ]
❉❛t❡✿ ✷✼✴✵✶✴✷✵✶✵✳
✶