Groups meet C

Groups meet C

-algebras, an appetizer

Wolfgang Lück Bonn Germany

email wolfgang.lueck@him.uni-bonn.de http://131.220.77.52/lueck/

June 18th, 2020

∗

Introduction

Conference Groups meet C*-algebras

7th Florianopolis - Münster - Ottawa Conference

in honour of Siegfried Echterhoff’s 60th birthday

Münster, June 15-19, 2020

∗

∗

Happy Birthday Siegfried

I will not present all the slides during the talk.

I will post all of them on my homepage, but without the additional drawings or comments given during the talk.

∗

Baby Example C

Letf:V →W be aC-linear map between finite-dimensional complex vector spaces.

Interesting numbers associated tof aredim_C(ker(f))and dim_C(coker(f)).

But they are nothomotopy invariantnotions.

However theindex, defined to be the integer

index(f):= dim_C(coker(f))−dim_C(ker(f)), is a homotopy invariant.

Namely, the additivity of the dimension under exact sequences implies the formula

index(f) = dim_C(W)−dim_C(V).

∗

More generally, letC_∗ be a finiteC-chain-complex.

· · · →0−^c−−^d+1→C_d −→^cd C_d−1−−−→ · · ·^cd−1 C₁−→^c1 C₀→0→ · · · .

Define itsEuler characteristic(orindex) χ(C∗):=X

i

(−1)ⁱ·dim_C(H_i(C∗)).

SinceH_n(C∗) = ker(c_n)/im(c_n+1)is aC-chain homotopy invariant, χ(C_∗)is a chain homotopy invariant.

We have

χ(C∗) =X

i

(−1)ⁱ·dim_C(C_i).

LetX be a finiteCW-complex.

It comes with a finiteC-chain complexC_∗^c(X).

Define itsEuler characteristicχ(X)to beχ(C_∗^c(X))which is a homotopy invariant ofX.

The formula above says

χ(X) =X

k

(−1)^k · |{cells of dimensionk}|.

∗

There is the well-known formula from combinatorics 1=

n

X

i=0

(−1)ⁱ·

n+1 i+1

.

We want to give a topological proof.

Let∆nbe then-dimensional simplexwhich is the convex hull of the points(1,0, . . . ,0),(0,1,0, . . . ,0),. . .(0,0, . . . ,1)inRⁿ⁺¹.

∆₀is a point

•

∆1is the interval[0,1].

• •

∆₂is a (solid) triangle

•

• •

∗

∆₃is the (solid) tetrahedron.

•

• •

•

∆nis a finiteCW-complex whose numbers ofi-cells is ⁿ⁺¹_i+1 . It is homotopy equivalent to∆₀.

Hence we get

1=χ(∆₀) =χ(∆n) =

n

X

i=0

(−1)ⁱ· |{i−cells}|=

n

X

i=0

(−1)ⁱ·

n+1 i+1

.

∗

Passing to infinite dimensions

For applications in analysis it is unrealistic to consider finite-dimensional vector spaces.

AnoperatorT:H₀→H₁between Hilbert spaces (or pre-Hilbert spaces) is a continuous linearC-map.

It is calledFredholm operatorif the dimension of its kernel is finite, its image is closed, and the dimension of its cokernel is finite.

Itsindexis defined as above to be the integer

index(f)= dim_C(coker(f))−dim_C(ker(f)).

This turns out to be a homotopy invariant and there is also a chain complex version.

A basic example is thedeRahm cochain complexΩ^∗(M) associated to a smooth closedn-dimensional manifoldM

· · · →0→Ω⁰(M)−→^d0 Ω¹(M)−→ · · ·^d1 −−−→ · · ·^dn−1 Ωⁿ(M)→0→ · · · . It is a Fredholm cochain complex in the sense thatHⁱ(Ω^∗(M))is finite dimensional for everyi.

We cannot define its index on the cochain complex level, since eachΩⁱ(M)is infinite-dimensional, but can define itsindexto be the integer

index(Ω^∗(M))=X

i

(−1)ⁱ·dim_C Hⁱ(Ω^∗(M)) .

The famousdeRham Theoremsays thatHⁱ(Ω^∗(M))agrees with thei-th cellular cohomology ofM withC-coefficients and hence in particular we get a kind of Index Theorem

index(Ω^∗(M)) =χ(M).

∗

Another important example is theDirac operatorDon a closed Spin manifoldM.

One can assign to it itsanalytic indexindex(D)∈Z.

One can assign to a closed Spin manifoldM by topological methods an (a priori rational) number, itsA-genusb A(M).b The famousAtiyah-Singer Index Theoremsays in particular

index(D) =A(M).b

The Atiyah-Singer Index Theorem was motivated and reproves the Hirzebruch Signature Theorem

sign(M) =bL(M).

Taking the action of a finite group into account

LetGbe a finite group.

LetV andW be finite-dimensional complex vector spaces with linearG-actions.

Letf:V →W be a linearG-map.

We want to improve our results from the beginning by a more sophisticated “counting”.

∗

Definition (Projective class groupK0(R))

Theprojective class groupK₀(R)of a ringR is defined to be the abelian group whose generators are isomorphism classes[P]of finitely generated projectiveR-modulesP and whose relations are [P₀] + [P₂] = [P₁]for every exact sequence 0→P₀→P₁→P₂→0 of finitely generated projectiveR-modules.

This is the same as theGrothendieck constructionapplied to the abelian monoid of isomorphism classes of finitely generated projectiveR-modules under direct sum.

IfF is a field, e.g.,F =C, the dimension induces an isomorphism K₀(F)−→^∼= Z.

IfRis any ring andGis any (discrete) group, thegroup ringRGis theR-algebra, whose underlyingR-module is the freeR-module generated byGand whose multiplication comes from the group structure.

An elementx ∈RGis a formal sumP

g∈Gr_g·g such that only finitely many of the coefficientsrg∈Rare different from zero.

The multiplication comes from the tautological formula g·h=g·h, more precisely



 X

g∈G

r_g·g



·



 X

g∈G

s_g·g



:=X

g∈G



 X

h,k∈G,hk=g

r_hs_k



·g.

ARG-moduleP is the same asG-representation with coefficients inR, i.e., aR-modulPtogether with aG-action byR-linear maps.

∗

IfGis finite, a finitely generated projectiveCG-module is the same as a finite-dimensional complexG-representation, and we get an identification ofK₀(CG)and thecomplex representation ring R_C(G).

We get fromWedderburn’s Theorem CG∼=Y

V

M_n(V)(C)

whereV runs through the isomorphism classes of irreducible G-representations.

This implies

R_C(G)∼=K₀(CG)∼=Y

V

Z∼=Z^c

wherec is the number of isomorphism classes of irreducible G-representations which agrees with the number of conjugacy classes of elements inG.

Now we will count inR_C(G)instead ofZ.

We go back to theG-equivariant linear mapf:V →W of finite-dimensionalG-representations.

We defineindex^G(f)= [ker(f)]−[cok(f)]∈R_C(G).

We get inR_C(G)

index^G(f) = [W]−[V] and henceindex^G(f)is aG-homotopy invariant.

∗

Definition (G-CW-complex)

LetGbe a topological group. AG-CW-complexX is aG-space together with aG-invariant filtration

∅=X−1⊆X₀⊆. . .⊆Xn⊆. . .⊆ [

n≥0

Xn=X

such thatX carries thecolimit topologywith respect to this filtration, andXnis obtained fromXn−1for eachn≥0 byattaching equivariant n-dimensional cells, i.e., there exists aG-pushout

`

i∈InG/H_i×Sⁿ⁻¹

`

i∈Inq_iⁿ

//

X_n−1

`

i∈InG/H_i×Dⁿ

`

i∈InQⁿ_i

//Xn

LetGbe a finite group and letX be aG-CW-complex which is finite, e.g., build by finitely manyG-cells, or, equivalently,X is compact.

Define theG-Euler characteristicofX χ^G(X)=X

i

(−1)ⁱ·[H_i(X)] ∈R_C(G) =K₀(CG).

We get by the same proof as before the equality in R_C(G) =K₀(CG)

χ^G(X) =X

c

(−1)^n(c)·[C[G/Hc]]

wherec runs through the equivariant cellsc =G/H_c×D^n(c).

∗

Also the index theorems mentioned above carry over in this fashion.

The passage to infinite groups and the role of the group C

-algebra

There are many reasons why one would like to consider also infinite groups.

One reason is that the fundamental groupπ of a closed smooth manifoldMis infinite and that one would like to carry out the analogues of some of the previous constructions for its universal coveringMe taking theπ-action into account.

This causes formidable problems concerning the analysis and one is forced to replace the complex group ring by certain larger completions.

∗

LetGbe (countable discrete) group andL²(G)be the associated Hilbert space. One obtains an embedding

CG⊆ B(L²(G),L²(G))

into the algebra of boundedG-operatorsL²(G)→L²(G)by the regular representation sendingg∈Gto the operator

R_g:L²(G)→L²(G), x 7→xg⁻¹.

We can define a string of subalgebras ofB(L²(G),L²(G))by completingCGwith respect to specific topologies or norms.

CG⊆F(G)⊆L¹(G)⊆C_r^∗(G)⊆

N(G)=B(L²(G),L²(G))^G ⊆ B(L²(G),L²(G))

L¹(G),C_r^∗(G), andN(G)areBanach algebras with involutions, i.e., normed complete complex vector spaces with the structure of aC-algebra satisfying||A·B|| ≤ ||A|| · ||B||and an isometric involution∗.

C_r^∗(G)andN(G)areC^∗-algebras, i.e., Banach algebras with involution satisfying the so calledC^∗-identity||xx^∗||=||x||². IfG=Z, one can find nice models, which is in general not possible

F(Z) = O(C\ {0});

C_r^∗(Z) = C(S¹);

N(Z) = L^∞(S¹).

One can think of the notion of aC^∗-algebra as anon-commutative spacesince any commutativeC^∗-algebra is of the formC(X) equipped with the supremums norm for a compact Hausdorff spaceX.

∗

A very important invariant of aC^∗-algebraAis itstopological K-theorywhich assigns toAaZ-graded abelian groupK∗(A).

K₀(A)is the projective class group and is independent of the topological structure onA;

Kn(A)isπn(GL(A))forn≥1 and does depend of the topological structure onA;

TopologicalK-theory satisfiesBott periodicity:Kn(A)∼=K_n+2(A);

IfX is a compact Hausdorff space, thenK_n(C(X))agrees with the classical topologicalK-theoryKⁿ(X)defined in terms of complex vector bundles overX.

There has been tremendous progress in the classification of C^∗-algebras in the sense that certain classes ofC^∗-algebras can be classified by their topologicalK-theory.

A lot of major contributions to the construction and classification of C^∗-algebras are due to mathematicians from Münster, notably Joachim Cuntz,Wilhelm Winter, andXin Li.

However, members of these classes are simple and nuclear, and groupC^∗-algebras do not have this property in general.

∗

The topologicalK-groupsK∗(C_r^∗(G))are the natural recipients for indices ofG-equivariant operators acting on proper cocompact smooth RiemannianG-manifolds by isometries.

An example is theDirac operatorDe acting on the universal coveringMe of ann-dimensional closed Spin manifoldM with fundamental groupπ.

We can assign to it itsC^∗-index

index_Cr^∗_(π)(M) = index_C^∗_r(π)(D)e ∈Kn(C_r^∗(π)).

and also its variant overRinstead ofC index_Cr^∗_(π;R)(M) = index_C^∗

r(π;R)(De_R)∈KO_n(C_r^∗(π;R)).

Next we illustrate its significance.

ABott manifoldis any simply connected closed Spin-manifoldBof dimension 8 whoseA-genusb A(B)b is 8;

We fix such a choice. (The particular choice does not matter.) We have

ind_Cr^∗_(π;R)(M) = ind_C^∗

r(π;R)(M×B).

∗

IfMcarries a Riemannian metric with positive scalar curvature, then the indexindex_Cr^∗_(π;R)(M)∈KO_n(C_r^∗(π;R))must vanish by theBochner-Lichnerowicz formula.

Conjecture ((Stable) Gromov-Lawson-Rosenberg Conjecture) Let M be a closed connected Spin-manifold of dimension n≥5.

Then M ×B^k carries for some integer k ≥0a Riemannian metric with positive scalar curvature if and only if

index_Cr^∗_(π;R)(M) =0 ∈KOn(C_r^∗(π;R)).

The requirementdim(M)≥5 is essential in the Stable

Gromov-Lawson-Rosenberg Conjecture, since in dimension four new obstructions, theSeiberg-Witten invariants, occur.

Theunstable versionof the Gromov-Lawson-Rosenberg

Conjecture says thatM carries a Riemannian metric with positive scalar curvature if and only ifind_Cr^∗_(π1(M);R)(M) =0.

Schick(1998)has constructed counterexamples to the unstable version using minimal hypersurface methods due toSchoen and YauforG=Z⁴×Z/3.

It is not known whether the unstable version is true or false for finite fundamental groups.

∗

Classifying space for proper G-actions

Definition (Classifying space for properG-actions) A model for theclassifying space for properG-actionsis a G-CW-complexEGwhich has the following properties:

All isotropy groups ofE Gare finite;

For every finite subgroupH⊆GtheH-fixed point setE G^H is weakly contractible.

There always exists a model forE G

For every properG-CW-complexX there is up toG-homotopy precisely oneG-mapX →E G.

Two models forE GareG-homotopy equivalent.

These spaces play a central role in equivariant homotopy theory over infinite groups.

The spaceE Ghave often very nice geometric models and

capture much more information about a groupGand its geometry thanEGifGis not torsionfree.

Here is a list of examples

group space

hyperbolic group Rips complex

Mapping class group Teichmüller space

Out(F_n) Outer space

lattice L in a connected Lie group G G/K

∗

Other nice models come from appropriate actions of a groupGon trees or manifolds with non-negative sectional curvature.

The Baum-Connes Conjecture

There is aG-homology theoryK_∗^G which assigns to every G-CW-complexX aZ-graded abelian groupK_∗^G(X).

We have for finiteH ⊆G Kn(G/H) =

(R_C(H) neven;

{0} nodd.

There is for everyn∈Zanassembly mapgiven essentially by takingC^∗-indices of operators

K_n^G(E G)→Kn(C_r^∗(G)).

Conjecture (Baum-Connes Conjecture)

A group G satisfies theBaum-Connes Conjectureif the assembly map is bijective for every n∈Z.

∗

The Baum-Connes Conjecture is one of the most important conjectures about groupC^∗-algebras.

It has many consequences for the theory ofC^∗-algebras, also for ones which do not come from groups.

It also has a lot of consequences for questions about groups, geometry, and topology.

For instance it implies the famousNovikov Conjectureabout the homotopy invariance of higher signatures.

It implies the Stable Gromov-Lawson-Rosenberg Conjecture as proved byStolz.

The Baum-Connes Conjecture (and its version with coefficients) is known for a large class of groups including groups having the Haagerup property and hyperbolic groups. This is due to Higson-KasparovandLafforgue.

There is a long list of mathematicians who made substantial contributions to the Baum-Connes Conjecture.

Permanence properties for the Baum-Connes Conjecture (with coefficients) have been established byChabert-Echterhoff.

The Baum-Connes Conjecture has been proved by

Chabert-Echterhoff-Nestfor almost connected second countable Hausdorff groups.

The Baum-Connes Conjecture is open forSLn(Z)forn≥3.

∗

Computations based on the Baum-Connes Conjecture

Most computations ofK_n(C_r^∗(G))are based on the Baum-Connes Conjecture since the source of the assembly map is much more accessible than the target.

This comes from certain techniques from equivariant homotopy theory, e.g. equivariant Atiyah-Hirzebruch spectral sequence, p-chain spectral sequence, or good models forE Gcoming from geometry.

RationallyK_n^G(E G)is rather well understood due toequivariant Chern characters, seeBaum-Connes,Lück

Integrally calculations can only be done in special cases, no general pattern is in sight and actually not expected.

Such calculations are interesting in their own right. Often they have interesting consequences for questions and the classification of certainC^∗-algebras, which are not necessarily themselves groupC^∗-algebras but in some sense connected, thanks to the meanwhile well-established classification of certain classes of C^∗-algebras by their topologicalK-theory.

An example is the classification of certainC^∗-algebras, which Cuntzassigned to the ring of integers in number fields, byLi-Lück.

It turned out that theseC^∗-algebras do not capture much from the number theory. This ledCuntz, Deninger, Lito the insight that one has to take certain dynamical systems into account.

∗

Another application of such computations to questions about C^∗-algebras is the analysis of the structure of crossed products of irrational rotation algebras by finite subgroups ofSL₂(Z)and the tracial Rokhlin property byEchterhoff-Lück-Phillipps-Walter.

Recall thatSchickdisproved the unstable

Gromov-Lawson-Rosenberg Conjecture forG=Z⁴×Z/3.

On the other handDavis-Lückproved it for certain semi-direct productsG=Z⁴o Z/3 based on calculatingKO(C_r^∗(Z⁴o Z/3)).

This shows that the class of groups, for which the unstable version holds, is not closed under extensions.

The following computation is due toLanger-Lück.

Consider the extension of groups 1→Zⁿ→Γ→Z/m→1 such that the conjugation action ofZ/monZⁿis free outside the origin 0∈Zⁿ.

We obtain an isomorphism

ω₁:K₁(C_r^∗(Γ))−^∼=→K₁(Γ\EΓ).

Restriction with the inclusionk:Zⁿ→Γinduces an isomorphism k^∗:K₁(C_r^∗(Γ))−^∼=→K₁(C_r^∗(Zⁿ))^Z/m.

∗

LetMbe the set of conjugacy classes of maximal finite subgroups ofΓ.

There is an exact sequence

0→ M

(M)∈M

Re_C(M)

L

(M)∈Mi_M

−−−−−−−→K₀(C_r^∗(Γ))−→^ω0 K₀(Γ\EΓ)→0,

whereRe_C(M)is the kernel of the mapR_C(M)→Zsending the class[V]of a complexM-representationV todim_C(C⊗_CMV)and the mapi_M comes from the inclusionM →Γand the identification R_C(M) =K₀(C_r^∗(M)).

We have

K_i(C_r^∗(Γ))∼=Z^si where

s_i = ( P

(M)∈M(|M| −1) +P

l∈Zrk_Z (Λ^2lZⁿ)^Z/m

ifi even;

P

l∈Zrk_Z (Λ^2l+1Zⁿ)^Z/m

ifi odd.

Ifmis even, thens₁=0 and

K₁(C_r^∗(Γ))∼={0}.

Ifmis a primep, then

s_i =















p^k ·(p−1) +^2n+p−1_2p +^{pk−1·(p−1)}₂ p6=2 andi even;

2ⁿ+p−1

2p −^{pk−1·(p−1)}₂ p6=2 andi odd;

3·2^k−1 p=2 andi even;

0 p=2 andi odd.

The groupΓis a crystallographic group. The computation of the topologicalK-theory of the groupC^∗-algebra seems to be out of reach for crystallographic groups in general.

∗

The proof of the results above is surprisingly complicated.

It is based on computations of the group homology ofZⁿo Z/m byLanger-Lück.

They prove a conjecture ofAdem-Ge-Pan-Petrosyanwhich says that the associated Lyndon-Hochschild-Serre spectral sequence collapses in the strongest sense, in the special case that the conjugation action ofZ/mofZⁿ is free outside the origin 0∈Zⁿ; Moreover, they use generalizations of the Atiyah-Segal

Completion Theorem for finite groups to infinite groups due to Lück-Oliver.

Interestingly, the conjecture ofAdem-Ge-Pan-Petrosyanis disproved in general byLanger-Lück.

How much does K

(C

(G)) tells us about G?

The answer is, roughly speaking, not much.

One can compute the topologicalK-theory of the group

C^∗-algebra for certain classes of groups and it turns out in many cases that the result does only depend on a few invariants of the group. (This is of course good news from the computational point of view.)

IfGis a finite abelian group, thenK∗(C_r^∗(G))isZ^|G|in even dimensions and{0}in odd dimensions and hence depends only on the order|G|ofG.

This phenomenon can be confirmed for instance for one-relator groups, right-angled Artin groups and right-angled Coxeter groups, where complete calculations are possible. Also the computations above forZⁿo Z/msupport this.

∗

There is atracehomomorphism

tr:K₀(C_r^∗(G))→R.

If the groupGcontains an element of ordern, then 1/nis in the imageim(tr).

In particularGis torsionfree only ifim(tr) =Z.

Suppose thatGsatisfies the Baum-Connes Conjecture. Then:

Gis torsionfree if and only ifim(tr) =Z.

Gcontains non-trivialp-torsion, if and only if 1/p∈im(tr).

LetZ⊆Λ⊆Qbe the ring obtained fromZby inverting the orders of all finite subgroups ofG. Thenim(tr)⊆Λ. This follows from Λ-valued Chern character ofLück.

There is a groupGsuch that any non-trivial finite subgroup is isomorphic toZ/3, but 1/9 is contained inim(tr), seeRoy.

How much does C

(G) tells us about G?

Also here the answer seems to be not much but of course more thanK∗(C_r^∗(G)). Here are some positive or negative results.

Two finite abelian groups have isomorphic complex group rings.

However, two finite abelian groups are isomorphic if and only if their rational group rings are isomorphic.

The quaternion group and the dihedral group of order eight have isomorphic complex group rings.

Actually,Hertweckgave in 2001 a counterexample to the

conjecture that two finite groups are isomorphic if and only if their integral group rings are isomorphic.

∗

Two finitely generated free groups are isomorphic if and only if the groupC^∗-algebras are isomorphic. This is a famous unsolved problem for group von Neumann algebras. Actually, one can read off the rank of a free group from the topologicalK-theory of its groupC^∗-algebra.

A groupGis amenable if and only if its groupC^∗-algebraC_r^∗(G)is nuclear.