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  1. ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❉✐♠❡♥s✐♦♥s ❛♥❞ ❈r♦ss❡❞ Pr♦❞✉❝t C ∗ ✲❆❧❣❡❜r❛s ❏✐❛♥❝❤❛♦ ❲✉ P❡♥♥ ❙t❛t❡ ❯♥✐✈❡rs✐t② ◆❊❆▼ ✷✱ ❯♥✐✈❡rs✐t② ❛t ❆❧❜❛♥②✱ ❙❯◆❨✱ ❖❝t♦❜❡r ✶✹✱ ✷✵✶✼ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✶ ✴ ✾

  2. C ∗ ✲❛❧❣❡❜r❛s ❆ C ∗ ✲❛❧❣❡❜r❛ A ❂ ❛ ♥♦r♠✲❝❧♦s❡❞ s❡❧❢✲❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛ ♦❢ B ( H ) ✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C ∗ ✲❛❧❣❡❜r❛ ∼ = C ( X ) ❢♦r s♦♠❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ X ✳ ■t r❡♠❡♠❜❡rs X ❛s ✐ts s♣❡❝tr✉♠ ✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ C ∗ ✲❛❧❣❡❜r❛s ← → ♥♦♥❝♦♠♠✉t❛t✐✈❡✴q✉❛♥t✉♠ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✳ C ∗ ✲❛❧❣❡❜r❛✐❝ ❝r♦ss❡❞ ♣r♦❞✉❝ts α : G � A ❜② ❛✉t♦♠♦r♣❤✐s♠s � ❝r♦ss❡❞ ♣r♦❞✉❝t C ∗ ✲❛❧❣❡❜r❛ A ⋊ α G ✳ α G � X ❜② ❤♦♠❡♦♠♦r♣❤✐s♠s � C ( X ) ⋊ α G ♦❢t❡♥ r❡♣r❡s❡♥ts t❤❡ q✉♦t✐❡♥t X/G ❛s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤❡② ❢♦r♠ ♦♥❡ ♦❢ t❤❡ r✐❝❤❡st s♦✉r❝❡s ♦❢ C ∗ ✲❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡✿ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s ❂ C ( T ) ⋊ α Z ✱ ✇❤❡r❡ Z � T ❜② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✷ ✴ ✾

  3. C ∗ ✲❛❧❣❡❜r❛s ❆ C ∗ ✲❛❧❣❡❜r❛ A ❂ ❛ ♥♦r♠✲❝❧♦s❡❞ s❡❧❢✲❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛ ♦❢ B ( H ) ✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C ∗ ✲❛❧❣❡❜r❛ ∼ = C ( X ) ❢♦r s♦♠❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ X ✳ ■t r❡♠❡♠❜❡rs X ❛s ✐ts s♣❡❝tr✉♠ ✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ C ∗ ✲❛❧❣❡❜r❛s ← → ♥♦♥❝♦♠♠✉t❛t✐✈❡✴q✉❛♥t✉♠ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✳ C ∗ ✲❛❧❣❡❜r❛✐❝ ❝r♦ss❡❞ ♣r♦❞✉❝ts α : G � A ❜② ❛✉t♦♠♦r♣❤✐s♠s � ❝r♦ss❡❞ ♣r♦❞✉❝t C ∗ ✲❛❧❣❡❜r❛ A ⋊ α G ✳ α G � X ❜② ❤♦♠❡♦♠♦r♣❤✐s♠s � C ( X ) ⋊ α G ♦❢t❡♥ r❡♣r❡s❡♥ts t❤❡ q✉♦t✐❡♥t X/G ❛s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤❡② ❢♦r♠ ♦♥❡ ♦❢ t❤❡ r✐❝❤❡st s♦✉r❝❡s ♦❢ C ∗ ✲❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡✿ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s ❂ C ( T ) ⋊ α Z ✱ ✇❤❡r❡ Z � T ❜② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✷ ✴ ✾

  4. C ∗ ✲❛❧❣❡❜r❛s ❆ C ∗ ✲❛❧❣❡❜r❛ A ❂ ❛ ♥♦r♠✲❝❧♦s❡❞ s❡❧❢✲❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛ ♦❢ B ( H ) ✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C ∗ ✲❛❧❣❡❜r❛ ∼ = C ( X ) ❢♦r s♦♠❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ X ✳ ■t r❡♠❡♠❜❡rs X ❛s ✐ts s♣❡❝tr✉♠ ✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ C ∗ ✲❛❧❣❡❜r❛s ← → ♥♦♥❝♦♠♠✉t❛t✐✈❡✴q✉❛♥t✉♠ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✳ C ∗ ✲❛❧❣❡❜r❛✐❝ ❝r♦ss❡❞ ♣r♦❞✉❝ts α : G � A ❜② ❛✉t♦♠♦r♣❤✐s♠s � ❝r♦ss❡❞ ♣r♦❞✉❝t C ∗ ✲❛❧❣❡❜r❛ A ⋊ α G ✳ α G � X ❜② ❤♦♠❡♦♠♦r♣❤✐s♠s � C ( X ) ⋊ α G ♦❢t❡♥ r❡♣r❡s❡♥ts t❤❡ q✉♦t✐❡♥t X/G ❛s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤❡② ❢♦r♠ ♦♥❡ ♦❢ t❤❡ r✐❝❤❡st s♦✉r❝❡s ♦❢ C ∗ ✲❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡✿ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s ❂ C ( T ) ⋊ α Z ✱ ✇❤❡r❡ Z � T ❜② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✷ ✴ ✾

  5. C ∗ ✲❛❧❣❡❜r❛s ❆ C ∗ ✲❛❧❣❡❜r❛ A ❂ ❛ ♥♦r♠✲❝❧♦s❡❞ s❡❧❢✲❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛ ♦❢ B ( H ) ✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C ∗ ✲❛❧❣❡❜r❛ ∼ = C ( X ) ❢♦r s♦♠❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ X ✳ ■t r❡♠❡♠❜❡rs X ❛s ✐ts s♣❡❝tr✉♠ ✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ C ∗ ✲❛❧❣❡❜r❛s ← → ♥♦♥❝♦♠♠✉t❛t✐✈❡✴q✉❛♥t✉♠ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✳ C ∗ ✲❛❧❣❡❜r❛✐❝ ❝r♦ss❡❞ ♣r♦❞✉❝ts α : G � A ❜② ❛✉t♦♠♦r♣❤✐s♠s � ❝r♦ss❡❞ ♣r♦❞✉❝t C ∗ ✲❛❧❣❡❜r❛ A ⋊ α G ✳ α G � X ❜② ❤♦♠❡♦♠♦r♣❤✐s♠s � C ( X ) ⋊ α G ♦❢t❡♥ r❡♣r❡s❡♥ts t❤❡ q✉♦t✐❡♥t X/G ❛s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤❡② ❢♦r♠ ♦♥❡ ♦❢ t❤❡ r✐❝❤❡st s♦✉r❝❡s ♦❢ C ∗ ✲❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡✿ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s ❂ C ( T ) ⋊ α Z ✱ ✇❤❡r❡ Z � T ❜② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✷ ✴ ✾

  6. C ∗ ✲❛❧❣❡❜r❛s ❆ C ∗ ✲❛❧❣❡❜r❛ A ❂ ❛ ♥♦r♠✲❝❧♦s❡❞ s❡❧❢✲❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛ ♦❢ B ( H ) ✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C ∗ ✲❛❧❣❡❜r❛ ∼ = C ( X ) ❢♦r s♦♠❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ X ✳ ■t r❡♠❡♠❜❡rs X ❛s ✐ts s♣❡❝tr✉♠ ✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ C ∗ ✲❛❧❣❡❜r❛s ← → ♥♦♥❝♦♠♠✉t❛t✐✈❡✴q✉❛♥t✉♠ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✳ C ∗ ✲❛❧❣❡❜r❛✐❝ ❝r♦ss❡❞ ♣r♦❞✉❝ts α : G � A ❜② ❛✉t♦♠♦r♣❤✐s♠s � ❝r♦ss❡❞ ♣r♦❞✉❝t C ∗ ✲❛❧❣❡❜r❛ A ⋊ α G ✳ α G � X ❜② ❤♦♠❡♦♠♦r♣❤✐s♠s � C ( X ) ⋊ α G ♦❢t❡♥ r❡♣r❡s❡♥ts t❤❡ q✉♦t✐❡♥t X/G ❛s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤❡② ❢♦r♠ ♦♥❡ ♦❢ t❤❡ r✐❝❤❡st s♦✉r❝❡s ♦❢ C ∗ ✲❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡✿ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s ❂ C ( T ) ⋊ α Z ✱ ✇❤❡r❡ Z � T ❜② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✷ ✴ ✾

  7. C ∗ ✲❛❧❣❡❜r❛s ❆ C ∗ ✲❛❧❣❡❜r❛ A ❂ ❛ ♥♦r♠✲❝❧♦s❡❞ s❡❧❢✲❛❞❥♦✐♥t s✉❜❛❧❣❡❜r❛ ♦❢ B ( H ) ✳ ❆ ❝♦♠♠✉t❛t✐✈❡ ✉♥✐t❛❧ C ∗ ✲❛❧❣❡❜r❛ ∼ = C ( X ) ❢♦r s♦♠❡ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ X ✳ ■t r❡♠❡♠❜❡rs X ❛s ✐ts s♣❡❝tr✉♠ ✳ ◆♦♥❝♦♠♠✉t❛t✐✈❡ C ∗ ✲❛❧❣❡❜r❛s ← → ♥♦♥❝♦♠♠✉t❛t✐✈❡✴q✉❛♥t✉♠ t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡s✳ C ∗ ✲❛❧❣❡❜r❛✐❝ ❝r♦ss❡❞ ♣r♦❞✉❝ts α : G � A ❜② ❛✉t♦♠♦r♣❤✐s♠s � ❝r♦ss❡❞ ♣r♦❞✉❝t C ∗ ✲❛❧❣❡❜r❛ A ⋊ α G ✳ α G � X ❜② ❤♦♠❡♦♠♦r♣❤✐s♠s � C ( X ) ⋊ α G ♦❢t❡♥ r❡♣r❡s❡♥ts t❤❡ q✉♦t✐❡♥t X/G ❛s ❛ ♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡✳ ❚❤❡② ❢♦r♠ ♦♥❡ ♦❢ t❤❡ r✐❝❤❡st s♦✉r❝❡s ♦❢ C ∗ ✲❛❧❣❡❜r❛s✳ ❊①❛♠♣❧❡✿ ❆ ♥♦♥❝♦♠♠✉t❛t✐✈❡ t♦r✉s ❂ C ( T ) ⋊ α Z ✱ ✇❤❡r❡ Z � T ❜② ✐rr❛t✐♦♥❛❧ r♦t❛t✐♦♥s✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✷ ✴ ✾

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