❘♦❦❤❧✐♥ ❉✐♠❡♥s✐♦♥ ❇❡②♦♥❞ ❘❡s✐❞✉❛❧❧② ❋✐♥✐t❡ ●r♦✉♣s ❏✐❛♥❝❤❛♦ ❲✉ P❡♥♥ ❙t❛t❡ ❯♥✐✈❡rs✐t② ❯❆❇✱ ❏✉❧② ✶✷✱ ✷✵✶✼ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✶ ✴ ✾
◆✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❛♥❞ t❤❡ ❊❧❧✐♦tt ❝❧❛ss✐✜❝❛t✐♦♥ ♣r♦❣r❛♠ ❲✐♥t❡r ❛♥❞ ❩❛❝❤❛r✐❛s ❞❡✈❡❧♦♣❡❞ ❛ ❦✐♥❞ ♦❢ ❞✐♠❡♥s✐♦♥ t❤❡♦r② ❢♦r ✭♥✉❝❧❡❛r✮ C ∗ ✲❛❧❣❡❜r❛s✳ dim nuc : CStarAlg → Z ≥ 0 ∪ {∞} ✳ ❙♦♠❡ ❜❛s✐❝ ♣r♦♣❡rt✐❡s✿ X t♦♣♦❧♦❣✐❝❛❧ s♣❛❝❡ ⇒ dim nuc ( C 0 ( X )) = dim( X ) ✭❝♦✈❡r✐♥❣ ❞✐♠✳✮✳ X ❛ ♠❡tr✐❝ s♣❛❝❡ ⇒ dim nuc ( C ∗ u ( X )) ≤ asdim( X ) ✭❛s②♠♣t♦t✐❝ ❞✐♠✳✮✳ → ( ✜♥✳❞✐♠✳ C ∗ ✲❛❧❣ ) ✮✳ dim nuc ( A ) = 0 ⇐ ⇒ A ✐s ❆❋ ✭ = lim − A ❑✐r❝❤❜❡r❣ ❛❧❣❡❜r❛ ✭❡✳❣✳ O n ✮ = ⇒ dim nuc ( 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✳ ❙✐♥❝❡ ♠❛♥② ❢✉♥ ❡①❛♠♣❧❡s ❛r❡ ❝♦♥str✉❝t❡❞ ❛s ❝r♦ss❡❞ ♣r♦❞✉❝ts✱ ✇❡ ❛s❦✿ ◗✉❡st✐♦♥✿ ❲❤❡♥ ❞♦❡s ❋❆❉ ♣❛ss t❤r♦✉❣❤ t❛❦✐♥❣ ❝r♦ss❡❞ ♣r♦❞✉❝ts❄ ▼♦r❡ ♣r❡❝✐s❡❧②✱ ✐❢ dim nuc ( A ) < ∞ ✫ G � A ✱ ✇❤❡♥ dim nuc ( A ⋊ G ) < ∞ ❄ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✷ ✴ ✾
➽✇❤❡♥❄ dim nuc ( A ) < ∞ = ⇒ dim nuc ( A ⋊ G ) < ∞ ■♥ ❣❡♥❡r❛❧✱ t❤✐s ✐s ❞✐✣❝✉❧t✳ ❇✉t t❤❡r❡ ❛r❡ s♦♠❡ s✉❝❝❡ss❢✉❧ str❛t❡❣✐❡s✱ ❡✳❣✳✱ ✉s✐♥❣ ❞✐♠❡♥s✐♦♥s ❞❡✜♥❡❞ ❢♦r ❛❝t✐♦♥s✳ ❋✐rst ❝❛s❡✿ ■❢ A = C ( X ) ✱ t❤❡r❡ ❛r❡ ❛ ♥✉♠❜❡r ♦❢ ❝❧♦s❡❧②✲r❡❧❛t❡❞ ❞✐♠❡♥s✐♦♥s t❤❛t ✇❡ ❝❛♥ ✉s❡✱ ❡✳❣✳✱ ❞②♥❛♠✐❝❛❧ ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ ✭●✉❡♥t♥❡r✲❲✐❧❧❡tt✲❨✉✮✱ ❛♠❡♥❛❜✐❧✐t② ❞✐♠❡♥s✐♦♥ ✭●✲❲✲❨✮✱ ✭✜♥❡✮ t♦✇❡r ❞✐♠❡♥s✐♦♥ ✭❑❡rr✮✳ ❲❡ ✇✐❧❧ ❢♦❝✉s ♦♥ t♦✇❡r ❞✐♠❡♥s✐♦♥ ✐♥ t❤✐s t❛❧❦✳ ▲❡t X ❜❡ ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ❛♥❞ α : G � X ❛♥ ❛❝t✐♦♥✳ ❉❡✜♥✐t✐♦♥ ✭❚♦✇❡r ❞✐♠❡♥s✐♦♥✮ ❆ t♦✇❡r ✐s ❛ ♣❛✐r ( B, S ) ✇❤❡r❡ B ⊂ X ✐s ♦♣❡♥✱ S ⋐ G ✱ ❛♥❞ { α g ( B ): g ∈ S } ✐s ❞✐s❥♦✐♥t✳ dim tow ( α ) ≤ d ✐✛ ∀ F ⋐ G ✱ ∃ t♦✇❡rs ( B i , S i ) ❢♦r i ∈ I = I (0) ∪ . . . ∪ I ( d ) ✱ s✉❝❤ t❤❛t ∀ l ∈ { 0 , . . . , d } ✱ { α g ( B i ): g ∈ S i , i ∈ I ( l ) } ✐s ❞✐s❥♦✐♥t❀ ✶ ∀ x ∈ X ✱ ∃ i ∈ I ❛♥❞ g ∈ S i s✉❝❤ t❤❛t Fg ⊂ S i ❛♥❞ x ∈ α g ( B i ) ✷ ✭ = ⇒ X = � g ∈ S i ,i ∈ I α g ( B i ) ✮✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✸ ✴ ✾
❉❡✜♥✐t✐♦♥ ✭❚♦✇❡r ❞✐♠❡♥s✐♦♥✮ dim tow ( α ) ≤ d ✐✛ ∀ F ⋐ G ✱ ∃ t♦✇❡rs ( B i , S i ) ❢♦r i ∈ I = I (0) ∪ . . . ∪ I ( d ) ✱ s✉❝❤ t❤❛t ✶ ∀ l ∈ { 0 , . . . , d } ✱ { α g ( B i ): g ∈ S i , i ∈ I ( l ) } ✐s ❞✐s❥♦✐♥t❀ ✷ ∀ x ∈ X ✱ ∃ i ∈ I ❛♥❞ g ∈ S i s✉❝❤ t❤❛t Fg ⊂ S i ❛♥❞ x ∈ α g ( B i ) ✳ ❚❤❡♦r❡♠ ✭❑❡rr✮ dim +1 nuc ( C ( X ) ⋊ α G ) ≤ dim +1 ( X ) · dim +1 tow ( α ) ✳ ■❢ G ✐s ✜♥✐t❡✱ ✇❡ ♠❛② ❝❤♦♦s❡ F = G ✳ ❚❤❡♥ ❲▲❖●✱ S i = G ✱ ∀ i ✱ ❛♥❞ ❡❛❝❤ t♦✇❡r ✐s ❛ ❧♦❝❛❧ tr✐✈✐❛❧✐③❛t✐♦♥ ♦❢ t❤❡ ❛❝t✐♦♥✳ ❚❤❡ t♦✇❡r ❞✐♠❡♥s✐♦♥ r❡❝♦✈❡rs ❛ ♥♦t✐♦♥ ❝❛❧❧❡❞ t❤❡ G ✲✐♥❞❡①✳ ❊①❛♠♣❧❡✿ ❚❤❡ ♦❞♦♠❡t❡r ❛❝t✐♦♥ α : Z � Z n ❤❛s dim tow ( α ) = 1 ✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ✐❢ G ✐s r❡s✐❞✉❛❧❧② ✜♥✐t❡ ❛♥❞ lim → G/G i ✐s ✐ts ♣r♦✜♥✐t❡ − ❝♦♠♣❧❡t✐♦♥ ❛ss♦❝✳ t♦ ❛ s❡♣❛r❛t✐♥❣ ❝❤❛✐♥ G 1 > G 2 > . . . ♦❢ ✜♥✐t❡✲✐♥❞❡① s✉❜❣r♦✉♣s✱ ✇❡ ❤❛✈❡ dim tow ( G � lim → G/G i ) ≤ asdim( � { G i } G ) ✱ t❤❡ − ❛s②♠♣t♦t✐❝ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ❜♦① s♣❛❝❡ ❛ss♦❝✳ t♦ { G i } ✳ dim tow ( G � βG ) = asdim( G ) ✳ ❊✳❣✳✱ t❤✐s ✇♦r❦s ❢♦r t❤❡ ❢r❡❡ ❣r♦✉♣✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✹ ✴ ✾
❚❤❡♦r❡♠ ✭❑❡rr✮ nuc ( C ( X ) ⋊ α G ) ≤ dim +1 ( X ) · dim +1 dim +1 tow ( α ) ✳ ■♥❣r❡❞✐❡♥ts ✐♥ t❤❡ ♣r♦♦❢✿ ❆ t♦✇❡r ( B, S ) ❞❡t❡r♠✐♥❡s ❛ s✉❜❛❧❣❡❜r❛ ♦❢ C ( X ) ⋊ G ✿ h : f ∈ C 0 ( B ) , h, g ∈ S } ∼ { u g fu ∗ = M | S | ( C 0 ( X )) ✳ ❚❤❡ ✏ F ✲♦✈❡r❧❛♣s✑ ♦❢ t❤❡ t♦✇❡rs ❤❡❧♣ ♣r♦❞✉❝❡ ❛ ♣❛rt✐t✐♦♥ ♦❢ ✉♥✐t② s✉❜❥❡❝t t♦ t❤❡ t♦✇❡rs ❝♦♥s✐st✐♥❣ ♦❢ ❢✉♥❝t✐♦♥s ✇❤✐❝❤ ❛r❡ ✏❛❧♠♦st ✢❛t ❛❧♦♥❣ ♦r❜✐ts✑✳ ❚❤✉s ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ✈❡r✐❢② dim am ( − ) < ∞ ✳ ❚❤❡♦r❡♠ ✭❙③❛❜♦✲❲✲❩❛❝❤❛r✐❛s✮ α ▲❡t G ❜❡ ❛ ❢✳❣✳ ♥✐❧♣♦t❡♥t ❣r♦✉♣ ❛♥❞ G � X ❛ ❢r❡❡ ❛❝t✐♦♥ ♦♥ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡ ✇✐t❤ dim( X ) < ∞ ✳ ❚❤❡♥ dim tow ( α ) < ∞ ✳ ❈♦r♦❧❧❛r② ❋♦r s✉❝❤ ❛ ( X, G, α ) ✱ t❤❡ ❝r♦ss❡❞ ♣r♦❞✉❝t C ( X ) ⋊ α G ❤❛s ✜♥✐t❡ ♥✉❝❧❡❛r ❞✐♠❡♥s✐♦♥ ❛♥❞ ✐s t❤✉s ✐♥ t❤❡ ❝❧❛ss ♦❢ s✐♠♣❧❡ s❡♣❛r❛❜❧❡ ✉♥✐t❛❧ ♥✉❝❧❡❛r C ∗ ✲❛❧❣❡❜r❛s t❤❛t ❛r❡ ❝❧❛ss✐✜❛❜❧❡ ❜② t❤❡ ❊❧❧✐♦tt ✐♥✈❛r✐❛♥t✳ ❏✐❛♥❝❤❛♦ ❲✉ ✭P❡♥♥ ❙t❛t❡✮ ❘♦❦❤❧✐♥ ❞✐♠❡♥s✐♦♥ ❜❡②♦♥❞ r✳❢✳ ❣r♦✉♣s ❯❆❇✱ ❏✉❧② ✶✷ ✺ ✴ ✾ ❲❤❛t ✐❢ A ✐s ♥♦♥❝♦♠♠✉t❛t✐✈❡❄ ❙❡❡ t❤❡ ♥❡①t s❧✐❞❡ − →
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