Homological Algebra Methods in the Theory of Operator Algebras

Homological algebra methods in the theory of Operator Algebras

Ryszard Nest Homological algebra methods in the theory of

Homological Operator Algebras II, algebra Ideals and projectives Homological functors, derived functors, Adams I-exact complexes Projective objects spectral sequence, assembly map and BC for The phantom tower Assembly map Derived functors compact quantum groups. The ABC spectral sequence

Example: Γ = Z Compact Quantum Ryszard Nest Groups

University of Copenhagen

25th July 2016 Homological algebra methods in the theory of Operator Algebras Let C be an abelian category Ryszard Nest We call a covariant functor F : T → C homological if

Homological algebra F (C) → F (A) → F (B) Ideals and projectives I-exact complexes Projective objects The phantom tower is exact for any distinguished triangle Assembly map Derived functors The ABC spectral ΣB → C → A → B. sequence

Example: Γ = Z n Compact We deﬁne Fn(A) := F (Σ A) for n ∈ . Quantum Z Groups Similarly, we call a contravariant functor F : T → C cohomological if F (B) → F (A) → F (C) is exact for any distinguished triangle, and we deﬁne F n(A) := F (ΣnA). Homological algebra methods in the theory of Operator Algebras

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Homological algebra We will always be in the following situation. Ideals and projectives I-exact complexes 1 C is some abelian category equipped with a shift Projective objects The phantom tower Σ: C → C (C is stable). Assembly map Derived functors The ABC spectral 2 Our functors F : T → C are homological and stable, i. e. sequence

Example: Γ = Z commute with Σ. Compact Quantum Groups Homological algebra methods in the theory of Operator Algebras We will do homological algebra relative to some ideal in T wich

Ryszard Nest satisﬁes the following property.

Homological Deﬁnition algebra Ideals and projectives An ideal I ⊆ T is called stable if the suspension isomorphisms I-exact complexes ∼ Projective objects = The phantom tower Σ: T(A, B) −→ T(ΣA, ΣB) for A, B ∈∈ T restrict to Assembly map isomorphisms Derived functors ∼ The ABC spectral = sequence Σ: I(A, B) −→ I(ΣA, ΣB).

Example: Γ = Z Compact Quantum Groups Deﬁnition An ideal I ⊆ T in a triangulated category is called homological if it is the kernel of a stable homological functor. Homological algebra methods in the theory of Operator Algebras

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Homological Before continuing with deﬁnitions, recall the picture of algebra distinguished triangle in T. Ideals and projectives I-exact complexes Projective objects f The phantom tower A / C Assembly map _ Derived functors The ABC spectral ◦ sequence h g Example: Γ = Z C Compact Quantum Groups Homological algebra methods in the theory of Operator Algebras Deﬁnition Ryszard Nest Let T be a triangulated category and let I ⊆ T be a

Homological homological ideal. Let f : A → B be a morphism in T and algebra f g h Ideals and projectives embed it in an exact triangle A → B → C → ΣA. I-exact complexes Projective objects • We call f I monic if h ∈ I. The phantom tower Assembly map • We call f I epic if g ∈ I. Derived functors The ABC spectral sequence • We call f an I equivalence if f is both I monic and I epic Example: Γ = Z or, equivalently, if g, h ∈ I. Compact Quantum • We call f an I phantom map if f ∈ I. Groups An object A ∈∈ T is called I contractible if idA ∈ I(A, A). g An exact triangle A →f B → C →h ΣA in T is called I exact if h ∈ I. Homological algebra methods in the theory of Operator Algebras Consider a chain complex C• = (Cn, dn). For each n ∈ N, we Ryszard Nest may embed the map dn in an exact triangle

Homological dn fn gn algebra Cn −→ Cn−1 −→ Xn −→ ΣCn, (1.1) Ideals and projectives I-exact complexes Projective objects which is determined uniquely up to (non-canonical) The phantom tower Assembly map isomorphism of triangles. Hence the following deﬁnition does Derived functors The ABC spectral not depend on auxiliary choices: sequence

Example: Γ = Z Compact Deﬁnition Quantum Groups The chain complex C• is called Iexact in degree n if the gn Σfn+1 composite map Xn −→ ΣCn −−−→ ΣXn+1 belongs to I. It is called Iexact if it is Iexact in degree n for all n ∈ Z. Homological algebra methods in the theory of Operator Algebras

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Homological If we recall from yesterday, in the diagram algebra Ideals and projectives I-exact complexes dn dn−1 Projective objects Cn / Cn−1 / Cn−2 The phantom tower ` c Assembly map ◦ ◦ Derived functors The ABC spectral | { sequence Xn o Xn−1 Example: Γ = Z Compact the composition given by the stipled arrow is in I. Quantum Groups Homological algebra methods in the theory of Operator Algebras

Ryszard Nest Lemma Let F : T → C be a stable homological functor into some stable Homological algebra Abelian category C. Let C• be a chain complex over T. The Ideals and projectives I-exact complexes complex C• is KerF exact in degree n if and only if the Projective objects The phantom tower sequence Assembly map Derived functors The ABC spectral F (dn+1) F (dn) sequence F (Cn+1) −−−−−→ F (Cn) −−−→ F (Cn−1) Example: Γ = Z Compact Quantum in C is exact at F (Cn). Groups Later we will meet homological ideals given as intersections kernels of a family of homological ideals. Homological algebra methods in the theory of Operator Algebras

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Homological algebra Ideals and projectives Deﬁnition I-exact complexes Projective objects The phantom tower An object A ∈∈ T is called I-projective if the functor Assembly map T(A, ␣): T → Ab is Iexact. Derived functors The ABC spectral sequence We write PI for the class of I-projective objects in T.

Example: Γ = Z Compact Quantum Groups Homological algebra methods in the theory of Operator Algebras

Ryszard Nest Lemma Homological algebra An object A ∈∈ T is I-projective if and only if I(A, B) = 0 for Ideals and projectives I-exact complexes all B ∈∈ T.. Projective objects The phantom tower Assembly map Derived functors Lemma The ABC spectral sequence The class PI of I-projective objects is closed under Example: Γ = Z Compact (de)suspensions, retracts, and possibly inﬁnite direct sums (as Quantum Groups far as they exist in T). Homological algebra methods in the theory of Operator Algebras

Ryszard Nest The following will supply us with projective objects.

Homological Theorem algebra Ideals and projectives I-exact complexes 1 Suppose that F is I-exact and Q ∈ T satisﬁes Projective objects The phantom tower Assembly map T(Q, A) = C(X, F (A)) Derived functors The ABC spectral sequence Example: Γ = Z for some object X in C. Then Q is I-projective. Compact Quantum 2 Suppose that I = Ker F . Then an object P of T is Groups projective iﬀ F (P) is projective in C Homological algebra methods in the theory of Operator Algebras Ryszard Nest T contains enough projectives if, for any object A in T, there Homological exists an exact triangle of the form algebra Ideals and projectives I-exact complexes j Projective objects A / N . The phantom tower _ Assembly map ◦ Derived functors The ABC spectral sequence P Example: Γ = Z Compact Quantum with P projective and j ∈ I(A, N). Then we can construct Groups projective resolutions. Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes AN N N N ··· Projective objects 0 1 2 3 The phantom tower Assembly map Derived functors The ABC spectral sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N N N N ··· Projective objects 0 1 2 3 The phantom tower Assembly map Derived functors The ABC spectral sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N0 N1 N2 N3 ··· Projective objects Y The phantom tower Assembly map π0 Derived functors The ABC spectral sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N0 / N1 N2 N3 ··· Projective objects Y The phantom tower Assembly map ◦ π0 Derived functors The ABC spectral Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of π0 Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 ι0 I-exact complexes A N0 / N1 N2 N3 ··· Projective objects Y Y The phantom tower Assembly map ◦ π0 π1 Derived functors The ABC spectral Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 2 ι0 ι1 I-exact complexes A N0 / N1 / N2 N3 ··· Projective objects Y Y The phantom tower Assembly map ◦ ◦ π0 π1 Derived functors The ABC spectral Õ Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of π1 Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 2 ι0 ι1 I-exact complexes A N0 / N1 / N2 N3 ··· Projective objects Y Y Y The phantom tower Assembly map ◦ ◦ π0 π1 π2 Derived functors The ABC spectral Õ Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 2 3 ι0 ι1 ι2 I-exact complexes A N0 / N1 / N2 / N3 ··· Projective objects Y Y Y The phantom tower Assembly map ◦ ◦ ◦ π0 π1 π2 Derived functors The ABC spectral Õ Õ Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of π2 Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 2 3 ι0 ι1 ι2 I-exact complexes A N0 / N1 / N2 / N3 / ··· Projective objects Y Y Y Y The phantom tower Assembly map ◦ ◦ ◦ π0 π1 π2 Derived functors The ABC spectral Õ Õ Õ sequence P0 P1 P2 P3 ··· Example: Γ = Z Compact Quantum mapping cone of etc. Groups Homological algebra methods in the theory of Operator Algebras The phantom tower.

Ryszard Nest We will use above to construct a projective resolution. Homological algebra Ideals and projectives 1 2 3 ι0 ι1 ι2 I-exact complexes A N0 / N1 / N2 / N3 / ··· Projective objects Y Y Y Y The phantom tower Assembly map ◦ ◦ ◦ π0 π1 π2 Derived functors The ABC spectral Õ Õ Õ sequence P0 o P1 o P2 o P3 o ··· Example: Γ = Z Compact Quantum mapping cone of Groups The succesive compositions produce the projective resolution of A Homological algebra methods in the theory of Operator Algebras Example

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Homological algebra Γ Ideals and projectives 1 T = KK for a discrete group Γ I-exact complexes H Projective objects 2 j ∈ I if, for all torsion subgroups H ⊂ Γ, j = 0 in KK The phantom tower Assembly map 3 PI coincides with the usual class of proper Γ-algebras. Derived functors The ABC spectral sequence Example: Γ = Z Deﬁnition Compact Quantum Given A, its projective cover is a I-projective object PA and Groups Γ Γ DA ∈ KK (PA, A) such that every k ∈ KK (Q, A) with Q ∈ PI factorizes through D. k Q / A ? DA PA Homological algebra methods in the theory of Operator Algebras Example continued

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Homological algebra Such a PA always exists! Ideals and projectives I-exact complexes In fact, it is of the form (PC ⊗ A, DC ⊗ 1), where DC is the usual Dirac Projective objects operator (mostly familiar if BΓ is a spin-manifold) The phantom tower Assembly map Derived functors The ABC spectral γ element sequence

Example: Γ = Z Γ has a γ-element if DC has the right inverse, i. e. there exists a Compact d ∈ KK Γ( , P )(dual Dirac) such that Quantum C C Groups d ◦ D = id|P . C C C Γ In this case the exterior Kasparov product with γ = DC ◦ dC ∈ KK (C, C) is the projection onto the complement of N , the I-contractible objects, and

Γ KK =< PI > ⊕N Homological algebra methods in the theory of Operator Algebras Example continued

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Homological algebra Ideals and projectives Given homological functor F , which vanishes on I, it has a total left I-exact complexes derived functor F with a natural transformation Projective objects L The phantom tower Assembly map LF → F . Derived functors The ABC spectral sequence In fact, LF (A) = F (PA). Example: Γ = Z Compact Theorem Quantum ∗ Groups Let F (A) = K∗(A ored Γ). Then LF (A) = KΓ (A) and

∗ KΓ (A) = LF (A) → F (A) = K∗(A ored Γ) is the assembly map. Homological algebra methods in the theory of Operator Algebras Example continued

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Homological algebra Theorem Ideals and projectives I-exact complexes Suppose that Γ satisﬁes strong Baum-Connes conjecture, i. e. it has the Projective objects Γ The phantom tower γ-element equal to one. Then KK coincides with the localizing Assembly map subcategory generated by PI. Derived functors The ABC spectral sequence Recall that f. ex. amenable groups satisfy the hypothesis. Moreover PI is Example: Γ = Z generated by homogeneous actions of Γ, hence any stable homological Compact Γ ∗ functor on KK which coincides with KΓ on homogeneous actions is the Quantum ∗ Groups same as KΓ . Corollary Suppose that Γ satisﬁes the strong Baum-Connes conjecture, Γ acts on X and that A is a Γ-C*-algebra in C∗alg(X). If A is in B(X), then so is A ored Γ. Homological algebra methods in the theory of Operator Algebras

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Homological algebra Ideals and projectives I-exact complexes Once we have a notion of exactness for chain complexes, we Projective objects The phantom tower can do homological algebra in the homotopy category Ho(T) of Assembly map Derived functors all chain complexes over T. The ABC spectral sequence

Example: Γ = Z Compact Quantum Groups Homological algebra methods in the theory of Operator Deﬁnition Algebras Let I be a homological ideal in a triangulated category T with Ryszard Nest enough projective objects. Let F : T → C be an additive Homological algebra functor with values in an Abelian category C. Applying F Ideals and projectives pointwise to chain complexes, we get an induced functor I-exact complexes Projective objects F¯ : Ho(T) → Ho(C). Let P : T → Ho(T) be the functor that The phantom tower Assembly map maps an object of T to a projective resolution of T. Let Derived functors The ABC spectral Hn : Ho(C) → C be the nth homology functor. The composite sequence

Example: Γ = Z functor ¯ Compact P F Hn Quantum LnF : T −→ Ho(T) −→ Ho(C) −→ C Groups for n ∈ N is called the nth left derived functor of F . If F : Top → C is a contravariant additive functor, then the n op corresponding composite functor R F : T → C is called the nth right derived functor of F . Homological algebra methods in the theory of Operator Algebras More concretely, nF (A) for a covariant functor F and A ∈∈ T Ryszard Nest L is the homology of the chain complex Homological algebra F (δn+1) F (δn) Ideals and projectives · · · → F (Pn+1) −−−−→ F (Pn) −−−→ F (Pn−1) → · · · → F (P0) I-exact complexes Projective objects The phantom tower Assembly map at F (Pn) in degree n, where (P•, δ•) is an Iprojective Derived functors resolution of A. Similarly, nF (A) for a contravariant The ABC spectral R sequence functor F is the cohomology of the chain complex Example: Γ = Z Compact Quantum F (δn+1) F (δn) Groups · · · ← F (Pn+1) ←−−−− F (Pn) ←−−− F (Pn−1) ← · · · ← F (P0)

at F (Pn) in degree −n. Homological algebra methods in the theory of Operator Algebras

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Homological algebra Ideals and projectives The phantom tower of A generates spectral sequences. For I-exact complexes Projective objects simplicity, we consider a homological functor F : T → C Since The phantom tower Assembly map we are not going to use it later, we’ll just describe the exact Derived functors The ABC spectral couple which generates it. sequence

Example: Γ = Z Compact Quantum Groups Homological algebra methods in the theory of Let Nn := A for n ∈ −N and deﬁne bigraded Abelian groups Operator Algebras X D := Dpq, Dpq := Fp+q+1(Np+1), Ryszard Nest p,q∈Z

Homological X E := Epq, Epq := Fp+q(Pp), algebra Ideals and projectives p,q∈Z I-exact complexes Projective objects and homogeneous group homomorphisms The phantom tower Assembly map i Derived functors p+2 D / D ipq := (ι )∗ : Dp,q → Dp+1,q−1, deg i = (1, −1), The ABC spectral Y p+1 sequence Example: Γ = jpq := (εp)∗ : Dp,q → Ep,q, deg j = (0, 0), Z k j Compact Õ kpq := (πp)∗ : Ep,q → Dp−1,q, deg k = (−1, 0). Quantum E Groups Since F is homological, the chain complexes

n+1 εn∗ πn∗ ιn∗ · · · → Fm+1(Nn+1) −−→ Fm(Pn) −−→ Fm(Nn) −−→ Fm(Nn+1) → · · ·

are exact for all m ∈ Z. This means that (D, E, i, j, k) is an exact couple. Homological algebra methods in the theory of Operator Algebras

Ryszard Nest Theorem Homological algebra Ideals and projectives The ABC spectral sequence for homological functor is I-exact complexes independent of auxiliary choices, functorial in Aand abuts at Projective objects The phantom tower F (A). The second tableaux involves only the derived functors: Assembly map Derived functors The ABC spectral 2 ∼ sequence Epq = LpFq(A), Example: Γ = Z Compact Quantum Groups There is a similar result for cohomological functors. Homological algebra • Γ = Z methods in Z the theory of • I = Ker : KK → KK Operator Algebras The I-projective resolution of C has the form Ryszard Nest

Homological 2 algebra K(l (Z)) / C ' Σc0(Z) / 0 Ideals and projectives e f I-exact complexes ◦ Projective objects π Σ The phantom tower x ~ Assembly map c0( ) o c0( ) Derived functors Z 1−σ Z The ABC spectral sequence Example: Γ = Z The projective cover of C is just the mapping cone Compact Quantum Groups c0(Z) → c0(Z) → ΣC1−σ. But this is just the rotated exact triangle associated to the extension

0 → Σc0(Z) → C0(R) → c0(Z) → 0,

the ∗-homomorphism C0(R) → c0(Z) given by the evaluation f → f |Z. Homological algebra methods in the theory of Operator Conclusion Algebras 2 Ryszard Nest PC = C0(R ), D = ∂, the usual Dirac operator (or rather its phase), Homological algebra ∗ 2 Ideals and projectives K (A) = K∗((A ⊗ C0( )) ) → K∗(A ), I-exact complexes Z R o Z o Z Projective objects The phantom tower Assembly map where the assembly map is given by the product with Dirac Derived functors operator. The ABC spectral sequence The spectral sequence computing K ∗(A) becomes the six term Example: Γ = Z Z Compact exact sequence in K-theory associated to the extension Quantum Groups 2 Σ(A ⊗ c0(Z)) o Z (A ⊗ C0(R )) o Z (A ⊗ c0(Z)) o Z

Since A ⊗ c0(Z)) o Z ' A ⊗ K, this is just the usual Pimsner-Voiculescu exact sequence. Homological algebra methods in the theory of Operator A discrete quantum group G is a unital C*-algebra C(G) with a Algebras compatible coalgebra structure Ryszard Nest ∆ : C(G) → C(G) ⊗min C(G). Homological algebra satsfying the non-degeneracy conditions Ideals and projectives I-exact complexes Projective objects ∆(C(G)(1 ⊗ C(G)) = C(G) ⊗min C(G) = (C(G) ⊗ 1) ∆(C(G), The phantom tower Assembly map The additional structures are antipode and Haar state µ. There is an Derived functors ˆ The ABC spectral associated dual compact quantum group (C0(Gb), ∆) which carries it’s own sequence Haar weight. C0( ) is nonunital (or ﬁnite dimensional) and the coproduct Example: Γ = Z Gb Compact has the form Quantum C0(Gb) → Mult(C0(Gb) ⊗min C0(Gb)) Groups An action of G on a C*-algebra A is the same as a coaction of Gb on A, i.e. a ∗-homomorphism ρ : A → A ⊗min C(Gb) satisfying the condition

ι ⊗ ∆ ◦ ρ = ρ ⊗ ι ◦ ρ : A → A ⊗min C(Gb) ⊗min C(Gb). and nondegeneracy ρ(A) 1 ⊗ C(Gb) = A ⊗min C(Gb). Homological algebra methods in the theory of Operator Algebras

Ryszard Nest There is a notion of the category of G-C*-algebras and Homological corresponding notion of the bivariant KK G functor satisfying all algebra Ideals and projectives the epected properties. In particular we get the corresponding I -exact complexes G Projective objects triangulated KK category. The phantom tower Assembly map Derived functors Examples The ABC spectral sequence Example: Γ = Z ∗ • A discrete group Γ. C(G) = Cred (Γ), the coproduct is given by Compact Quantum γ 7→ γ ⊗ γ and and C0(bΓ) = c0(Γ). Groups • A compact group G. C(G) = C(G), the coproduct is given by ∗ ∆(f )(g, h) = f (gh) and C0(bΓ) = C (G). Homological algebra methods in the theory of Operator Algebras

Ryszard Nest We assume that G is a compact, connected group. Gˆ Homological • T = KK , the KK-category of G-coalgebras, algebra Gˆ Ideals and projectives • I = Ker : KK → KK I-exact complexes ∗ Projective objects • F (A) = K (A Gˆ) The phantom tower o Assembly map Derived functors The ABC spectral Theorem sequence Example: Γ = Z Baum-Connes for coactions Compact Quantum Gˆ Groups < PI >= KK Homological algebra methods in The spectral sequence computing K ∗ (A) = F (A) becomes a the theory of Gˆ L Operator ˆ Algebras spectral sequence for the K-theory of B = A o G: Ryszard Nest ∗ ˆ Ep,q =⇒ Kp+q(A o G) Homological algebra 2 Ideals and projectives and the E -term has form I-exact complexes Projective objects 2 p The phantom tower E−p,q = H (RG , Kq(A)) Assembly map Derived functors The ABC spectral ˆ sequence In terms of a G-algebra B = A o G we get Example: Γ = Z Compact Corollary Quantum Groups Let G be a connected compact group and B a separable G- C* algebra. Then

G K∗ (B) = K∗(B o G) = 0 =⇒ K∗(B) = 0. Homological algebra methods in the theory of Operator Algebras

Ryszard Nest Let Gb = SUq(2). In this paticular case G is "torsion free" and the BC says that the trivial G-module C is contained in the Homological algebra localising category generated by C(G). More generally the Ideals and projectives following stronger statement holds. I-exact complexes Projective objects The phantom tower Theorem Assembly map Derived functors \ G The ABC spectral Let G = SUq(2). The KK -category coincides with the sequence subcategory generated by C*-algebras with trivial actions Example: Γ = Z G Compact Quantum Note that, in distinction to the group case, the KK G-category Groups has no tensor product, so the Dirac - dual Dirac method does not work without major modiﬁcation.