aa r X i v : . [ m a t h . K T ] J a n A twisted Version of controlled K-Theory
Elisa HartmannJanuary 28, 2019
Abstract
This paper studies controlled operator K − theory on coarse spaces in light of new studiesin [2]. Contents
Controlled operator K − theory is one of the most popular homological invariants on coarse metricspaces. Meanwhile a new cohomological invariant on coarse spaces recently appeared in [2].The paper [2] studies sheaf cohomology on coarse spaces. Note that cohomology theories thatare derived functors are immensely more powerful than those that do not.In this paper we study the controlled K − theory of a proper metric space X which is intro-duced in [3, Chapter 6.3]. Note that this theory does not appear as a derived functor as far aswe know.The Theorem 12 shows if X is a proper metric space a modified version of the Roe-algebra C ∗ ( X ) is a cosheaf on X . This result gives rise to new computational tools one of which is anew Mayer-Vietoris six-term exact sequence which is Corollary 13.Note that in a general setting cosheaves with values in Ab do not give rise to a derivedfunctor. In [1] is explained that the dual version of sheafification, cosheafification, does not workin general.The outline of this paper is as follows: • The Chapter 1 introduces cosheaves on coarse spaces. • The main part of the study is in Chapter 2 • and Chapter 3 computes examples. 1 COSHEAVES Elisa Hartmann
We recall [2, Definition 45]:
Definition 1. ( coarse cover ) If X is a coarse space and U ⊆ X a subset a finite family ofsubsets U , . . . , U n ⊆ U is said to coarsely cover U if for every entourage E ⊆ U the set E [ U c ] ∩ · · · ∩ E [ U cn ]is bounded in U . Coarse covers on X determine a Grothendieck topology X ct on X . Definition 2. ( precosheaf ) A precosheaf on X ct with values in a category C is a covariantfunctor Cat ( X ct ) → C . Definition 3. ( cosheaf ) Let C be a category with finite limits and colimits. A precosheaf F on X ct with values in C is a cosheaf on X ct with values in C if for every coarse cover { U i → U } i there is a coequalizer diagram: M ij F ( U i ∩ U j ) ⇒ M i F ( U i ) → F ( U )Here the two arrows on the left side relate to the following 2 diagrams: L i,j F ( U i ∩ U j ) / / L i F ( U i ) F ( U i ∩ U j ) O O / / F ( U i ) O O and L i,j F ( U i ∩ U j ) / / L i F ( U j ) F ( U i ∩ U j ) / / O O F ( U j ) O O where L denotes the coproduct over the index set. Notation 4.
If we write • P i a i ∈ L i F ( U i ) then a i is supposed to be in F ( U i ) • P ij b ij ∈ L ij F ( U i ∩ U j ) then b ij is supposed to be in F ( U i ∪ U j ) Proposition 5. If F is a precosheaf on X ct with values in a category C with finite limits andcolimits and for every coarse cover { U i → U } i
1. and every a ∈ F ( U ) there is some P i a i ∈ L i F ( U i ) such that P i a i | U = a
2. and for every P i a i ∈ L i F ( U i ) such that P i a i | U = 0 there is some P ij b ij ∈ L ij F ( U i ∩ U j ) such that ( P j b ij − b ji ) | U i = a i for every i .then F is a cosheaf.Proof. easy. Remark . Denote by
CStar the category of C ∗ -algebras. According to [4] all finite limits andfinite colimits exist in CStar . 2
MODIFIED ROE-ALGEBRA Elisa Hartmann
Lemma 7. If X is a proper metric space and Y ⊆ X is a closed subspace then • the subset I ( Y ) = { f : f | Y = 0 } is an ideal of C ( X ) and we have C ( Y ) = C ( X ) /I ( Y ) • we can restrict the non-degenerate representation ρ X : C ( X ) → B ( H X ) to a representation ρ Y : C ( Y ) → B ( H Y ) in a natural way. • the inclusion i Y : H Y → H X covers the inclusion i : Y → X . • if the representation ρ X : C ( X ) → B ( H X ) is ample then the representation ρ Y : C ( Y ) → B ( H Y ) is ample.Proof. • This one follows by Gelfand duality. • We define H I ( Y ) = ρ X ( I ( Y )) H X . Then H X = H I ( Y ) ⊕ H ⊥ I ( Y ) is the direct sum of reducing subspaces for ρ X ( C ( X )). We define H Y = H ⊥ I ( Y ) and a representation of C ( Y ) on H Y by ρ Y ([ a ]) = ρ X ( a ) | H Y for every [ a ] ∈ C ( Y ). Note that ρ X ( · ) | H Y annihilates I ( Y ) so this is well defined. • Note that the support of i Y is supp( i Y ) = ∆ Y ⊆ X × Y • We recall a representation ρ X : C ( X ) → B ( H X ) is ample if it is non-degenerate and forevery f ∈ C ( X ) with f = 0 the operator ρ X ( f ) is not compact. If f ∈ C ( Y ) is such that ρ Y ( f ) : H Y → H Y is a compact operator and i Y : H Y → H Y covers the inclusion Y ⊆ X then i Y ◦ ρ Y ( f ) ◦ i ∗ Y = ρ X ( f )is a compact operator. Thus f as an element in C ( X ) acts compactly on B ( H X ). Lemma 8. If X is a proper metric space, B ⊆ X a compact subset and T ∈ C ∗ ( X ) an operatorwith supp T ⊆ B then T is a compact operator. MODIFIED ROE-ALGEBRA Elisa HartmannProof. • Suppose there is a non-degenerate representation ρ : C ( X ) → B ( H X ). For every f ∈ C ( B c ) , g ∈ C ( X ) the equations ρ ( f ) T ρ ( g ) = 0 and ρ ( g ) T ρ ( f ) = 0 hold. • This implies T ( I ( B )) = 0 and im T ∩ I ( B ) = 0. Thus T : H B → H B is the same map. • Thus T ∈ C ∗ ( B ) already. Now T is locally compact, B is compact thus T is a compactoperator. Definition 9. ( modified Roe-algebra ) Let X be a proper metric space thenˆ C ∗ ( X ) = C ∗ ( X ) / K ( H X )where K ( H X ) denotes the compact operators of B ( H X ) is called the modified Roe-algebra of X . Remark . If U ⊆ X is a subset of a proper metric space then U is coarsely dense in ¯ U . Wedefine ˆ C ∗ ( U ) := ˆ C ∗ ( ¯ U )Note that makes sense because if U , U is a coarse cover of U then ¯ U , ¯ U is a coarse cover of ¯ U also. Lemma 11. If Y ⊆ X is a closed subspace and i Y : H Y → H X the inclusion operator ofLemma 7 then • the operator Ad ( i Y ) : C ∗ ( Y ) → C ∗ ( X ) T i Y T i ∗ Y is well defined and maps compact operators to compact operators. • Then the induced operator on quotients ˆ Ad ( i Y ) : ˆ C ∗ ( Y ) → ˆ C ∗ ( X ) is the dual version of a restriction map.Proof. • i Y covers the inclusion the other statement is obvious. • easy. Theorem 12. If X is a proper metric space then the assignment U ˆ C ∗ ( U ) for every subspace U ⊆ X is a cosheaf with values in CStar .Proof.
Let U , U ⊆ U be subsets that coarsely cover U ⊆ X and V : H U → H U and V : H U → H U the corresponding inclusion operators.4 MODIFIED ROE-ALGEBRA Elisa Hartmann
1. Let T ∈ C ∗ ( U ) be a locally compact controlled operator. We need to construct T ∈ C ∗ ( U ) , T ∈ C ∗ ( U ) such that V T V ∗ + V T V ∗ = T modulo compacts. Denote by E = supp( T )the support of T in U . Define T := V ∗ T V and T := V ∗ T V then it is easy to check that T , T are locally compact and controlled operators, thuselements in C ∗ ( U ) , C ∗ ( U ). Now supp( V T V ∗ ) = U ∩ E and supp( V T V ∗ ) = U ∩ E .Thus supp( V T V ∗ + V T V ∗ − T ) = E ∩ ( U ∪ U ) c ⊆ B where B is bounded. This implies T | U + T | U = T .2. Suppose there are T ∈ C ∗ ( U ) , T ∈ C ∗ ( U ) such that V T V ∗ + V T V ∗ = 0modulo compacts. That implies that supp( V T V ∗ ) ⊆ ( U ∩ U ) modulo bounded setsand supp( V T V ∗ ) ⊆ ( U ∩ U ) modulo bounded sets. Also V T V ∗ = − V T V ∗ modulocompacts. Denote by V i : H U ∩ U → H U i the inclusion for i = 1 ,
2. Define T = V ∗ T V Then V T V ∗ = V V ∗ T V V ∗ = T Then V ◦ V = V ◦ V implies V T V ∗ = V ∗ V V T V ∗ V ∗ V = V ∗ V V T V ∗ V ∗ V = V ∗ V T V ∗ V = − T modulo compacts. 5 COMPUTING EXAMPLES Elisa Hartmann
Corollary 13. If U , U coarsely cover a subset U of a proper metric space X then there is asix-term Mayer-Vietoris exact sequence K ( ˆ C ∗ ( U ∩ U )) / / K ( ˆ C ∗ ( U )) ⊕ K ( ˆ C ∗ ( U )) / / K ( ˆ C ∗ ( U )) (cid:15) (cid:15) K ( ˆ C ∗ ( U )) O O K ( ˆ C ∗ ( U )) ⊕ K ( ˆ C ∗ ( U )) o o K ( ˆ C ∗ ( U ∩ U )) o o Proof.
By Theorem 12 there is a pullback diagram of C ∗ − algebras and ∗ -homomorphismsˆ C ∗ ( U ∩ U ) (cid:15) (cid:15) / / ˆ C ∗ ( U ) (cid:15) (cid:15) ˆ C ∗ ( U ) / / ˆ C ∗ ( U )The result is an application of [3, Exercise 4.10.22]. Remark . Now for every proper metric space there is a short exact sequence0 → K ( H X ) → C ∗ ( X ) → ˆ C ∗ ( X ) → K ( K ( H X )) / / K ( C ∗ ( X )) / / K ( ˆ C ∗ ( X )) (cid:15) (cid:15) K ( ˆ C ∗ ( X )) O O K ( C ∗ ( X )) o o K ( K ( H X )) o o If X is flasque then K i ( ˆ C ∗ ( X )) = ( i = 0 Z i = 1 Example 15. ( Z ) Now Z is the coarse disjoint union of two copies of Z + which is a flasquespace. By Corollary 13 there is an isomorphism K i ( ˆ C ∗ ( Z )) = ( i = 0 Z ⊕ Z i = 1Then it is a result of Remark 14 that there is an isomorphism K i ( C ∗ ( Z )) = ( i = 0 Z i = 1no surprise. 6 EFERENCES Elisa Hartmann
Example 16. ( Z ) We coarsely cover Z with V = Z + × Z ∪ Z × Z + and V = Z − × Z ∪ Z × Z . then again V is coarsely covered by U = Z + × Z and U = Z × Z + and V is coarsely covered in a similar fashion. We first compute modified controlled K-theoryof V and then of Z . Note that the inclusion U ∩ U → U is split by r : Z + × Z → Z ( x, y ) ( x, | y | )Thus using Corollary 13 we conclude that K i ( ˆ C ∗ ( V j )) = ( i = 0 Z i = 1for j = 1 ,
2. Then again using Corollary 13 and that the inclusion Z → V i is split we cancompute K i ( ˆ C ∗ ( Z )) = ( i = 00 i = 1Translating back we get that K i ( C ∗ ( Z )) = ( Z i = 00 i = 1This one also fits previous computations. References [1] Justin Michael Curry.
Sheaves, cosheaves and applications . ProQuest LLC, Ann Arbor, MI,2014. Thesis (Ph.D.)–University of Pennsylvania.[2] E. Hartmann. Coarse Cohomology with twisted Coefficients.
ArXiv e-prints , September 2017.[3] Nigel Higson and John Roe.