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❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆♦♥❝♦♠♠✉t❛t✐✈❡ ❉✐♠❡♥s✐♦♥s

❏✐❛♥❝❤❛♦ ❲✉ ✭❥♦✐♥t ✇✐t❤ ■❧❛♥ ❍✐rs❤❜❡r❣✮

P❡♥♥ ❙t❛t❡ ❯♥✐✈❡rs✐t②

❋✐❡❧❞s ■♥st✐t✉t❡✱ ❚♦r♦♥t♦✱ ❆✉❣✉st ✹✱ ✷✵✶✼

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✶ ✴ ✶✷

◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❛♥❞ t❤❡ ❊❧❧✐♦tt ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❣r❛♠

❲✐♥t❡r ❛♥❞ ❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ❛ ❦✐♥❞ ♦❢ ❞✐♠❡♥s✐♦♥ t❤❡♦r② ❢♦r ✭♥✉❝❧❡❛r✮ C∗✲❛❧❣❡❜r❛s✳ dimnuc : CStarAlg → Z≥0 ∪ {∞}✳ ❙♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s✿ X t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ⇒ dimnuc(C0(X)) = dim(X) ✭❝♦✈❡r✐♥❣ ❞✐♠✳✮✳ X ❛ ♠❡tr✐❝ s♣❛❝❡ ⇒ dimnuc(C∗

u(X)) ≤ asdim(X) ✭❛s②♠♣t♦t✐❝ ❞✐♠✳✮✳

dimnuc(A) = 0 ⇐ ⇒ A ✐s ❆❋ ✭= lim − →(✜♥✳❞✐♠✳ C∗✲❛❧❣)✮✳ A ❑✐r❝❤❜❡r❣ ❛❧❣❡❜r❛ ✭❡✳❣✳ On✮ = ⇒ dimnuc(A) = 1✳ ❋✐♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ✐s ♣r❡s❡r✈❡❞ ✉♥❞❡r t❛❦✐♥❣✿ ⊕✱ ⊗✱ q✉♦t✐❡♥ts✱ ❤❡r❡❞✐t❛r② s✉❜❛❧❣❡❜r❛s✱ ❞✐r❡❝t ❧✐♠✐ts✱ ❡①t❡♥s✐♦♥s✱ ❡t❝✳

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

❚❤❡ ❝❧❛ss ♦❢ ✉♥✐t❛❧ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ 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❝❡ ♦❢ ✐♥t❡r❡st✐♥❣ C∗✲❛❧❣❡❜r❛s✳ ❲❡ ❛s❦✿

◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s ❋❆❉ ♣❛ss t❤r♦✉❣❤ t❛❦✐♥❣ ❝r♦ss❡❞ ♣r♦❞✉❝ts❄

▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ dimnuc(A) < ∞ ✫ G A✱ ✇❤❡♥ dimnuc(A ⋊ G) < ∞❄

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✷ ✴ ✶✷

dimnuc(A) < ∞

➽✇❤❡♥❄

= ⇒ dimnuc(A ⋊ G) < ∞

❆ ♣r♦♠✐♥❡♥t ❝❛s❡ ✐s ✇❤❡♥ A = C(X) ❢♦r ♠❡tr✐❝ s♣❛❝❡ X ❛♥❞ G ✐s ♥♦♥❝♣t✳

❚❤❡♦r❡♠ ✭❚♦♠s✲❲✐♥t❡r✱ ❍✐rs❤❜❡r❣✲❲✐♥t❡r✲❩❛❝❤❛r✐❛s✮

■❢ Z X ♠✐♥✐♠❛❧❧② ❛♥❞ dim(X) < ∞✱ t❤❡♥ dimnuc(C(X) ⋊ Z) < ∞✳ ❍✐rs❤❜❡r❣✲❲✐♥t❡r✲❩❛❝❤❛r✐❛s ♣r♦✈✐❞❡❞ ❛ ♠♦r❡ ❝♦♥❝❡♣t✉❛❧ ❛♣♣r♦❛❝❤ ❜② ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ✭♠♦r❡ ♦♥ t❤❛t ❧❛t❡r✮✳ ◆♦t❡✿ ■❢ X ✐s ✐♥✜♥✐t❡✱ ❛ ♠✐♥✐♠❛❧ Z✲❛❝t✐♦♥ ✐s ❢r❡❡✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✮

■❢ Zm X ❢r❡❡❧② ❛♥❞ dim(X) < ∞✱ t❤❡♥ dimnuc(C(X) ⋊ Zm) < ∞✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

■❢ ❛ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t ❣r♦✉♣ G X ❢r❡❡❧② ❛♥❞ dim(X) < ∞✱ t❤❡♥ dimnuc(C(X) ⋊ G) < ∞✳ ④❋✳❣✳ ✈✐r✳♥✐❧♣✳ ❣♣s⑥ ●r♦♠♦✈ = ④❢✳❣✳ ❣♣s ✇✐t❤ ♣♦❧②♥♦♠✐❛❧ ❣r♦✇t❤⑥ ∋ ✜♥✐t❡ ❣♣s✱ Zm✱ t❤❡ ❞✐s❝r❡t❡ ❍❡✐s❡♥❜❡r❣ ❣r♦✉♣ 1

a c 1 b 1

• : a, b, c ∈ Z
• ✱ ❡t❝✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✸ ✴ ✶✷

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮ r❡♣❡❛t❡❞

❋✳❣✳ ✈✐r✳♥✐❧♣✳ G X ❢r❡❡❧② ✫ dim(X) < ∞ ⇒ dimnuc(C(X) ⋊ G) < ∞✳ ■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✿

✶ ❚❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α)✱ ❞❡✜♥❡❞ ❢♦r ❛ C∗✲❞②♥❛♠✐❝❛❧ s②st❡♠

α: G A✱ ✇❤❡r❡ G ✐s ✜♥✐t❡ ✭❍✲❲✲❩✮✱ Z ✭❍✲❲✲❩✮✱ Zm ✭❙③❛❜ó✮✱ r❡s✐❞✉❛❧❧② ✜♥✐t❡ ✭❙✲❲✲❩✮✱ ❝♦♠♣❛❝t ✭❍✐rs❤❜❡r❣✲P❤✐❧❧✐♣s✱ ●❛r❞❡❧❧❛✮✱ R ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✮✱ ✳✳✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

dim+1

nuc(A ⋊α,w G) ≤ asdim+1(G) · dim+1 nuc(A) · dim+1 Rok(α)

✳

✷ ❚❤❡ ♠❛r❦❡r ♣r♦♣❡rt② ✭❛♥❞ t❤❡ t♦♣♦❧♦❣✐❝❛❧ s♠❛❧❧ ❜♦✉♥❞❛r② ♣r♦♣❡rt②✮✱

st✉❞✐❡❞ ❜② ▲✐♥❞❡♥str❛✉ss✱ ●✉t♠❛♥✱ ❙③❛❜ó✱ ❛♥❞ ♦t❤❡rs✳

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

❋✳❣✳ ✈✐r✳♥✐❧♣✳ G

α

X ❢r❡❡❧② ✫ dim(X) < ∞ ⇒ dimRok(G C(X)) < ∞✳

✸ ❇♦✉♥❞ asdim+1(G) ❢♦r ❢✳❣✳ ✈✐r✳♥✐❧♣ G ✭❙✲❲✲❩✱ ❉❡❧❛❜✐❡✲❚♦✐♥t♦♥✮✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✹ ✴ ✶✷

P❛r❛❧❧❡❧ ❛♣♣r♦❛❝❤❡s

❙✐♠✐❧❛r ❛♣♣r♦❛❝❤❡s ♠❛❦❡ ✉s❡ ♦❢ ♦t❤❡r ❞✐♠❡♥s✐♦♥s ❞❡✜♥❡❞ ❢♦r t♦♣♦❧♦❣✐❝❛❧ ❞②♥❛♠✐❝❛❧ s②st❡♠s✱ ❡✳❣✳✱

❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ DAD(−) ✭●✉❡♥t♥❡r✲❲✐❧❧❡tt✲❨✉✮✱ ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ dimam(−) ✭●✲❲✲❨✱ ❙✲❲✲❩✱ ❛❢t❡r ❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✮✱ ❛♥❞ ✭✜♥❡✮ t♦✇❡r ❞✐♠❡♥s✐♦♥ dimtow(−) ✭❑❡rr✮✳

❚❤❡② ❛r❡ ❝❧♦s❡❧② r❡❧❛t❡❞ t❤r♦✉❣❤ ✐♥t❡rt✇✐♥✐♥❣ ✐♥❡q✉❛❧✐t✐❡s s✉❝❤ ❛s✿

❚❤❡♦r❡♠ ✭❙③❛❜ó✲❲✲❩❛❝❤❛r✐❛s✮

dim+1

Rok(α) ≤ dim+1 am(α) ≤ dim+1 Rok(α) · asdim+1(G)

✳ ❘❡♠❛r❦❛❜❧②✱ t❤❡ ♦r✐❣✐♥❛❧ ♠♦t✐✈❛t✐♦♥s ❢♦r ✐♥tr♦❞✉❝✐♥❣ dimam ❛♥❞ DAD ✇❡r❡ t♦ ❢❛❝✐❧✐t❛t❡ ❝♦♠♣✉t❛t✐♦♥s ♦❢ K✲t❤❡♦r② ❢♦r A ⋊ G✱ ✐♥ ♦r❞❡r t♦ ♣r♦✈❡ K✲t❤❡♦r❡t✐❝ ✐s♦♠♦r♣❤✐s♠ ❝♦♥❥❡❝t✉r❡s ✭t❤❡ ❇❛✉♠✲❈♦♥♥❡s ❝♦♥❥❡❝t✉r❡ ❛♥❞ t❤❡ ❋❛rr❡❧❧✲❏♦♥❡s ❝♦♥❥❡❝t✉r❡✮✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✺ ✴ ✶✷

❚❤❡ ❝❛s❡ ♦❢ ✢♦✇s

❲❤❡♥ G = R X ❝♦♥t✐♥✉♦✉s❧②✱ ✇❡ ❛❧s♦ ❤❛✈❡

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❙③❛❜ó✲❲✐♥t❡r✲❲✮

■❢ R X ❢r❡❡❧② ❛♥❞ dim(X) < ∞✱ t❤❡♥ dimnuc(C(X) ⋊ R) < ∞✳ ■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✿

✶ ❚❤❡ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ dimRok(α) ❞❡✜♥❡❞ ❢♦r ❛♥② C∗✲✢♦✇ α: R A✳

❚❤❡♦r❡♠ ✭❍✲❙✲❲✲❲✮

dim+1

nuc(A ⋊α R) ≤ 2 · dim+1 nuc(A) · dim+1 Rok(α)

✳

✷ ❚❤❡ ❡①✐st❡♥❝❡ ♦❢ ✏❧♦♥❣ t❤✐♥ ❝♦✈❡rs✑ ♦♥ ✢♦✇ s♣❛❝❡s ✱❞✉❡ t♦

❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤ ❛♥❞ ✐♠♣r♦✈❡❞ ❜② ❑❛s♣r♦✇s❦✐✲❘ü♣✐♥❣✳

❚❤❡♦r❡♠ ✭❇❛rt❡❧s✲▲ü❝❦✲❘❡✐❝❤✱ ❑❛s♣r♦✇s❦✐✲❘ü♣✐♥❣✱ ❍✲❙✲❲✲❲✮

R X ❢r❡❡❧② ❛♥❞ dim(X) < ∞ ⇒ dimRok(G C(X)) < ∞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✻ ✴ ✶✷

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

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✮

Z X ❧♦❝✳ ❝♣t ❍❛✉s❞✳ ✇✐t❤ dim(X) < ∞ ⇒ dimnuc(C0(X) ⋊ Z) < ∞✳

= ⇒ ❊①❛♠♣❧❡s ♦❢ ❣r♦✉♣s C∗✲❛❧❣❡❜r❛s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥s

dimnuc(C∗(Z2 ⋊A Z)) < ∞✱ ✇❤❡r❡ A = 2

1 1 1

• ∈ SL(2, Z)✳ ❚❤✐s ✐s ❛♥

❡①❛♠♣❧❡ ♦❢ ❛ ❣r♦✉♣ ✇❤✐❝❤ ✐s ♣♦❧②❝②❝❧✐❝ ❜✉t ♥♦t ♥✐❧♣♦t❡♥t✳ dimnuc(C∗(L)) < ∞ ❢♦r L = Z2 ≀ Z = Z

Z 2

⋊shift Z ✭❧❛♠♣❧✐❣❤t❡r ❣♣✮✳ ❇♦t❤ ❛r❡ ◗❉ ❜✉t ◆❖❚ str♦♥❣❧② ◗❉ ✭⇒ ❤❛✈❡ ✐♥✜♥✐t❡ ❞❡❝♦♠♣♦s✐t✐♦♥ r❛♥❦✮✦

❚❤❡♦r❡♠ ✭❊❝❦❤❛r❞t✲▼❝❑❡♥♥❡② ✇✐t❤♦✉t ✏✈✐rt✉❛❧❧②✑✱ ❊✲●✐❧❧❛s♣②✲▼✮

dimnuc(C∗(❛♥② ❢✳❣✳ ✈✐r✳♥✐❧♣✳ ❣♣))

• ≤ dr(C∗(❛♥② ❢✳❣✳ ✈✐r✳♥✐❧♣✳ ❣♣))
• < ∞✳

❚❤❡♦r❡♠ ✭❊❝❦❤❛r❞t✮

❉❡❝♦♠♣♦s✐t✐♦♥ r❛♥❦ dr(C∗(Zm ⋊A Z)) < ∞ ⇔ Zm ⋊A Z ✈✐r✳♥✐❧♣♦t❡♥t✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✼ ✴ ✶✷

❋♦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)) < ∞
• ✈✐rt✉❛❧❧② ♣♦❧②❝②❧✐❝ G
• ❡❧❡♠✳ ❛♠❡♥❛❜❧❡ G

✇✐t❤ ✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤

❚r✉❡ ❢♦r G = ❆❜❡❧✐❛♥⋊Z

✭❍✐rs❤❜❡r❣✲❲✮

• ???

dimnuc(C∗(G)) < ∞ ❘❡♠❛r❦✿ dimnuc(C∗(Z ≀ Z)) = ∞✳ Z ≀ Z ❛❧s♦ ❤❛s ✐♥✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤✳

◗✉❡st✐♦♥ ✭❊❝❦❤❛r❞t✲●✐❧❧❛s♣②✲▼❝❑❡♥♥❡②✮

❋♦r ❢✳❣✳ ❣r♦✉♣ G✱ dr(C∗(G)) < ∞ ⇒ G ✐s ✈✐rt✉❛❧❧② ♥✐❧♣♦t❡♥t❄

◗✉❡st✐♦♥

❲❤❛t ✐s t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❡❧❡♠❡♥t❛r② ❛♠❡♥❛❜❧❡ ❣r♦✉♣s ✇✐t❤ ✜♥✐t❡ ❍✐rs❝❤ ❧❡♥❣t❤ ❛♥❞ ❣r♦✉♣s ✇✐t❤ ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥❄

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✽ ✴ ✶✷

◆♦♥✲❢r❡❡ ✢♦✇s

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✮

R X ❧♦❝✳ ❝♣t ❍❛✉s❞✳ ✇✐t❤ dim(X) < ∞ ⇒ dimnuc(C0(X) ⋊ R) < ∞✳ ❘♦✉❣❤ s❦❡t❝❤ ♦❢ t❤❡ ♣r♦♦❢✿ P✐❝❦ ❛ ✏t❤r❡s❤♦❧❞✑ R > 0 ✭t♦ ❜❡ ❞❡t❡r♠✐♥❡❞✮✳ X≤R := ✉♥✐♦♥ ♦❢ ✭♣❡r✐♦❞✐❝✮ ♦r❜✐ts ♦❢ ❧❡♥❣t❤s ≤ R✱ ❛♥❞ X>R := ✉♥✐♦♥ ♦❢ ✭♣♦ss✐❜❧② ♥♦♥✲♣❡r✐♦❞✐❝✮ ♦r❜✐ts ♦❢ ❧❡♥❣t❤s > R✳ ✐♥✈❛r✐❛♥t ❞❡❝♦♠♣♦s✐t✐♦♥ X = X≤R ⊔ X>R ⇒ ❊①❛❝t s❡q✉❡♥❝❡ 0 → C0(X>R) ⋊ R → C0(X) ⋊ R → C0(X≤R) ⋊ R → 0 ❋❛❝t✿ dimnuc < ∞ ♣❛ss❡s t❤r♦✉❣❤ ❡①t❡♥s✐♦♥s ⇒ ▲♦♦❦ ❛t t❤❡ t✇♦ ❡♥❞s✦

✶ R X≤R ✐s ✇❡❧❧✲❜❡❤❛✈❡❞ ✭✐♥ ♣❛rt✐❝✉❧❛r✱ X≤R/R ✐s ❍❛✉s❞♦r✛✮ ⇒

dim+1

nuc(C0(X≤R) ⋊ R) ≤ 2 dim+1(X≤R) ≤ 2 dim+1(X)✳

■♠♣♦rt❛♥t✿ ❚❤✐s ❜♦✉♥❞ ❞♦❡s ♥♦t ❞❡♣❡♥❞ ♦♥ R✦

✷ ❋❛❝t✿ dimnuc < ∞ ✐s ❛ ✏❧♦❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✑ ♣r♦♣❡rt② ⇒ ✇❤❡♥ R ✐s

❝❤♦s❡♥ ❧❛r❣❡ ❡♥♦✉❣❤ ✭❞❡♣❡♥❞✐♥❣ ♦♥ t❤❡ ❞❡s✐r❡❞ ♣r❡❝✐s✐♦♥ ♦❢ t❤❡ ❧♦❝❛❧ ❛♣♣r♦①✐♠❛t✐♦♥✮✱ R X>R ❜❡❤❛✈❡s ❧✐❦❡ ❛ ❢r❡❡ ❛❝t✐♦♥ ❢♦r t❤❡ ♣✉r♣♦s❡ ♦❢ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ ⇒ ❲❡ ♠✐♠✐❝ t❤❡ ❛♣♣r♦❛❝❤ ❢♦r ❢r❡❡ ❛❝t✐♦♥s✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✾ ✴ ✶✷

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

❆ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ♦♥ X ❝♦♥s✐sts ♦❢ ❛♥ ❛t❧❛s ♦❢ ❝♦♠♣❛t✐❜❧❡ ❝❤❛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♦♥✮

❋❛❝t✿ ❡❛❝❤ ❧❡❛❢ ♦❢ ❛ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ∼ = R ♦r S1✳ ❆ ✢♦✇ R X ✇✐t❤♦✉t ✜①❡❞ ♣♦✐♥ts ❛♥ ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥✳ ❖r✐❡♥t❛t✐♦♥ ❢♦r ❛ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ❂ ❣❧♦❜❛❧ ❝❤♦✐❝❡ ♦❢ ❞✐r❡❝t✐♦♥s ❢♦r ❛❧❧ ❧✐♥❡s✳

❚❤❡♦r❡♠ ✭❲❤✐t♥❡②✮

❊✈❡r② ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ ✐s ✐♥❞✉❝❡❞ ❜② ❛ ✢♦✇ R X✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✶✵ ✴ ✶✷

❆ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ F ♦♥ X ❞❡✜♥❡s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥ ∼F ♦♥ X ♦❢ ✏❜❡✐♥❣ ♦♥ t❤❡ s❛♠❡ ❧❡❛❢✑✳ X/ ∼F ✐s t②♣✐❝❛❧❧② ♣❛t❤♦❧♦❣✐❝❛❧✳ ❈♦♥♥❡s✿ ❝♦♥s✐❞❡r t❤❡ ✏♥♦♥❝♦♠♠✉t❛t✐✈❡ q✉♦t✐❡♥t✑❀ ♠♦r❡ ♣r❡❝✐s❡❧②✱ ❝♦♥s✐❞❡r C∗(GF)✱ t❤❡ ❣r♦✉♣♦✐❞ C∗✲❛❧❣❡❜r❛ ♦❢ t❤❡ ❤♦❧♦♥♦♠② ❣r♦✉♣♦✐❞ GF ❛ss♦❝✐❛t❡❞ t♦ F✳ ❚❤❡ K✲t❤❡♦r② ♦❢ C∗(GF) ♣❧❛②s ❛ ❢✉♥❞❛♠❡♥t❛❧ r♦❧❡ ✐♥ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ✐♥❞❡① t❤❡♦r❡♠ ✭❈♦♥♥❡s✲❙❦❛♥❞❛❧✐s✮✳

Pr♦♣♦s✐t✐♦♥

■❢ F ✐s ✐♥❞✉❝❡❞ ❢r♦♠ ❛ ✢♦✇ R X✱ t❤❡♥ C∗(GF) ✐s ❛ q✉♦t✐❡♥t ♦❢ C0(X) ⋊ R✳

❚❤❡♦r❡♠ ✭❍✐rs❤❜❡r❣✲❲✮

❋♦r ❛♥② ♦r✐❡♥t❛❜❧❡ ❧✐♥❡ ❢♦❧✐❛t✐♦♥ F ♦♥ X ✇✐t❤ dim(X) < ∞✱ ✇❡ ❤❛✈❡ dimnuc(C∗(GF)) < ∞✳ Pr♦♦❢✿ dimnuc(C∗(GF)) ≤ dimnuc(C0(X) ⋊ R) < ∞✳

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✶✶ ✴ ✶✷

❚❤❛♥❦ ②♦✉✦

❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❈r♦ss❡❞ Pr♦❞✉❝ts ❛♥❞ ◆❈ ❉✐♠❡♥s✐♦♥s ❚♦r♦♥t♦✱ ❆✉❣✉st ✹ ✶✷ ✴ ✶✷