A sheaf cohomology theory for C*-algebras › sites › default › files › content › ...A sheaf...

Topology Exact categories Sheaves Our kind of categories OMod∞A (X ) is exact The next steps

A sheaf cohomology theory for C*-algebras

Martin Mathieu

(Queen’s University Belfast)

Banach Algebras 2017 at Ouluon 6 July 2017

Partially supported by UK Engineering and Physical Sciences Research CouncilGrant No. EP/M02461X/1.

Martin Mathieu (Queen’s University Belfast)



(Commutative) Topology

C∗1com

Comp

category of unital commutative C*-algebras with unital*-homomorphisms as morphisms;

category of compact Hausdorff spaces with continuousmappings as morphisms.




(Commutative) Topology

C∗1com

Comp

category of unital commutative C*-algebras with unital*-homomorphisms as morphisms;

category of compact Hausdorff spaces with continuousmappings as morphisms.

Comp

X ; Xf−→ Y

A; Bπ−→ A

π(ϕ) = ϕ ◦ π, ϕ ∈ B

C−→

ˆ←−

C∗1com

C (X ); C (Y )C(f )−→ C (X )

C (f )(g) = g ◦ f , g ∈ C (Y )

A; Aπ−→ B




Noncommutative Topology

C∗1 category of unital C*-algebras with unital*-homomorphisms as morphisms;





C∗1 category of unital C*-algebras with unital*-homomorphisms as morphisms;

? Martin Mathieu (Queen’s University Belfast)




A C*-algebra





A C*-algebra

Prim(A) space of primitive ideals of A with hull-kernel topology

OPrim(A) 3 U A(U) =⋂

t /∈U t;

I � A U(I ) = {t ∈ Prim(A) | I * t} ∈ OPrim(A).





A C*-algebra



t /∈U t;


hence V ⊆ U ⇒ A(V ) ⊆ A(U);

ringed space (A,X = Spec(A),OX )

“C* -ringed space” (A,Prim(A),A), A is a sheaf of C*-algebras





A C*-algebra



t /∈U t;


hence V ⊆ U ⇒ A(V ) ⊆ A(U);

ringed space (A,X = Spec(A),OX )

“C* -ringed space” (A,Prim(A),A), A is a sheaf of C*-algebras

Note A separable ⇒ closed prime ideals are primitive

P. Ara, M. Mathieu, Sheaves of C*-algebras, Math. Nachrichten 283 (2010), 21–39.




Sheaves of C*-algebras

Example 1. The multiplier sheaf

A C*-algebra with primitive ideal space Prim(A);

MA : OPrim(A) → C∗1 , MA(U) = M(A(U)),

where M(A(U)) denotes the multiplier algebra of the closedideal A(U) of A associated to the open subset U ⊆ Prim(A).

M(A(U))→ M(A(V )), V ⊆ U, the restriction homomorphisms.

Proposition

The above functor MA defines a sheaf of C*-algebras.




Sheaves of C*-algebras

Example 2. The injective envelope sheaf

let I (B) denote the injective envelope of B;

IA : OPrim(A) → C∗1 , IA(U) = pU I (A) = I (A(U)),

where pU = pA(U) denotes the unique central open projectionin I (A) such that pA(U)I (A) is the injective envelope of A(U).

I (A(U))→ I (A(V )), V ⊆ U, given by multiplication by pV

(as pV ≤ pU).

{pU | U ∈ OPrim(A)} is a complete Boolean algebra isomorphic tothe Boolean algebra of regular open subsets of Prim(A), and it isprecisely the set of projections of the AW*-algebra Z (I (A)).




The Aim: Sheaf Cohomology

H i(X ,F), i ∈ N

where X is a topological space, A a sheaf of C*-algebras on X andF ∈ OMod∞A (X ).




The Aim: Sheaf Cohomology

H i(X ,F), i ∈ N

where X is a topological space, A a sheaf of C*-algebras on X andF ∈ OMod∞A (X ).

joint work in progress with Pere Ara




A key property in abelian categories

in an abelian category,

• the morphism set between any pair of objects is an abeliangroup;

• every morphism has a kernel and a cokernel;

• every morphism can be uniquely factorised as

Eϕ //

π

F

G

µ

>>

where π is an epimorphism and µ is a monomorphism.









Eϕ //

π

F

G

µ

>>










Eϕ //

π

F

G

µ

>>





Operator modules

Definition

A right A-module E which at the same time is an operator space isa right operator A-module if it satisfies either:

(a) There exist a complete isometry Φ: E −→ B(H,K ), for someHilbert spaces H, K , and a *-homomorphism π : A −→ B(H)such that Φ(x · a) = Φ(x)π(a) for all x ∈ E , a ∈ A.

(b) The bilinear mapping E × A −→ E , (x , a) 7→ x · a extends to acomplete contraction E ⊗h A −→ E .

(c) For each n ∈ N, Mn(E ) is a right Banach Mn(A)-module inthe canonical way.

E is nondegenerate if the linear span of {x · a | x ∈ E , a ∈ A} isdense in E , and it is unital if A is unital and x · 1 = x for all x ∈ E .




Our kind of categories

Categories we need to consider (for a C*-algebra A):

OMod∞A

OMod1A

the category with objects the nondegenerate rightoperator A-modules and morphisms the completelybounded A-module maps;

the subcategory of OMod∞A with morphisms thecompletely contractive A-module maps.





Categories we need to consider (for a C*-algebra A):

OMod∞A

OMod1A

the category with objects the nondegenerate rightoperator A-modules and morphisms the completelybounded A-module maps;

additive, finitely bicomplete

the subcategory of OMod∞A with morphisms thecompletely contractive A-module maps.

non-additive, bicomplete

None of these categories is abelian!




A kernel

Let f ∈ MorC(A,B) for some A,B ∈ C.

A morphism i : K → A is a kernel of f if fi = 0 and for eachD ∈ C and g ∈ MorC(D,A) with fg = 0 there is a uniqueh ∈ MorC(D,K ) making the diagram below commutative

D

g

��

0

��

h

��K

0

66i // A

f // B

Any kernel is a monomorphism and is, up to isomorphism, unique.




A cokernel

Let f ∈ MorC(A,B) for some A,B ∈ C.

A morphism p : B → C is a cokernel of f if pf = 0 and for eachD ∈ C and g ∈ MorC(B,D) with gf = 0 there is a uniqueh ∈ MorC(C ,D) making the diagram below commutative

A

0

((

0&&

f// B p

//

g

��

C

h��

D

Any cokernel is an epimorphism and is, up to isomorphism, unique.





the situation in our categories:

monomorphisms are injective A-module maps

epimorphisms are A-module maps with dense range





the situation in our categories:

monomorphisms are injective A-module maps

epimorphisms are A-module maps with dense range

OMod1A

OMod∞A

T : E → F kernel iff is it a complete isometry;

T : E → F cokernel iff it is a complete quotient map.

T : E → F kernel iff it is a completely boundedisomorphism onto its image;

T : E → F cokernel iff it is surjective and completelyopen.




Homological Algebra

fundamental concept “short exact sequence”




Homological Algebra


ModA

OModA

0 // Eα // F

β // G // 0

where E ,F ,G are modules over a ring A andkerα = 0, imβ = G and imα = ker β.

E // M // FP // // G

where E ,F ,G are right operator A-modules over theC*-algebra A and M = Ker P and P = Coker M.




Homological Algebra


ModA

OModA

0 // Eα // F

β // G // 0

where E ,F ,G are modules over a ring A andkerα = 0, imβ = G and imα = ker β.

E // M // FP // // G “kernel–cokernel pair”

where E ,F ,G are right operator A-modules over theC*-algebra A and M = Ker P and P = Coker M.




Exact categories

let C be an additive category;




Exact categories


if C is abelian

every monomorphism is a kernel and every epimorphism is acokernel;




Exact categories


A kernel–cokernel pair (M,P) consists of two composablemorphisms in C such that M = Ker P and P = Coker M,

E1// M // E2

P // // E3

where Ei ∈ C.A monomorphism M arising in such a pair is called admissible

E // //F

and an epimorphism arising in such a pair is called admissible

E // //F




Exact categories (a la Quillen)

An exact structure on an additive category C is a class of kernel–cokernelpairs, closed under isomorphisms, satisfying the following axioms.

[E0] ∀ E ∈ C : 1E is an admissible monomorphism;

[E0op ] ∀ E ∈ C : 1E is an admissible epimorphism;

[E1] the class of admissible monomorphisms is closed undercomposition;

[E1op ] the class of admissible epimorphisms is closed under composition;

[E2] the push-out of an admissible monomorphism along an arbitrarymorphism exists and yields an admissible monomorphism;

[E2op ] the pull-back of an admissible epimorphism along an arbitrarymorphism exists and yields an admissible epimorphism.

Th. Buhler, Exact categories, Expo. Math. 28 (2010), 1–69.




An exact structure

Definition

Let A be a C*-algebra. We endow the additive category OMod∞Awith the exact structure of all kernel–cokernel pairs

E1// M // E2

P // // E3

where Ei ∈ obj(OMod∞A ), 1 ≤ i ≤ 3, M is a monomorphism inOMod∞A with closed range and completely bounded inverse, P is acompletely open mapping in OMod∞A (in particular, surjective)and ker P = im M.




An exact structure

Theorem

Let A be a C*-algebra. The class of all kernel–cokernel pairs inOMod∞A is an exact structure on OMod∞A .




what is next?

. . . now do the same for sheaves of operator modules




Sheaves in categories

X topological space;OX category of open subsets (with open subsets U as objectsand V → U if and only if V ⊆ U).

C bicomplete category with equalisers and coequalisers.

Definition

A presheaf on X in C is a contravariant functor F : OX → C.

A sheaf with values in C is a presheaf F such that F(∅) = 0 and,for every open subset U of X and every open cover U =

⋃i Ui ,

the maps F(U)→ F(Ui ) are the limit of the diagramsF(Ui )→ F(Ui ∩ Uj) for all i , j .




Limits in categories

Let I −→ C be a small diagram; we write Ei ∈ C for the imageof an object i ∈ I and, if ϕij : i → j is a morphism, we denote itsimage by Tϕij : Ei → Ej . An object L ∈ C together with morphismsπi : L→ Ei , i ∈ I is a limit of the diagram if they make thediagram below commutative and is final with this property.

Ei

Tϕij

��

L′

ρi

55

ρj))

// Lπi

::

πj

$$Ej

Also called an inverse limits or projective limit.





Let U ∈ OX . Let I be a set and U =⋃

i∈I Ui with Ui ∈ OX . Let

the index category I (2) consist of unordered pairs of elements in I(we allow the case of singleton sets in I (2))where {i , j} → {k, `} if {i , j} ⊆ {k , `}. Consider the diagram

I (2) → {Uij = Ui ∩ Uj | {i , j} ∈ I (2)} ⊆ OU , {i , j} 7→ Uij

composed with the functor F : OX → C. Then F(U) is the limit ofthe diagram

F(Ui )

))F(U)

77

''

F(Ui ∩ Uj)

F(Uj)

55





F(Ui0) // F(Ui0 ∩ Ui1)

F(U)

::

$$

ρ //∏i∈I

F(Ui )

OO

��

µi0 i1

77

νi0 i1''

µ //ν

//∏

(i0,i1)∈I×IF(Ui0 ∩ Ui1)

OO

��F(Ui1) // F(Ui0 ∩ Ui1)

The sheaf property is the requirement that the morphism ρ is theequaliser of the pair (µ, ν).




Sheaves in concrete categories

Notation and Terminology:

suppose C is a concrete category

the elements of F(U) are called sections over U ∈ OX ;

by s|V , V ⊆ U open, we mean the “restriction” of s ∈ F(U) to V ;i.e., the image of s in F(V ) under ρVU : F(U)→ F(V );

the unique gluing property of a sheaf can be expressed as follows:

for each compatible family of sections si ∈ F(Ui ), i.e.,si |Ui∩Uj

= sj |Ui∩Ujfor all i , j , there is a unique section s ∈ F(U)

such that s|Ui= si for all i .





F, G (pre)sheaves on X

morphism ϕ : F→ G is a natural transformation, i.e.,

F(U)

ρVU��

ϕU // G(U)

ρ′VU��

F(V ) ϕV

// G(V )

is commutative for all U,V ∈ OX , V ⊆ U;

hence we have the categories of sheaves on X , Sh(X ,C),and of presheaves, PSh(X ,C).





Stalks

F presheaf on X , t ∈ X ;

stalk of F at t is defined as Ft = lim−→UtF(U),

where Ut denotes the downward directed family of openneighbourhoods of t and lim−→ denotes the (directed) colimit in C.

With stalks we can build bundles and with bundles we can getsheaves again.





An Example

Let F be a presheaf on X . For each U ∈ OX , putFp(U) =

∏t∈U Ft and, for V ⊆ U, set ρVU the canonical

morphism from∏

t∈U Ft →∏

t∈V Ft .

In this way we obtain the product sheaf associated with F.

We shall assume that the canonical morphism σU : F(U)→ Fp(U)is a monomorphism whenever F is a sheaf.





The stalk functor

Suppose F and G are presheaves on X and ϕ : F→ G is amorphism of presheaves.For each t ∈ X there is a unique morphism ϕt : Ft → Gt .In this way, we obtain the stalk functor at t : PSh(X ,C)→ C.





The stalk functor


Properties:

Let ϕ(1), ϕ(2) : F→ G be morphisms of sheaves. Then

ϕ(1) = ϕ(2) if and only if ϕ(1)t = ϕ

(2)t for all t ∈ X .

Let ϕ : F→ G be a morphism of sheaves. Then

(i) ϕ is a monomorphism if ϕt is a monomorphism for all t ∈ X ;(ii) ϕ is an epimorphism if ϕt is an epimorphism for all t ∈ X ;(iii) ϕ is an isomorphism only if ϕt is an isomorphism for all t ∈ X .





The stalk functor


Properties:

Let ϕ(1), ϕ(2) : F→ G be morphisms of sheaves. Then

ϕ(1) = ϕ(2) if and only if ϕ(1)t = ϕ

(2)t for all t ∈ X .

Let ϕ : F→ G be a morphism of sheaves. Then

(i) ϕ is a monomorphism if ϕt is a monomorphism for all t ∈ X ;(ii) ϕ is an epimorphism if ϕt is an epimorphism for all t ∈ X ;(iii) ϕ is an isomorphism only if ϕt is an isomorphism for all t ∈ X .




“C*-ringed spaces”

Sheaves of operator modules over sheaves of C*-algebras

X topological space, A sheaf of C*-algebras on X

E : OX −→Ban1 (sheaf in Ban1) right operator A-module on X

if, for each U ∈ OX , E(U) is a (nondegenerate) right operatorA(U)-module and, for U,V ∈ OX with V ⊆ U,

TVU : E(U)→ E(V ) is completely contractive and

TVU(x · a) = TVU(x) · πVU(a) (x ∈ E(U), a ∈ A(U)),

where πVU : A(U)→ A(V ) are the restriction maps in A.




The categories we are interested in

Categories we need to consider(for a given sheaf A of C*-algebras on X ):

OMod∞A (X )

OMod1A(X )

the category with objects the right operatorA-modules and morphisms (at U) the completelybounded A(U)-module maps;

the subcategory of OMod∞A (X ) with the sameobjects and morphisms (at U) the completelycontractive A(U)-module maps.






OMod∞A (X )

OMod1A(X )


additive, finitely bicomplete


non-additive, bicomplete






OMod∞A (X )

OMod1A(X )


we need this one for homological algebra


we need this one for constructions






OMod∞A (X )

OMod1A(X )



None of these categories is abelian!




An exact structure on OMod∞A (X )

Definition

Let X be a topological space and let A be a sheaf of C*-algebrason X . Let ExA(X ) denote the collection of all kernel–cokernelspairs in OMod∞A (X )

E1// µ // E2

$ // // E3 .

We call this the canonical exact structure on OMod∞A (X ).




An exact structure on OMod∞A (X )

Definition

Let X be a topological space and let A be a sheaf of C*-algebrason X . Let ExA(X ) denote the collection of all kernel–cokernelspairs in OMod∞A (X )

E1// µ // E2

$ // // E3 .

We call this the canonical exact structure on OMod∞A (X ).

Theorem

The class ExA(X ) of all kernel–cokernel pairs defines an exactstructure on OMod∞A (X ).




Exact sequences

in ExA(X ), every kernel–cokernel pair

E1// µ // E2

$ // // E3

where Ei ∈ OMod∞A (X ) is called a short exact sequence.

Definition

The morphism ϕ ∈ CBA(E,F), E,F ∈ OMod∞A (X ) is calledadmissible if it can be factorised as

Eϕ //

$ ��

F

G?? µ

??

for some admissible monomorphism µ and some admissibleepimorphism $ in OMod∞A (X ).




Exact sequences

Definition

A sequence of admissible morphisms in OMod∞A (X )

E1ϕ1 //

$1 $$ $$

E2ϕ2 //

$2 $$ $$

E3

G1

:: µ1

::

G2

:: µ2

::

is said to be exact if the short sequence G1// µ1 // E2

$2 // // G2

is exact. An arbitrary sequence of admissible morphisms inOMod∞A (X ) is exact if the sequences given by any twoconsecutive morphisms are exact.




what are the next steps in our programme?

♦ to introduce injective sheaves;

♦ to construct injective resolutions;

♦ to define the homology of a complex F• in OMod∞A (X )

. . . // Fi−1δi−1

// Fiδi // Fi+1

// . . .

♦ to use homological algebra (such as the Horseshoe Lemma) inOMod∞A (X ) to ensure that homotopic injective resolutions yieldthe same homology

♦ to introduce the cohomology groups as the right derived functorof the global section functor applied to an injective resolution ofF ∈ OMod∞A (X )








. . . // Fi−1δi−1

// Fiδi // Fi+1

// . . .










. . . // Fi−1δi−1

// Fiδi // Fi+1

// . . .










. . . // Fi−1δi−1

// Fiδi // Fi+1

// . . .










. . . // Fi−1δi−1

// Fiδi // Fi+1

// . . .










. . . // Fi−1δi−1

// Fiδi // Fi+1

// . . .






Progress!






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