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❇✐❛➟♦✇✐❡➺❛ ✷✵✶✻ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s ❝❛♥♦♥✐❝❛❧❧② r❡❧❛t❡❞ t♦ ❛ W ∗ ✲❛❧❣❡❜r❛ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❯♥✐✈❡rs✐t② ✐♥ ❇✐❛➟②st♦❦ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❘❊❋❊❘❊◆❈❊❙✿ ✶ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ❚✳ ❘❛t✐✉✳ ❇❛♥❛❝❤ ▲✐❡✲P♦✐ss♦♥ s♣❛❝❡s ❛♥❞ r❡❞✉❝t✐♦♥ ✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✹✸ ✭✷✵✵✸✮ ✶✲✺✹ ✷ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ❆✳ ❙❧✐➺❡✇s❦❛✳ ❇❛♥❛❝❤ ▲✐❡ ❣r♦✉♣♦✐❞s ❛ss♦❝✐❛t❡❞ t♦ W ∗ ✲❛❧❣❡❜r❛ ✱ t♦ ❛♣♣❡❛r ✐♥ ❏✳ ❙②♠♣❧✳ ●❡♦♠✳ ✸ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ●✳ ❏❛❦✐♠♦✇✐❝③✱ ❆✳ ❙❧✐➺❡✇s❦❛✳ ❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r♦✐❞s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❣r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ ❛ W ∗ ✲❛❧❣❡❜r❛ ✳ ❏✳●❡♦♠✳P❤②s✳✱ ✾✺ ✭✷✵✶✺✮ ✶✵✽✲✶✷✻ ✹ ❆✳ ❖❞③✐❥❡✇✐❝③✱ ●✳ ❏❛❦✐♠♦✇✐❝③✱ ❆✳ ❙❧✐➺❡✇s❦❛✳ ❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r P♦✐ss♦♥ str✉❝t✉r❡s r❡❧❛t❡❞ t♦ W ∗ ✲❛❧❣❡❜r❛ ✭✐♥ ♣r❡♣❛r❛t✐♦♥✮ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤ P♦✐ss♦♥ ♠❛♥✐❢♦❧❞ P ✲ ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ ✭♠♦❞❡❧❡❞ ♦♥ ♥♦♥✲r❡✢❡❦s✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡ ✐♥ ❣❡♥❡r❛❧✮ π ∈ Γ ∞ ( � 2 T ∗∗ P ) • Γ ∞ ( T ∗ P ) ∋ α �→ # α := π ( α, · ) ∈ Γ ∞ ( � 2 T ∗∗ P ) • # ✲ T ∗ P T ∗∗ P ✻ ✻ # ✲ , ✭✶✮ T ♭ P TP T ∗ P ⊃ T ♭ P ✲ q✉❛s✐ ❇❛♥❛❝❤ ✈❡❝t♦r s✉❜❜✉♥❞❧❡ ✭✇✐t❤♦✉t ❇❛♥❛❝❤ • ❝♦♠♣❧❡♠❡♥t ✐♥ ❣❡♥❡r❛❧✮ # T ♭ P = TP • P ∞ ( P ) := { f ∈ C ∞ ( P ) : # d f ∈ Γ ∞ TP } • P ∞ ( P ) ✲ P♦✐ss♦♥ ❛❧❣❡❜r❛ ✇✐t❤ r❡s♣❡❝t t♦ { f, g } := π ( d f, dg } ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞ G , B ✲ ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ ✇✐t❤ ❍❛✉s❞♦r✛ ✉♥❞❡r❧②✐♥❣ t♦♣♦❧♦❣② ❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞ G ⇒ B ✿ ✶ s♦✉r❝❡ ♠❛♣ s : G → B ❛♥❞ t❛r❣❡t ♠❛♣ t : G → B ✲ s✉❜♠❡rs✐♦♥s ✷ ♣r♦❞✉❝t m : G (2) → G m ( g, h ) =: gh, ❞❡✜♥❡❞ ♦♥ t❤❡ s❡t ♦❢ ❝♦♠♣♦s❛❜❧❡ ♣❛✐rs G (2) := { ( g, h ) ∈ G × G : s ( g ) = t ( h ) } , ✸ ✐❞❡♥t✐t② s❡❝t✐♦♥ ε : B → G ✲ ✐♠♠❡rs✐♦♥ ✹ ✐♥✈❡rs❡ ♠❛♣ ι : G → G ✱ ι ◦ ι = id ✱ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤✲▲✐❡ ❣r♦✉♣♦✐❞ ✇❤✐❝❤ s❛t✐s❢② t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s✿ s ( gh ) = s ( h ) , t ( gh ) = t ( g ) , ✭✷✮ k ( gh ) = ( kg ) h, ✭✸✮ ε ( t ( g )) g = g = gε ( s ( g )) , ✭✹✮ ι ( g ) g = ε ( s ( g )) , gι ( g ) = ε ( t ( g )) , ✭✺✮ ✇❤❡r❡ g, k, h ∈ G . ❚❤❡ s❤♦rt❡r ❞❡✜♥✐t✐♦♥✿ ❆ ❣r♦✉♣♦✐❞ ✐s ❛ s♠❛❧❧ ❝❛t❡❣♦r② ✇✐t❤ ✐♥✈❡rt✐❜❧❡ ♠♦r♣❤✐s♠s✳ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r♦✐❞ ❆ ❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r♦✐❞ ♦♥ ❇❛♥❛❝❤ ♠❛♥✐❢♦❧❞ M ✐s ❛ ❇❛♥❛❝❤ ✈❡❝t♦r ❜✉♥❞❧❡ q : A → M t♦❣❡t❤❡r ✇✐t❤✿ ✶ a : A → TM ✭❛♥❝❤♦r ♠❛♣✮ ✷ [ , ] : Γ A × Γ A → Γ A ✭▲✐❡ ❜r❛❝❦❡t✮ s✉❝❤ t❤❛t [ X, fY ] = f [ X, Y ] + a ( X )( f ) Y a ([ X, Y ]) = [ a ( X ) , a ( Y )] ❢♦r ❛❧❧ X, Y ∈ Γ A, f ∈ C ∞ ( M ) ✳ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r P♦✐ss♦♥ str✉❝t✉r❡ ❋✐❜r❡✲✇✐s❡ ❧✐♥❡❛r P♦✐ss♦♥ str✉❝t✉r❡ ❖♥❡ t❛❦❡s ❛s ❛ ✭s✉❜✮ P♦✐ss♦♥ ♠❛♥✐❢♦❧❞ P = E ✱ ✇❤❡r❡ E ✐s t❤❡ t♦t❛❧ q → M s✉❝❤ t❤❛t s♣❛❝❡ ♦❢ ❛ ❇❛♥❛❝❤ ✈❡❝t♦r ❜✉♥❞❧❡ E • P ∞ ( E ) ⊃ P ∞ lin ( E ) ✲ ✜❜r❡✲✇✐s❡ ❧✐♥❡❛r • P ∞ ( E ) ⊃ P ∞ B ( E ) ✲ ❝♦♥st❛♥t ♦♥ t❤❡ ✜❜r❡s ♦❢ q • { P ∞ B ( E ) , P ∞ B ( E ) } = 0 • { P ∞ B ( E ) , P ∞ lin ( E ) } ⊂ P ∞ B ( E ) • { P ∞ lin ( E ) , P ∞ lin ( E ) } ⊂ P ∞ lin ( E ) ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

W ∗ ✲❛❧❣❡❜r❛ ❆ C ∗ ✲❛❧❣❡❜r❛ M ✐s ❝❛❧❧❡❞ W ∗ ✲❛❧❣❡❜r❛ ✭✈♦♥ ◆❡✉♠❛♥♥ ❛❧❣❡❜r❛✮ ✐❢ t❤❡r❡ ❡①✐sts ❛ ❇❛♥❛❝❤ s♣❛❝❡ M ∗ s✉❝❤ t❤❛t ( M ∗ ) ∗ = M , ✭✻✮ M ∗ ⊂ M ∗ M ∗ ✲ ♣r❡❞✉❛❧ ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ M ✱ σ ( M , M ∗ ) ✲ t♦♣♦❧♦❣② ♦♥ M • M ∗ ∋ ρ ≥ 0 and � ρ � = 1 ρ ✲ st❛t❡ ✭♥♦r♠❛❧✮ ♦❢ t❤❡ q✉❛♥t✉♠ s②st❡♠ p 2 = p = p ∗ ∈ M • L ( M ) ∋ p ⇔ L ( M ) ✲ ❧❛tt✐❝❡ ❝♦♠♣❧❡t❡ ✐♥ σ ( M , M ∗ ) ✲t♦♣♦❧♦❣② L ( M ) ✲ ✒q✉❛♥t✉♠ ❧♦❣✐❝✑ ✭♣r♦♣♦s✐t✐♦♥s ❝❛❧❝✉❧✉s✮ • ♠♦r♣❤✐s♠ ♦❢ ❧❛tt✐❝❡s Σ : B → L ( M ) ≡ q✉❛♥t✉♠ ♦❜s❡r✈❛❜❧❡s M ∗ = L 1 ( M ) ✲ st❛♥❞❛r❞ ♠♦❞❡❧ ♦❢ M = L ∞ ( M ) ❊①❛♠♣❧❡ ✿ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

❚❤❡ ❇❛♥❛❝❤✲▲✐❡ P♦✐ss♦♥ str✉❝t✉r❡ ♦♥ M ∗ Df ( ρ ) , Dg ( ρ ) ∈ M ❢♦r f, g ∈ C ∞ ( M ∗ ) • ad ∗ : M ∗ → M ∗ ad ∗ M ∗ ⊂ M ∗ ✱ • and ad X Y := [ X, Y ] for X, Y ∈ M • ( M , [ · , · ]) ✲ ❇❛♥❛❝❤✲▲✐❡ ❛❧❣❡❜r❛ ❍❡♥❝❡ ♦♥❡ ❤❛s ▲✐❡✲P♦✐ss♦♥ ❜r❛❝❦❡t { f, g } LP ( ρ ) := � ρ, [ Df ( ρ ) , Dg ( ρ )] � ❍❛♠✐❧t♦♥✐❛♥ ❡q✉❛t✐♦♥ d dtρ = ad ∗ DH ( ρ ) ρ ❢♦r ❛ ❍❛♠✐❧t♦♥✐❛♥ H ∈ C ∞ ( M ∗ ) M = L ∞ ( M ) M ∗ = L 1 ( M ) ❊①❛♠♣❧❡ ✿ ✭✶✮ d dt ρ = [ DH ( ρ ) , ρ ] ✲ ♥♦♥✲❧✐♥❡❛r ✈♦♥ ◆❡✉♠❛♥♥✲▲✐♦✉✈✐❧❧❡ ❡q✉❛t✐♦♥ ✭✷✮ ❚❤❡ ❝❛s❡s ♦❢ t❤❡ ✐♥✜♥✐t❡ ❚♦❞❛ ❧❛tt✐❝❡ ❛♥❞ t❤❡ ♥♦♥✲❧✐♥❡❛r ❙❝❤rö❞✐♥❣❡r ❝❛♥ ❜❡ ❛❧s♦ ✇r✐tt❡♥ ✐♥ t❤✐s ✇❛②✳ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

●r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ W ∗ ✲❛❧❣❡❜r❛ M ▲❡❢t s✉♣♣♦rt l ( x ) ∈ L ( M ) ✭ r✐❣❤t s✉♣♣♦rt r ( x ) ∈ L ( M ) ✮ ♦❢ x ∈ M ✐s t❤❡ ❧❡❛st ♣r♦❥❡❝t✐♦♥ ✐♥ M ✱ s✉❝❤ t❤❛t l ( x ) x = x (resp . x r ( x ) = x ) . ✭✼✮ ■❢ x ∈ M ✐s s❡❧❢❛❞❥♦✐♥t✱ t❤❡♥ s✉♣♣♦rt s ( x ) s ( x ) := l ( x ) = r ( x ) . P♦❧❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ❢♦r x ∈ M x = u | x | , ✭✽✮ √ x ∗ x ∈ M + ✱ s✉❝❤ t❤❛t ✇❤❡r❡ u ∈ M ✐s ♣❛rt✐❛❧ ✐s♦♠❡tr② ❛♥❞ | x | := l ( x ) = s ( | x ∗ | ) = uu ∗ , r ( x ) = s ( | x | ) = u ∗ u. ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳

●r♦✉♣♦✐❞ ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ W ∗ ✲❛❧❣❡❜r❛ M ▲❡t G ( p M p ) ❜❡ t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ W ∗ ✲s✉❜❛❧❣❡❜r❛ p M p ⊂ M ✳ ❲❡ ❞❡✜♥❡ t❤❡ s❡t G ( M ) ♦❢ ♣❛rt✐❛❧❧② ✐♥✈❡rt✐❜❧❡ ❡❧❡♠❡♥ts ✐♥ M G ( M ) := { x ∈ M ; | x | ∈ G ( p M p ) , where p = s ( | x | ) } ❘❡♠❛r❦ ✿ G ( M ) � M ✐♥ ❛ ❣❡♥❡r❛❧ ❝❛s❡✳ ❆♥❛t♦❧ ❖❞③✐❥❡✇✐❝③ ●❡♦♠❡tr✐❝ str✉❝t✉r❡s✳✳✳