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◆♦♥❝♦♠♠✉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 ✶✹ ✶ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

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✳ G

α

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 ✶✹ ✷ ✴ ✾

◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥

❲✐♥t❡r✲❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❢♦r C∗✲❛❧❣❡❜r❛s✿ dimnuc : {C∗✲❛❧❣❡❜r❛s} → Z≥0 ∪ {∞}✳ dimnuc(C(X)) = dim(X) ✭❝♦✈❡r✐♥❣ ❞✐♠❡♥s✐♦♥ ♦❢ X✮✳ ❋✐♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✐s ❛ ❝r✉❝✐❛❧ r❡❣✉❧❛r✐t② ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✏t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❤❛♠♠❡r✑ ✭●♦♥❣✲▲✐♥✲◆✐✉✱ ❊❧❧✐♦tt✲●♦♥❣✲▲✐♥✲◆✐✉✱ ❚✐❦✉✐s✐s✲❲❤✐t❡✲❲✐♥t❡r✱ ❑✐r❝❤❜❡r❣✲P❤✐❧❧✐♣s✱ ✳ ✳ ✳ ✮

❚❤❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ ✉♥✐t❛❧ C∗✲❛❧❣❡❜r❛s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✭❋◆❉✮ ❛♥❞ s❛t✐s❢②✐♥❣ ❯❈❚ ✐s ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❙✐♥❝❡ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❛r❡ ❛ ♠❛❥♦r s♦✉r❝❡ ♦❢ C∗✲❛❧❣❡❜r❛s✱ ✇❡ ❛s❦✿

▼❛✐♥ q✉❡st✐♦♥

❲❤❡♥ dimnuc(C(X) ⋊α G) < ∞❄ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥ X✱ G✱ ❛♥❞ t❤❡ ❛❝t✐♦♥ α✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✸ ✴ ✾

◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥

❲✐♥t❡r✲❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❢♦r C∗✲❛❧❣❡❜r❛s✿ dimnuc : {C∗✲❛❧❣❡❜r❛s} → Z≥0 ∪ {∞}✳ dimnuc(C(X)) = dim(X) ✭❝♦✈❡r✐♥❣ ❞✐♠❡♥s✐♦♥ ♦❢ X✮✳ ❋✐♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✐s ❛ ❝r✉❝✐❛❧ r❡❣✉❧❛r✐t② ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✏t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❤❛♠♠❡r✑ ✭●♦♥❣✲▲✐♥✲◆✐✉✱ ❊❧❧✐♦tt✲●♦♥❣✲▲✐♥✲◆✐✉✱ ❚✐❦✉✐s✐s✲❲❤✐t❡✲❲✐♥t❡r✱ ❑✐r❝❤❜❡r❣✲P❤✐❧❧✐♣s✱ ✳ ✳ ✳ ✮

❚❤❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ ✉♥✐t❛❧ C∗✲❛❧❣❡❜r❛s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✭❋◆❉✮ ❛♥❞ s❛t✐s❢②✐♥❣ ❯❈❚ ✐s ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❙✐♥❝❡ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❛r❡ ❛ ♠❛❥♦r s♦✉r❝❡ ♦❢ C∗✲❛❧❣❡❜r❛s✱ ✇❡ ❛s❦✿

▼❛✐♥ q✉❡st✐♦♥

❲❤❡♥ dimnuc(C(X) ⋊α G) < ∞❄ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥ X✱ G✱ ❛♥❞ t❤❡ ❛❝t✐♦♥ α✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✸ ✴ ✾

◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥

❲✐♥t❡r✲❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❢♦r C∗✲❛❧❣❡❜r❛s✿ dimnuc : {C∗✲❛❧❣❡❜r❛s} → Z≥0 ∪ {∞}✳ dimnuc(C(X)) = dim(X) ✭❝♦✈❡r✐♥❣ ❞✐♠❡♥s✐♦♥ ♦❢ X✮✳ ❋✐♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✐s ❛ ❝r✉❝✐❛❧ r❡❣✉❧❛r✐t② ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✏t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❤❛♠♠❡r✑ ✭●♦♥❣✲▲✐♥✲◆✐✉✱ ❊❧❧✐♦tt✲●♦♥❣✲▲✐♥✲◆✐✉✱ ❚✐❦✉✐s✐s✲❲❤✐t❡✲❲✐♥t❡r✱ ❑✐r❝❤❜❡r❣✲P❤✐❧❧✐♣s✱ ✳ ✳ ✳ ✮

❚❤❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ ✉♥✐t❛❧ C∗✲❛❧❣❡❜r❛s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✭❋◆❉✮ ❛♥❞ s❛t✐s❢②✐♥❣ ❯❈❚ ✐s ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❙✐♥❝❡ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❛r❡ ❛ ♠❛❥♦r s♦✉r❝❡ ♦❢ C∗✲❛❧❣❡❜r❛s✱ ✇❡ ❛s❦✿

▼❛✐♥ q✉❡st✐♦♥

❲❤❡♥ dimnuc(C(X) ⋊α G) < ∞❄ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥ X✱ G✱ ❛♥❞ t❤❡ ❛❝t✐♦♥ α✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✸ ✴ ✾

◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥

❲✐♥t❡r✲❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❢♦r C∗✲❛❧❣❡❜r❛s✿ dimnuc : {C∗✲❛❧❣❡❜r❛s} → Z≥0 ∪ {∞}✳ dimnuc(C(X)) = dim(X) ✭❝♦✈❡r✐♥❣ ❞✐♠❡♥s✐♦♥ ♦❢ X✮✳ ❋✐♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✐s ❛ ❝r✉❝✐❛❧ r❡❣✉❧❛r✐t② ♣r♦♣❡rt②✿

❚❤❡♦r❡♠ ✏t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ❤❛♠♠❡r✑ ✭●♦♥❣✲▲✐♥✲◆✐✉✱ ❊❧❧✐♦tt✲●♦♥❣✲▲✐♥✲◆✐✉✱ ❚✐❦✉✐s✐s✲❲❤✐t❡✲❲✐♥t❡r✱ ❑✐r❝❤❜❡r❣✲P❤✐❧❧✐♣s✱ ✳ ✳ ✳ ✮

❚❤❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ ✉♥✐t❛❧ C∗✲❛❧❣❡❜r❛s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✭❋◆❉✮ ❛♥❞ s❛t✐s❢②✐♥❣ ❯❈❚ ✐s ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❙✐♥❝❡ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❛r❡ ❛ ♠❛❥♦r s♦✉r❝❡ ♦❢ C∗✲❛❧❣❡❜r❛s✱ ✇❡ ❛s❦✿

▼❛✐♥ q✉❡st✐♦♥

❲❤❡♥ dimnuc(C(X) ⋊α G) < ∞❄ ■♥ ❣❡♥❡r❛❧✱ t❤❡ ❛♥s✇❡r ❞❡♣❡♥❞s ♦♥ X✱ G✱ ❛♥❞ t❤❡ ❛❝t✐♦♥ α✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✸ ✴ ✾

❋♦r G X ❝♦♠♣❛❝t ♠❡tr✐❝ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ●❆❋❆✱ ✷✵✶✸✮

✳✳✳✐❢ G = Z✱ ❛♥❞ G X ♠✐♥✐♠❛❧❧② ❛♥❞ ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = R✱ ❛♥❞ G X ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❊r❣♦❞✳ ❚❤❡♦r② ❉②♥✳ ❙②st✳✱ t♦ ❛♣♣❡❛r✮

✳✳✳✐❢ G ∈ FGVNilp✱ ❛♥❞ G X ❢r❡❡❧②✳ ❍❡r❡ FGVNilp ✿❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥

• r♦♠♦✈

= ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥✳ ■t ✐♥❝❧✉❞❡s Zd✱ ✜♥✐t❡ ❣r♦✉♣s✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

Z Z 1 Z 1

• ✳✳✳

❈♦r♦❧❧❛r②

❙✉❝❤ C(X) ⋊α G ❛r❡ ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✱ ✐❢ G X ♠✐♥✐♠❛❧❧②✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✹ ✴ ✾

❋♦r G X ❝♦♠♣❛❝t ♠❡tr✐❝ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ●❆❋❆✱ ✷✵✶✸✮

✳✳✳✐❢ G = Z✱ ❛♥❞ G X ♠✐♥✐♠❛❧❧② ❛♥❞ ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = R✱ ❛♥❞ G X ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❊r❣♦❞✳ ❚❤❡♦r② ❉②♥✳ ❙②st✳✱ t♦ ❛♣♣❡❛r✮

✳✳✳✐❢ G ∈ FGVNilp✱ ❛♥❞ G X ❢r❡❡❧②✳ ❍❡r❡ FGVNilp ✿❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥

• r♦♠♦✈

= ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥✳ ■t ✐♥❝❧✉❞❡s Zd✱ ✜♥✐t❡ ❣r♦✉♣s✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

Z Z 1 Z 1

• ✳✳✳

❈♦r♦❧❧❛r②

❙✉❝❤ C(X) ⋊α G ❛r❡ ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✱ ✐❢ G X ♠✐♥✐♠❛❧❧②✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✹ ✴ ✾

❋♦r G X ❝♦♠♣❛❝t ♠❡tr✐❝ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ●❆❋❆✱ ✷✵✶✸✮

✳✳✳✐❢ G = Z✱ ❛♥❞ G X ♠✐♥✐♠❛❧❧② ❛♥❞ ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = R✱ ❛♥❞ G X ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❊r❣♦❞✳ ❚❤❡♦r② ❉②♥✳ ❙②st✳✱ t♦ ❛♣♣❡❛r✮

✳✳✳✐❢ G ∈ FGVNilp✱ ❛♥❞ G X ❢r❡❡❧②✳ ❍❡r❡ FGVNilp ✿❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥

• r♦♠♦✈

= ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥✳ ■t ✐♥❝❧✉❞❡s Zd✱ ✜♥✐t❡ ❣r♦✉♣s✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

Z Z 1 Z 1

• ✳✳✳

❈♦r♦❧❧❛r②

❙✉❝❤ C(X) ⋊α G ❛r❡ ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✱ ✐❢ G X ♠✐♥✐♠❛❧❧②✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✹ ✴ ✾

❋♦r G X ❝♦♠♣❛❝t ♠❡tr✐❝ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ●❆❋❆✱ ✷✵✶✸✮

✳✳✳✐❢ G = Z✱ ❛♥❞ G X ♠✐♥✐♠❛❧❧② ❛♥❞ ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = R✱ ❛♥❞ G X ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❊r❣♦❞✳ ❚❤❡♦r② ❉②♥✳ ❙②st✳✱ t♦ ❛♣♣❡❛r✮

✳✳✳✐❢ G ∈ FGVNilp✱ ❛♥❞ G X ❢r❡❡❧②✳ ❍❡r❡ FGVNilp ✿❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥

• r♦♠♦✈

= ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥✳ ■t ✐♥❝❧✉❞❡s Zd✱ ✜♥✐t❡ ❣r♦✉♣s✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

Z Z 1 Z 1

• ✳✳✳

❈♦r♦❧❧❛r②

❙✉❝❤ C(X) ⋊α G ❛r❡ ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✱ ✐❢ G X ♠✐♥✐♠❛❧❧②✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✹ ✴ ✾

❋♦r G X ❝♦♠♣❛❝t ♠❡tr✐❝ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ●❆❋❆✱ ✷✵✶✸✮

✳✳✳✐❢ G = Z✱ ❛♥❞ G X ♠✐♥✐♠❛❧❧② ❛♥❞ ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = R✱ ❛♥❞ G X ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❊r❣♦❞✳ ❚❤❡♦r② ❉②♥✳ ❙②st✳✱ t♦ ❛♣♣❡❛r✮

✳✳✳✐❢ G ∈ FGVNilp✱ ❛♥❞ G X ❢r❡❡❧②✳ ❍❡r❡ FGVNilp ✿❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥

• r♦♠♦✈

= ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥✳ ■t ✐♥❝❧✉❞❡s Zd✱ ✜♥✐t❡ ❣r♦✉♣s✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

Z Z 1 Z 1

• ✳✳✳

❈♦r♦❧❧❛r②

❙✉❝❤ C(X) ⋊α G ❛r❡ ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✱ ✐❢ G X ♠✐♥✐♠❛❧❧②✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✹ ✴ ✾

❋♦r G X ❝♦♠♣❛❝t ♠❡tr✐❝ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ●❆❋❆✱ ✷✵✶✸✮

✳✳✳✐❢ G = Z✱ ❛♥❞ G X ♠✐♥✐♠❛❧❧② ❛♥❞ ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✱ ❈♦♠♠✳ ▼❛t❤✳ P❤②s✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = R✱ ❛♥❞ G X ❢r❡❡❧②✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✱ ❊r❣♦❞✳ ❚❤❡♦r② ❉②♥✳ ❙②st✳✱ t♦ ❛♣♣❡❛r✮

✳✳✳✐❢ G ∈ FGVNilp✱ ❛♥❞ G X ❢r❡❡❧②✳ ❍❡r❡ FGVNilp ✿❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞✱ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥

• r♦♠♦✈

= ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ❣r♦✉♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥✳ ■t ✐♥❝❧✉❞❡s Zd✱ ✜♥✐t❡ ❣r♦✉♣s✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

Z Z 1 Z 1

• ✳✳✳

❈♦r♦❧❧❛r②

❙✉❝❤ C(X) ⋊α G ❛r❡ ❝❧❛ss✐✜❡❞ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✱ ✐❢ G X ♠✐♥✐♠❛❧❧②✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✹ ✴ ✾

❆ ✷✲st❡♣ str❛t❡❣② t♦ s❤♦✇ dimnuc(C(X) ⋊α G) < ∞✳

✶ ❉❡✈❡❧♦♣ ❛ ❞✐♠❡♥s✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✱ s✉❝❤ ❛s✿

❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α) ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD(α) ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dimam(α) t♦✇❡r ❞✐♠❡♥s✐♦♥ dimtow(α)

❛♥❞ s❤♦✇✱ ❢♦r ✐♥st❛♥❝❡✱ dimRok(α) < ∞ = ⇒ dimnuc(C(X) ⋊α G) < ∞ .

✷ ❆❞❛♣t t❡❝❤♥✐q✉❡s ✐♥ t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝s ✭▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱

❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮ t♦ ❜♦✉♥❞ t❤❡ ❛❜♦✈❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡s❡ ❞✐♠❡♥s✐♦♥s ❛♣♣❧② t♦ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s ❜✉t ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✳

❘❡♠❛r❦✿ ❛♣♣❧✐❝❛t✐♦♥s t♦ K✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s

❚❤❡s❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❛♣♣❧✐❡❞ t♦✿ t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ♦♣❡r❛t♦r K✲t❤❡♦r②✮✱ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ❛❧❣❡❜r❛✐❝ K✲t❤❡♦r②✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✺ ✴ ✾

❆ ✷✲st❡♣ str❛t❡❣② t♦ s❤♦✇ dimnuc(C(X) ⋊α G) < ∞✳

✶ ❉❡✈❡❧♦♣ ❛ ❞✐♠❡♥s✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✱ s✉❝❤ ❛s✿

❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α) ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD(α) ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dimam(α) t♦✇❡r ❞✐♠❡♥s✐♦♥ dimtow(α)

❛♥❞ s❤♦✇✱ ❢♦r ✐♥st❛♥❝❡✱ dimRok(α) < ∞ = ⇒ dimnuc(C(X) ⋊α G) < ∞ .

✷ ❆❞❛♣t t❡❝❤♥✐q✉❡s ✐♥ t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝s ✭▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱

❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮ t♦ ❜♦✉♥❞ t❤❡ ❛❜♦✈❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡s❡ ❞✐♠❡♥s✐♦♥s ❛♣♣❧② t♦ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s ❜✉t ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✳

❘❡♠❛r❦✿ ❛♣♣❧✐❝❛t✐♦♥s t♦ K✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s

❚❤❡s❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❛♣♣❧✐❡❞ t♦✿ t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ♦♣❡r❛t♦r K✲t❤❡♦r②✮✱ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ❛❧❣❡❜r❛✐❝ K✲t❤❡♦r②✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✺ ✴ ✾

❆ ✷✲st❡♣ str❛t❡❣② t♦ s❤♦✇ dimnuc(C(X) ⋊α G) < ∞✳

✶ ❉❡✈❡❧♦♣ ❛ ❞✐♠❡♥s✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✱ s✉❝❤ ❛s✿

❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α) ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD(α) ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dimam(α) t♦✇❡r ❞✐♠❡♥s✐♦♥ dimtow(α)

❛♥❞ s❤♦✇✱ ❢♦r ✐♥st❛♥❝❡✱ dimRok(α) < ∞ = ⇒ dimnuc(C(X) ⋊α G) < ∞ .

✷ ❆❞❛♣t t❡❝❤♥✐q✉❡s ✐♥ t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝s ✭▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱

❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮ t♦ ❜♦✉♥❞ t❤❡ ❛❜♦✈❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡s❡ ❞✐♠❡♥s✐♦♥s ❛♣♣❧② t♦ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s ❜✉t ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✳

❘❡♠❛r❦✿ ❛♣♣❧✐❝❛t✐♦♥s t♦ K✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s

❚❤❡s❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❛♣♣❧✐❡❞ t♦✿ t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ♦♣❡r❛t♦r K✲t❤❡♦r②✮✱ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ❛❧❣❡❜r❛✐❝ K✲t❤❡♦r②✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✺ ✴ ✾

❆ ✷✲st❡♣ str❛t❡❣② t♦ s❤♦✇ dimnuc(C(X) ⋊α G) < ∞✳

✶ ❉❡✈❡❧♦♣ ❛ ❞✐♠❡♥s✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✱ s✉❝❤ ❛s✿

❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α) ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD(α) ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dimam(α) t♦✇❡r ❞✐♠❡♥s✐♦♥ dimtow(α)

❛♥❞ s❤♦✇✱ ❢♦r ✐♥st❛♥❝❡✱ dimRok(α) < ∞ = ⇒ dimnuc(C(X) ⋊α G) < ∞ .

✷ ❆❞❛♣t t❡❝❤♥✐q✉❡s ✐♥ t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝s ✭▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱

❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮ t♦ ❜♦✉♥❞ t❤❡ ❛❜♦✈❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡s❡ ❞✐♠❡♥s✐♦♥s ❛♣♣❧② t♦ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s ❜✉t ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✳

❘❡♠❛r❦✿ ❛♣♣❧✐❝❛t✐♦♥s t♦ K✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s

❚❤❡s❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❛♣♣❧✐❡❞ t♦✿ t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ♦♣❡r❛t♦r K✲t❤❡♦r②✮✱ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ❛❧❣❡❜r❛✐❝ K✲t❤❡♦r②✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✺ ✴ ✾

❆ ✷✲st❡♣ str❛t❡❣② t♦ s❤♦✇ dimnuc(C(X) ⋊α G) < ∞✳

✶ ❉❡✈❡❧♦♣ ❛ ❞✐♠❡♥s✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✱ s✉❝❤ ❛s✿

❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α) ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD(α) ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dimam(α) t♦✇❡r ❞✐♠❡♥s✐♦♥ dimtow(α)

❛♥❞ s❤♦✇✱ ❢♦r ✐♥st❛♥❝❡✱ dimRok(α) < ∞ = ⇒ dimnuc(C(X) ⋊α G) < ∞ .

✷ ❆❞❛♣t t❡❝❤♥✐q✉❡s ✐♥ t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝s ✭▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱

❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮ t♦ ❜♦✉♥❞ t❤❡ ❛❜♦✈❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡s❡ ❞✐♠❡♥s✐♦♥s ❛♣♣❧② t♦ ❞✐✛❡r❡♥t s✐t✉❛t✐♦♥s ❜✉t ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞✳

❘❡♠❛r❦✿ ❛♣♣❧✐❝❛t✐♦♥s t♦ K✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s

❚❤❡s❡ ✏❞②♥❛♠✐❝❛❧✑ ❞✐♠❡♥s✐♦♥s ❤❛✈❡ ❛❧s♦ ❜❡❡♥ ❛♣♣❧✐❡❞ t♦✿ t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ♦♣❡r❛t♦r K✲t❤❡♦r②✮✱ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡ ✭❢♦r ❛❧❣❡❜r❛✐❝ K✲t❤❡♦r②✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✺ ✴ ✾

◆♦♥✲❢r❡❡ ❛❝t✐♦♥s ⇒ Pr♦❜❧❡♠✿ dimRok(α) = ∞✱ ❜✉t✳✳✳

❋♦r G X ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ❆❞✈✳ ▼❛t❤✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = Z✳ ❘❡♠❛r❦✿ t❤❡r❡ ✐s ♥♦ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G = R✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G ∈ FGVNilp✳

❘❡❝❛❧❧ FGVNilp ❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥ ∋ Zd✱ ✜♥✐t❡ ❣r♦✉♣s✳✳✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✻ ✴ ✾

◆♦♥✲❢r❡❡ ❛❝t✐♦♥s ⇒ Pr♦❜❧❡♠✿ dimRok(α) = ∞✱ ❜✉t✳✳✳

❋♦r G X ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ❆❞✈✳ ▼❛t❤✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = Z✳ ❘❡♠❛r❦✿ t❤❡r❡ ✐s ♥♦ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G = R✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G ∈ FGVNilp✳

❘❡❝❛❧❧ FGVNilp ❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥ ∋ Zd✱ ✜♥✐t❡ ❣r♦✉♣s✳✳✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✻ ✴ ✾

◆♦♥✲❢r❡❡ ❛❝t✐♦♥s ⇒ Pr♦❜❧❡♠✿ dimRok(α) = ∞✱ ❜✉t✳✳✳

❋♦r G X ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ❆❞✈✳ ▼❛t❤✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = Z✳ ❘❡♠❛r❦✿ t❤❡r❡ ✐s ♥♦ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G = R✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G ∈ FGVNilp✳

❘❡❝❛❧❧ FGVNilp ❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥ ∋ Zd✱ ✜♥✐t❡ ❣r♦✉♣s✳✳✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✻ ✴ ✾

◆♦♥✲❢r❡❡ ❛❝t✐♦♥s ⇒ Pr♦❜❧❡♠✿ dimRok(α) = ∞✱ ❜✉t✳✳✳

❋♦r G X ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ❆❞✈✳ ▼❛t❤✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = Z✳ ❘❡♠❛r❦✿ t❤❡r❡ ✐s ♥♦ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G = R✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G ∈ FGVNilp✳

❘❡❝❛❧❧ FGVNilp ❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥ ∋ Zd✱ ✜♥✐t❡ ❣r♦✉♣s✳✳✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✻ ✴ ✾

◆♦♥✲❢r❡❡ ❛❝t✐♦♥s ⇒ Pr♦❜❧❡♠✿ dimRok(α) = ∞✱ ❜✉t✳✳✳

❋♦r G X ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C(X) ⋊α G) < ∞ ✐❢✳✳✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ❆❞✈✳ ▼❛t❤✳✱ ✷✵✶✼✮

✳✳✳✐❢ G = Z✳ ❘❡♠❛r❦✿ t❤❡r❡ ✐s ♥♦ ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ❛❝t✐♦♥ α✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G = R✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

✳✳✳✐❢ G ∈ FGVNilp✳

❘❡❝❛❧❧ FGVNilp ❂ ④✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣s⑥ ∋ Zd✱ ✜♥✐t❡ ❣r♦✉♣s✳✳✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✻ ✴ ✾

❆♣♣❧✐❝❛t✐♦♥✿ dimnuc ❢♦r ❣r♦✉♣ C∗✲❛❧❣❡❜r❛s C∗(G) := C ⋊ G

❋♦r ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ G✱ ✇❡ ❤❛✈❡✿ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t G

❊❝❦❤❛r❞t✲●✐❧❧❛s♣②✲▼❝❑❡♥♥❡②

• ❚r✉❡ ❢♦r G = Zm⋊AZ

✭❊❝❦❤❛r❞t✮

• dr(C∗(G)) < ∞
• ❡❧❡♠❡♥t❛r② ❛♠❡♥❛❜❧❡ G

✇✐t❤ ✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤

❚r✉❡ ❢♦r G ∈ Abelian⋊FGVNilp

✭❍✐rs❤❜❡r❣✲❲✮

• ???

dimnuc(C∗(G)) < ∞ ❍❡r❡ ❞r ❂ ❞❡❝♦♠♣♦s✐t✐♦♥ r❛♥❦✱ ❛ str♦♥❣❡r ✈❡rs✐♦♥ ♦❢ dimnuc✳ ❆ s✉♣♣♦rt✐♥❣ ♥♦♥✲❡①❛♠♣❧❡✿ G = Z ≀ Z ✭✇r❡❛t❤ ♣r♦❞✉❝t✮✳ ❚❤❡♥ dimnuc(C∗(G)) = ∞ ❛♥❞ G ❛❧s♦ ❤❛s ✐♥✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✼ ✴ ✾

❆♣♣❧✐❝❛t✐♦♥✿ dimnuc ❢♦r ❣r♦✉♣ C∗✲❛❧❣❡❜r❛s C∗(G) := C ⋊ G

❋♦r ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ G✱ ✇❡ ❤❛✈❡✿ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t G

❊❝❦❤❛r❞t✲●✐❧❧❛s♣②✲▼❝❑❡♥♥❡②

• ❚r✉❡ ❢♦r G = Zm⋊AZ

✭❊❝❦❤❛r❞t✮

• dr(C∗(G)) < ∞
• ❡❧❡♠❡♥t❛r② ❛♠❡♥❛❜❧❡ G

✇✐t❤ ✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤

❚r✉❡ ❢♦r G ∈ Abelian⋊FGVNilp

✭❍✐rs❤❜❡r❣✲❲✮

• ???

dimnuc(C∗(G)) < ∞ ❍❡r❡ ❞r ❂ ❞❡❝♦♠♣♦s✐t✐♦♥ r❛♥❦✱ ❛ str♦♥❣❡r ✈❡rs✐♦♥ ♦❢ dimnuc✳ ❆ s✉♣♣♦rt✐♥❣ ♥♦♥✲❡①❛♠♣❧❡✿ G = Z ≀ Z ✭✇r❡❛t❤ ♣r♦❞✉❝t✮✳ ❚❤❡♥ dimnuc(C∗(G)) = ∞ ❛♥❞ G ❛❧s♦ ❤❛s ✐♥✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✼ ✴ ✾

❆♣♣❧✐❝❛t✐♦♥✿ dimnuc ❢♦r C∗✲❛❧❣❡❜r❛s ♦❢ ❧✐♥❡ ❢♦❧✐❛t✐♦♥s

❆ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ♦♥ X ❝♦♥s✐sts ♦❢ ❛♥ ❛t❧❛s ♦❢ ❝❤❛rts ♦❢ t❤❡ ❢♦r♠ (0, 1) × U✳

✭❋✐❣✉r❡s t❛❦❡♥ ❢r♦♠ ●r♦✉♣♦✐❞s✱ ■♥✈❡rs❡ ❙❡♠✐❣r♦✉♣s✱ ❛♥❞ t❤❡✐r ❖♣❡r❛t♦r ❆❧❣❡❜r❛s ❜② ❆❧❛♥ P❛t❡rs♦♥✮

❆ ✢♦✇ R X ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts ❛♥ ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥✳ ❈♦♥♥❡s✿ ✉s❡ ❛ C∗✲❛❧❣❡❜r❛ C∗(GF) t♦ ❞❡s❝r✐❜❡ t❤❡ ✏♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡ ♦❢ ❧❡❛✈❡s✑ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥❞❡① t❤❡♦r②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

❋♦r ❛♥② ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ F ♦♥ X ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C∗(GF)) < ∞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✽ ✴ ✾

❆♣♣❧✐❝❛t✐♦♥✿ dimnuc ❢♦r C∗✲❛❧❣❡❜r❛s ♦❢ ❧✐♥❡ ❢♦❧✐❛t✐♦♥s

❆ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ♦♥ X ❝♦♥s✐sts ♦❢ ❛♥ ❛t❧❛s ♦❢ ❝❤❛rts ♦❢ t❤❡ ❢♦r♠ (0, 1) × U✳

✭❋✐❣✉r❡s t❛❦❡♥ ❢r♦♠ ●r♦✉♣♦✐❞s✱ ■♥✈❡rs❡ ❙❡♠✐❣r♦✉♣s✱ ❛♥❞ t❤❡✐r ❖♣❡r❛t♦r ❆❧❣❡❜r❛s ❜② ❆❧❛♥ P❛t❡rs♦♥✮

❆ ✢♦✇ R X ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts ❛♥ ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥✳ ❈♦♥♥❡s✿ ✉s❡ ❛ C∗✲❛❧❣❡❜r❛ C∗(GF) t♦ ❞❡s❝r✐❜❡ t❤❡ ✏♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡ ♦❢ ❧❡❛✈❡s✑ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥❞❡① t❤❡♦r②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

❋♦r ❛♥② ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ F ♦♥ X ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C∗(GF)) < ∞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✽ ✴ ✾

❆♣♣❧✐❝❛t✐♦♥✿ dimnuc ❢♦r C∗✲❛❧❣❡❜r❛s ♦❢ ❧✐♥❡ ❢♦❧✐❛t✐♦♥s

❆ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ♦♥ X ❝♦♥s✐sts ♦❢ ❛♥ ❛t❧❛s ♦❢ ❝❤❛rts ♦❢ t❤❡ ❢♦r♠ (0, 1) × U✳

✭❋✐❣✉r❡s t❛❦❡♥ ❢r♦♠ ●r♦✉♣♦✐❞s✱ ■♥✈❡rs❡ ❙❡♠✐❣r♦✉♣s✱ ❛♥❞ t❤❡✐r ❖♣❡r❛t♦r ❆❧❣❡❜r❛s ❜② ❆❧❛♥ P❛t❡rs♦♥✮

❆ ✢♦✇ R X ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts ❛♥ ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥✳ ❈♦♥♥❡s✿ ✉s❡ ❛ C∗✲❛❧❣❡❜r❛ C∗(GF) t♦ ❞❡s❝r✐❜❡ t❤❡ ✏♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡ ♦❢ ❧❡❛✈❡s✑ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥❞❡① t❤❡♦r②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

❋♦r ❛♥② ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ F ♦♥ X ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C∗(GF)) < ∞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✽ ✴ ✾

❆♣♣❧✐❝❛t✐♦♥✿ dimnuc ❢♦r C∗✲❛❧❣❡❜r❛s ♦❢ ❧✐♥❡ ❢♦❧✐❛t✐♦♥s

❆ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ♦♥ X ❝♦♥s✐sts ♦❢ ❛♥ ❛t❧❛s ♦❢ ❝❤❛rts ♦❢ t❤❡ ❢♦r♠ (0, 1) × U✳

✭❋✐❣✉r❡s t❛❦❡♥ ❢r♦♠ ●r♦✉♣♦✐❞s✱ ■♥✈❡rs❡ ❙❡♠✐❣r♦✉♣s✱ ❛♥❞ t❤❡✐r ❖♣❡r❛t♦r ❆❧❣❡❜r❛s ❜② ❆❧❛♥ P❛t❡rs♦♥✮

❆ ✢♦✇ R X ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts ❛♥ ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥✳ ❈♦♥♥❡s✿ ✉s❡ ❛ C∗✲❛❧❣❡❜r❛ C∗(GF) t♦ ❞❡s❝r✐❜❡ t❤❡ ✏♥♦♥❝♦♠♠✉t❛t✐✈❡ s♣❛❝❡ ♦❢ ❧❡❛✈❡s✑ ❛♣♣❧✐❝❛t✐♦♥s t♦ ✐♥❞❡① t❤❡♦r②✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✱ ✐♥ ♣r❡♣❛r❛t✐♦♥✮

❋♦r ❛♥② ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ F ♦♥ X ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C∗(GF)) < ∞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✽ ✴ ✾

❚❤❛♥❦ ②♦✉✦

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ◆❈ ❉✐♠❡♥s✐♦♥s ✫ ❈r♦ss❡❞ Pr♦❞✉❝ts ❆❧❜❛♥②✱ ◆❨✱ ❖❝t♦❜❡r ✶✹ ✾ ✴ ✾