Download Dimension theory of arbitrary modules over finite von Neumann

Journal f}r die reine und
angewandte Mathematik
J. reine angew. Math. 495 (1998), 135_162
9 Walter de Gruyter
Berlin New York 1998
Dimension theory of arbitrary modules over nite
von Neumann algebras and L2-Betti numbers I:
Foundations
The paper is dedicated to Martin Kneser on the occasion of his seventieth birthday
By Wolfgang L}ck at M}nster
Abstract. We dene for arbitrary modules over a nite von Neumann algebra A a
dimension taking values in G0, 8H which extends the classical notion of von Neumann
dimension for nitely generated projective A-modules and inherits all its useful properties
such as Additivity, Conality and Continuity. This allows to dene L2-Betti numbers for
arbitrary topological spaces with an action of a discrete group ! extending the well-known
denition for regular coverings of compact manifolds. We show for an amenable group
! that the p-th L2-Betti number depends only on the C!-module given by the p-th singular
homology.
0. Introduction
Let us recall the original denition of L2-Betti numbers by Atiyah G2H. Let M
1 M
be a regular covering of a closed Riemannian manifold M with ! as group of deck
transformations. We lift the Riemannian metric to a !-invariant Riemannian metric on M
.
Let L2 :p (M
) be the Hilbert space completion of the space C' :p (M
) of smooth R-valued
0
p-forms on M
with compact support and the standard L2-pre-Hilbert structure. The Laplace
operator " is essentially selfadjoint in L2:p (M
). Let " ^ O jdE p be the spectral decomp
p
J
position with right-continuous spectral family JE p j ` RK. Let E p (x, y) be the Schwartz
J
J
kernel of E p. Since E p (x , x) is an endomorphism of a nite-dimensional real vector space,
J
J
its trace trR (E p (x , x )) ` R is dened. Let F be a fundamental domain for the !-action on M
.
J
Dene the analytic L2-Betti number by
(0.1)
b(2) (M
) : O trR (E p (x, x )) d vol 0
x
p
F
` G0, 8) .
By means of a Laplace transformation this can also be expressed in terms of the heat
kernel e+tBp (x , y) on M
by
136
L }ck, von Neumann algebras and L2-Betti numbers I
(0.2)
b(2) (M
) : lim O trR (e+tBp (x , x)) d vol p
x
tU' F
` G0, 8) .
The p-th L2-Betti number measures the size of the space of smooth harmonic L2-integrable
p-forms on M
and vanishes precisely if there is no such non-trivial form. For a survey on
L2-Betti numbers and related invariants like Novikov-Shubin invariants and L2-torsion
and their applications and relations to geometry, spectral theory, group theory and K-theory
we refer for instance to G14H, section 8, G18H, G22H and G27H. In this paper, however, we
will not deal with the analytic side, but take an algebraic point of view.
The L2-Betti numbers can also be dened in an algebraic manner. Farber G12H, G13H
has shown that the category of nitely generated Hilbert A-modules for a nite von
Neumann algebra A can be embedded in an appropriate abelian category and that one
can treat L2-homology from a homological algebraic point of view. Farber gives as an
application for instance an improvement of the Morse inequalites of Novikov and Shubin
G25H, G26H in terms of L2-Betti numbers by taking the minimal number of generators into
account. An equivalent more algebra oriented approach is developed in G21H where it is
shown that the category of nitely generated projective modules over A, viewed just as a
ring, is equivalent to the category of nitely generated Hilbert A-modules and that the
category of nitely presented A-modules is an abelian category. This allows to dene for
a nitely generated projective A-module P its von Neumann dimension
(0.3)
dim (P) ` G0, 8)
A
by using the classical denition for nitely generated Hilbert A-modules in terms of the
von Neumann trace of a projector. This will be reviewed in Section 1.
In Section 2 we will prove the main technical result of this paper that this dimension
can be extended to arbitrary A-modules if one allows that the value may be innite (what
fortunately does not happen in a lot of interesting situations). Moreover, this extension
inherits all good properties from the original denition for nitely generated Hilbert Amodules such as Additivity, Conality and Continuity and is uniquely determined by these
properties. More precisely, we will introduce
Denition 0.4. Dene for an A-module M
dimF (M ) : sup Jdim(P) P ! M nitely generated projective A-submoduleK ` G0, 8H . ;
Recall that the dual module M< of a left A-module is the left A-module hom (M, A)
A
where the A-multiplication is given by (a f ) (x) ^ f (x) a< for f ` M<, x ` M and a ` A.
Denition 0.5. Let K be an A-submodule of the A-module M. Dene the closure
of K in M to be the A-submodule of M
K
: Jx ` M f (x) ^ 0 for all f ` M< with K ! ker ( f )K .
For a nitely generated A-module M dene the A-submodule TM and the A-quotient
module PM by:
L }ck, von Neumann algebras and L2-Betti numbers I
137
TM : Jx ` M f (x) ^ 0 for all f ` M<K ;
PM : MTM . ;
The notion of TM and PM corresponds in G12H to the torsion part and the projective
part. Notice that TM is the closure of the trivial submodule in M. It is the same as the
kernel of the canonical map i (M ) : M 1 (M)< which sends x ` M to the map M< 1 A,
f ) f (x). We will prove for a nite von Neumann algebra A in Section 2
Theorem 0.6. (1) A is semi-hereditary, i.e. any nitely generated submodule of a
projective module is projective.
(2) If K ! M is a submodule of the nitely generated A-module M, then MK
is nitely
generated and projective and K
is a direct summand in M.
(3) If M is a nitely generated A-module, then PM is nitely generated projective and
M PM TM .
(4) The dimension dimF has the following properties:
(a) Continuity.
If K ! M is a submodule of the nitely generated A-module M, then:
dimF (K ) ^ dimF (K
) .
(b) Conality.
Let JM i ` I K be a conal system of submodules of M, i.e. M ^ T M and for two
i
i
i`I
indices i and j there is an index k in I satisfying M , M ! M . Then:
i
j
k
(c) Additivity.
dimF (M ) ^ sup JdimF (M ) i ` I K .
i
p
i
If 0 __B M __B M __B M __B 0 is an exact sequence of A-modules, then:
1
2
0
dimF (M ) ^ dimF (M ) ^ dimF (M ) ,
1
0
2
where r ^ s for r, s ` G0, 8H is the ordinary sum of two real numbers if both r and s are not
8 and is 8 otherwise.
(d) Extension Property.
If M is nitely generated projective, then:
dimF (M) ^ dim (M) .
(e) If M is a nitely generated A-module, then:
138
L }ck, von Neumann algebras and L2-Betti numbers I
dimF (M) ^ dim (PM) ;
dimF (TM) ^ 0 .
(f) The dimension dimF is uniquely determined by Continuity, Conality, Additivity and
the Extension Property. ;
In the sequel we write dim instead of dimF. In Section 3 we will show for an inclusion
i : " 1 ! that the dimension function is compatible with induction with the induced ring
homomorphism i : N(") 1 N(!) and that N(!) is faithfully at over N(") (Theorem
3.3). This is important if one wants to relate the L2-Betti numbers of a regular covering
to the ones of the universal covering. We will prove that ! is non-amenable if and only
if N(!) C! C is trivial (Lemma 3.4.2). This generalizes the result of Brooks G5H, Remark
4.11.
In Section 4 we use this generalized dimension function to dene for a (discrete)
group ! and a !-space X its p-th L2-Betti number by
(0.7)
(HA (X; N(!)))
b(2) (X; N(!)) : dim
N(A) p
p
` G0, 8H ,
where HA (X; N(!)) denotes the N(!)-module given by the singular homology of X with
p
coeicients in the N(!)-Z!-bimodule N(!) (Denition 4.1). This denition agrees with
Atiyah's denition 0.1 if X is the total space and ! the group of deck transformations of
a regular covering of a closed Riemannian manifold. We will compare our denition also
with the one of Cheeger and Gromov G7H, section 2, Remark 4.12. In particular we can
dene for an arbitrary (discrete) group ! its p-th L2-Betti number
(0.8)
b(2) (!) : b(2) (E!; N(!))
p
p
` G0, 8H ,
where E! 1 B! is the universal !-principle bundle. These generalizations inherit all the
useful properties from the original versions and it pays o to have them at hand in this
generality. For instance if one is only interested in the L2-Betti numbers of a group ! for
which B! is a CW-complex of nite type and hence the original (simplicial) denition does
apply, it is important to have the more general denition available because such a group
! may contain an interesting normal subgroup " which is not even nitely generated. A
typical situation is when ! contains a normal innite amenable subgroup ". Then all the
L2-Betti numbers of B! are trivial by a result of Cheeger and Gromov G7H, Theorem 0.2
on page 191. This result was the main motivation for our attempt to construct the extensions
of dimension and of L2-Betti numbers described above.
In Section 5 we will get the theorem of Cheeger and Gromov mentioned above as a
corollary of the following result. If ! is amenable and M is a C!-module, then
(0.9)
(TorCA(N(!), M )) ^ 0 for p ~ 1 ,
dim
p
N(A)
where we consider N(!) as an N(!)-C!-bimodule (Theorem 5.1). We get from (0.9) by
a spectral sequence argument that the L2-Betti numbers of a !-space X depend only on
its singular homology with complex coeicients viewed as C!-module, namely
L }ck, von Neumann algebras and L2-Betti numbers I
(0.10)
139
b(2) (X; N(!)) ^ dim
(N(!) C! H sing (X; C)) ,
p
N(A)
p
provided that ! is amenable (Theorem 5.11). The result of Cheeger and Gromov mentioned
above follows from (0.10) since the singular homology of E! is trivial in all dimensions
except for dimension 0 where it is C.
We will discuss applications of this generalized dimension function to the Grothendieck group G (C!) of nitely generated C!-modules and L2-Euler characteristics and
0
the Burnside group in part II of the paper G23H.
1. Review of von Neumann dimension
In this section we recall some basic facts about nitely generated Hilbert-modules
and nitely generated projective modules over a nite von Neumann algebra. We x for
the sequel
Notation 1.1. Let A be a nite von Neumann algebra and tr : A 1 C be a normal
nite faithful trace. Denote by ! an (arbitrary) discrete group. Let N(!) be the group
von Neumann algebra with the standard trace tr .
N(A)
Module means always left-module and group actions on spaces are from the left
unless explicitly stated dierently. We will always work in the category of compactly
generated spaces (see G29H and G31H, I. 4). ;
Next we recall our main example for A and tr, namely the group von Neumann
algebra N(!) with the standard trace. The reader who is not familiar with the general
concept of nite von Neumann algebras may always think of this example. Let l2 (!) be
the Hilbert space of formal sums \ c c with complex coeicients j which are squareJ
C
C`A
summable, i.e. \ j 2~8. Dene the group von Neumann algebra and the standard trace by
C
C`A
(1.2)
N(!) : B (l2 (!), l2 (!))A ;
(1.3)
tr (a) : Va (e), eW
;
N(A)
l2(A)
where B (l2 (!), l2 (!))A is the space of bounded !-equivariant operators from l2 (!) to
itself, a ` N(!) and e ` ! ! l2 (!) is the unit element. The given trace on A extends to a
trace on square-matrices over A in the usual way
(1.4)
tr : M (n, n, A) 1 C ,
n
A ) \ tr (A ) .
i,i
i:1
Taking adjoints induces the structure of a ring with involution on A, i.e. we obtain a
map : A 1 A, a ) a<, which satises (a ^ b)< ^ a< ^ b<, (ab)< ^ b<a< and (a<)< ^ a
and 1< ^ 1 for all a, b ` A. This involution induces an involution on matrices
(1.5)
: M (m, n, A) 1 M (n, m, A), A ^ (A ) ) A< ^ (A< ) .
i,j
j,i
140
L }ck, von Neumann algebras and L2-Betti numbers I
Denition 1.6. Let P be a nitely generated projective A-module. Let A ` M (n, n, A)
be a matrix such that A ^ A<, A2 ^ A and the image of the A-linear map A : An 1 An
induced by right multiplication with A is A-isomorphic to P. Dene the von Neumann
dimension of
dim (P) ^ dim (P) : tr (A)
A
A
` G0, 8) .
;
It is not hard to check that this denition is independent of the choice of A and
depends only on the isomorphism class of P. Moreover the dimension is faithful, i.e.
dim (P) ^ 0 implies P ^ 0, is additive under direct sums and satises dim (An) ^ n.
We recall that we have dened K
, TM and PM for K ! M in Denition 0.5. A sef
g
quence L __B M __B N of A-modules is weakly exact resp. exact at M if im ( f ) ^ ker(g)
resp. im ( f ) ^ ker (g) holds. A morphism f : M 1 N of A-modules is called a weak isomorphism if its kernel is trivial and the closure of its image is N.
Next we explain how these concepts above correspond to their analogues for nitely
generated Hilbert A-modules. Let l2 (A) be the Hilbert space completion of A which is
viewed as a pre-Hilbert space by the inner product Va, bW ^ tr (ab<). A nitely generated
Hilbert A-module V is a Hilbert space V together with a left operation of A by C-linear
n
maps such that there exists a unitary A-embedding of V in l2 (A) for some n. A
i:1
morphism of nitely generated Hilbert A-modules is a bounded A-equivariant operator.
Denote by Jn. gen. Hilb. A-mod.K the category of nitely generated Hilbert A-modules.
f
g
A sequence U __B V __B W of nitely generated Hilbert A-modules is exact resp.
weakly exact at V if im ( f ) ^ ker (g) resp. im ( f ) ^ ker (g) holds. A morphism f : V 1 W
is a weak isomorphism if its kernel is trivial and its image is dense. For a survey on nite
von Neumann algebras and Hilbert A-modules we refer for instance to G19H, section 1,
G24H, section 1.
The right regular representation A 1 B (l2 (A), l2 (A))A from A into the space of
bounded A-equivariant operators from l2 (A) to itself sends a ` A to the extension of the
map A 1 A, b ) ba<. It is known to be bijective G9H, Theorem 1 in I. 5.2 on page 80,
Theorem 2 in I. 6.2 on page 99. Hence we obtain a bijection
(1.7)
l : M (m, n, A) 1 B (l2 (A)m, l2 (A)n)A,
which is compatible with the C-vector space structures, the involutions and composition.
The details of the following theorem and its proof can be found in G21H, section 2.
It is essentially a consequence of (1.7) and the construction of the idempotent completion
of a category. It allows us to forget the Hilbert-module-structures and simply work with
the von Neumann algebra as a plain ring. An equivalent approach is given by Farber G12H,
G13H and is identied with the one here in G21H, Theorem 0.9. An inner product on a nitely
generated projective A-module P is a map k : P^P 1 A which is linear in the rst
variable, symmetric in the sense k (x, y) ^ k (y, x)< and positive in the sense k (x, x) ~0 *
x @ 0 such that the induced map P 1 P< sending y ` P to k (^, y) is bijective.
L }ck, von Neumann algebras and L2-Betti numbers I
141
Theorem 1.8. (1) There is a functor
l : J n. gen. proj. A-mod. with inner prod.K 1 J n. gen. Hilb. A-mod.K
which is an equivalence of C-categories with involutions.
(2) Any nitely generated projective A-module has an inner product. Two nitely
generated projective A-modules with inner product are unitarily A-isomorphic if and only
if the underlying A-modules are A-isomorphic.
(3) Let l+1 be an inverse of l which is well-dened up to unitary natural equivalence.
The composition of l+1 with the forgetful functor induces an equivalence of C-categories
J fin. gen. Hilb. A-mod.K 1 J n. gen. proj. A-mod.K .
(4) l and l+1 preserve weak exactness and exactness.
;
Of course Denition 1.6 of dim (P) for a nitely generated projective A-module
agrees with the usual von Neumann dimension of the associated Hilbert A-module l (P)
after any choice of inner product on P.
2. The generalized dimension function
In this section we give the proof of Theorem 0.6 and investigate the behaviour of
the dimension under colimits. We recall that we have introduced dimF (M) for an arbitrary
A-module M in Denition 0.4 and K
, TM and PM for K ! M in Denition 0.5. We begin
with the proof of Theorem 0.6.
Proof. (1) is proven in G21H, Corollary 2.4 for nite von Neumann algebras. However, Pardo pointed out to us that any von Neumann algebra is semi-hereditary. This
follows from the facts that any von Neumann algebra is a Baer -ring and hence in
particular a Rickart C<-algebra G4H, Denition 1, Denition 2 and Proposition 9 in Chapter
1.4, and that a C<-algebra is semi-hereditary if and only if it is Rickart G1H, Corollary 3.7
on page 270.
(2) and (4) (a) in the special case that M ^ P for a nitely generated projective
A-module P.
Let P ^ JP i ` IK be the directed system of nitely generated projective A-submoi
dules of K. Notice that P is indeed directed by inclusion since the submodule of P generated
by two nitely generated projective submodules is again nitely generated and hence by
(1) nitely generated projective. Let j : P 1 P be the inclusion. Equip P and each P with
i i
i
a xed inner product and let pr : l (P) 1 l (P) be the orthogonal projection satisfying
i
im (pr ^ im (l ( j )) and pr : l (P) 1 l (P) be the orthogonal projection satisfying
i
i
im (pr) ^ T im(pr ). Next we show
i
i`I
(2.1)
im (l+1(pr)) ^ K
.
142
L }ck, von Neumann algebras and L2-Betti numbers I
Let f : P 1 A be an A-map with K ! ker ( f ). Then f j ^ 0 and therefore l ( f ) l ( j ) ^ 0
i
i
for all i ` I. We get im (pr ) ! ker (l ( f )) for all i ` I. Because the kernel of l ( f ) is
i
closed we conclude im (pr) ! ker (l ( f )). This shows im (l+1(pr)) ! ker ( f ) and hence
im (l+1(pr)) ! K
. As K ! ker (id ^ l+1(pr)) ^ im (l+1(pr)), we conclude K
! im (l+1(pr)).
This nishes the proof of (2.1) and of (2) in the special case M ^ P.
Next we prove
(2.2)
dimF (K ) ^ dim (K
) .
The inclusion j induces a weak isomorphism l (P ) 1 im (pr ) of nitely generated Hilbert
i
i
i
A-modules. If we apply the Polar Decomposition Theorem to it we obtain a unitary
A-isomorphism from l (P ) to im (pr ). This implies dim (P ) ^ tr (pr ). Therefore it remains
i
i
i
i
to prove
(2.3)
tr (pr) : sup Jtr (pr ) i ` IK .
i
As tr is normal, it suices to show for x ` l (P) that the net Jpr (x) i ` I K converges
i
to pr (x). Let e ~0 be given. Choose i (e) ` I and x ` im (pr ) with pr (x) ^ x ~ e2.
i(E)
i(E)
i(E)
We conclude for all i ~ i (e)
pr (x) ^ pr (x) ~ pr (x) ^ pr (x) i
i(E)
~ pr (x) ^ pr (x ) ^ pr (x ) ^ pr (x) i(E) i(E)
i(E) i(E)
i(E)
~ pr (x) ^ x ^ pr (x ^ pr (x)) i(E)
i(E) i(E)
~ pr (x) ^ x ^ pr x ^ pr (x) i(E)
i(E)
i(E)
~ 2 pr (x) ^ x i(E)
~e.
Now (2.3) and hence (2.2) follow. In particular we get from (2.2) for any nitely generated
projective submodule Q of a nitely generated projective A-module Q
0
(2.4)
dim (Q ) ~ dim (Q) ,
0
since by denition dim (Q ) ~ dimF (Q ) and dim(Q
) ~ dim (Q) follows from additivity of
0
0
0
dim under direct sums and that we have already proven that Q
is a direct summand in Q.
0
This implies for a nitely generated projective A-module Q
(2.5)
dim (Q) ^ dimF (Q) .
Now (2.2) and (2.5) imply (4) (a) in the special case that M ^ P for a nitely generated
projective A-module P.
(4) (d) has been already proven in 2.5.
L }ck, von Neumann algebras and L2-Betti numbers I
143
(4) (b) If P ! M is a nitely generated projective submodule, then there is an index
i ` I with P ! M by conality.
i
(4) (c) Let P ! M be a nitely generated projective submodule. We obtain an exact
2
sequence 0 1 M 1 p+1(P) 1 P 1 0. Since p+1(P) M P, we conclude
0
0
dimF(M )^ dim(P) ~ dimF ( p+1(P)) ~ dimF(M ) .
1
0
Since this holds for all nitely generated projective sumodules P ! M , we get
2
(2.6)
dimF(M ) ^ dimF(M ) ~ dimF(M ) .
0
2
1
Let Q ! M be nitely generated projective. Let i (M ) 3 Q be the closure of i (M ) 3 Q
0
1
0
in Q. We obtain exact sequences
0 1 i (M ) 3 Q 1 Q 1
p (Q)
1 0;
0
0 1 i (M ) 3 Q 1 Q 1 Qi (M ) 3 Q 1 0 .
0
0
By the special case of (2) which we have already proven above i (M ) 3 Q is a direct
0
summand in Q. We conclude
dim (Q) ^ dim ( i (M ) 3 Q ) ^ dim (Qi (M ) 3 Q ) .
0
0
From the special case (4) (a) we have already proven above, (4) (d) and the fact that there
is an epimorphism from p (Q) onto the nitely generated projective A-module Qi (M ) 3 Q,
0
we conclude
dim ( i (M ) 3 Q ) ^ dimF (i (M ) 3 Q) ;
0
0
dim (Qi (M ) 3 Q ) ~ dimF ( p (Q)) .
0
Since obviously dimF (M ) ~ dimF (N ) holds for A-modules M and N with M ! N, we get
dim(Q) ^ dim ( i (M ) 3 Q ) ^ dim (Qi (M ) 3 Q )
0
0
~ dimF (i (M ) 3 Q) ^ dimF ( p (Q))
0
~ dimF (M ) ^ dimF (M ) .
0
2
Since this holds for all nitely generated projective submodules Q ! M , we get
1
(2.7)
dimF(M ) ~ dimF(M ) ^ dimF (M ) .
1
0
2
Now (4) (c) follows from (2.6) and (2.7).
(2) and (4) (a) Choose a nitely generated free A-module F together with an epimorphism q : F 1 M. One easily checks that q+1(K
) is q+1(K ) and that Fq+1(K
) and
MK
are isomorphic. From the special case of (2) and (4) (a) which we have already proven
above we conclude that Fq+1(K ) and hence MK
are nitely generated projective and
144
L }ck, von Neumann algebras and L2-Betti numbers I
dimF (q+1(K )) ^ dimF ( q+1(K ) ) ^ dimF (q+1(K
)) .
If L is the kernel of q, we conclude from Additivity
dimF (q+1 (K
)) ^ dimF (L) ^ dimF (K
) ;
dimF (q+1 (K )) ^ dimF (L) ^ dimF (K ) .
Now (2) and (4) (a) follow in general.
(3) follows from (2), as J0 K ^ TM and MTM ^ PM by denition.
(4) (e) From (2), (4) (c) and (4) (d) we get: dimF (M ) ^ dimF (TM ) ^ dim (PM ). If we
apply (4) (a) to J0K ! M we get dimF (TM ) ^ 0 because of J0 K ^ TM.
(4) (f) Let dimFF be another function satisfying Continuity, Conality, Additivity and
the Extension Property. We want to show for an A-module M
dimFF (M ) ^ dimF (M ) .
Since (4) (e) is a consequence of Continuity, Additivity and the Extension Property alone,
this is obvious provided M is nitely generated. Since the system of nitely generated
submodules of a module is conal, the claim follows from Conality. This nishes the
proof of Theorem 0.6. ;
Notation 2.8. In view of Theorem 0.6 we will not distinguish between dimF and dim
in the sequel. ;
Next we investigate the behaviour of dimension under colimits indexed by a directed
set. We mention that colimit is sometimes called in the literature also inductive limit or
direct limit. The harder case of inverse limits which is not needed in this paper will be
treated at a dierent place (see also G7H, Appendix).
Theorem 2.9. Let I be a category such that between two objects there is at most one
morphism and for two objects i and i there is an object i with i ~ i and i ~ i where
1
2
0
1
0
2
0
we write i ~ k for two objects i and k if and only if there is a morphism from i to k. Let M
i
be a covariant functor from I to the category of A-modules. For i ~ j let : M 1 M be
i,j
i
j
the associated morphism of A-modules. For i ` I let t : M 1 colim M be the canonical
i
i
I i
morphism of A-modules. Then:
(1) We get for the dimension of the A-module given by the colimit colim M
I i
dim (colim M ) ^ sup Jdim (im (t )) i ` I K .
i
I i
(2) Suppose for each i ` I that there is i ` I with i ~ i such that dim (im ( )) ~8
0
0
i,i0
holds. Then:
dim (colim M ) ^ sup Jinf Jdim (im( : M 1 M )) j ` I, i ~ j K i ` I K .
i,j
i
j
I i
L }ck, von Neumann algebras and L2-Betti numbers I
145
Proof.
(1) Recall that colim M can be constructed as ] M ~ for the equivalence
I i
i
i`I
relation for which x ` M ~ y ` M holds precisely if there is k ` I with i ~ k and j ~ k with
i
j
the property (x) ^ ( y). With this description one easily checks
i,k
j,k
colim M ^ T im (t : M 1 colim M ) .
I i
i
i
I i
i`I
Now apply Conality of dim (see Theorem 0.6 (4)).
(2) It remains to show for i ` I
(2.10)
dim (im (t )) ^ inf Jdim (im( : M 1 M )) j ` I, i ~ j K .
i
i,j
i
j
By assumption there is i ` I with i ~ i such that dim (im ( )) is nite. Let K be the
0
0
i,i0
i0,j
kernel of the map im ( ) 1 im ( ) induced by for i ~ j and K be the kernel of
i,i0
i,j
i0,j
0
i0
the map im ( ) 1 im (t ) induced by t . Then K ^ T K and hence by Conality
i0,j
i,i0
i
i0
i0
(see Theorem 0.6 (4))
j ` I,i0~ j
dim(K ) ^ sup Jdim (K ) j ` I, i ~ jK .
i0
i0,j
0
Since dim (im ( )) is nite, we get from Additivity (see Theorem 0.6 (4))
i,i0
(2.11)
dim (im (t )) ^ dim (im (t : im ( ) 1 colim M ))
i
i0 im( i,i0)
i,i0
I i
^ dim (im ( )) ^ dim (K )
i,i0)
i0
^ dim (im ( )) ^ sup Jdim (K ) j ` I, i ~ jK
i,i0)
i0,j
0
^ inf Jdim (im( )) ^ dim (K ) j ` I, i ~ jK
i,i0)
i0,j
0
^ inf Jdim (im ( : im ( ) 1 im ( ))) j ` I, i ~ jK
i0,j im( i,i0)
0
i,i0
i,j
^ inf Jdim (im( )) j ` I, i ~ jK .
i,j
0
Given j ` J with i ~ j , there is j ` I with i ~ j and j ~ j and hence with
0
0
0
0
dim (im ( )) ~ dim (im ( )) .
i,j0
i,j
This implies
(2.12)
inf Jdim (im ( )) j ` J, i ~ jK ^ inf Jdim (im ( )) j ` J, i ~ jK .
i,j
i,j
0
Now (2.10) follows from (2.11) and (2.12). This nishes the proof of Theorem 2.9. ;
Examples 2.13. The condition in Theorem 2.9 (2) that for each i ` I there is i ` I
0
with i ~ i with dim (im ( )) ~8 is necessary as the following example shows.
0
i,i0
'
'
'
Take I ^ N. Dene M ^ A and : A 1 A to be the projection. Then
j
j,k
n:j
m:j
m:k
dim (im ( )) ^8 for all j ~ k, but colim M is trivial and hence has dimension zero. ;
j,k
I i
146
L }ck, von Neumann algebras and L2-Betti numbers I
Remark 2.14. From an axiomatic point of view we have only needed the following
basic properties of A. Namely, let R be an associative ring with unit which has the following
properties:
1. There is a dimension function dim which assigns to any nitely generated projective
R-module P an element
dim (P) ` G0, 8)
such that dim (P Q) ^ dim (P) ^ dim(Q) holds and dim (P) depends only on the isomorphism class of P.
2. If K ! P is a submodule of the nitely generated projective A-module P, then K
is a direct summand in P. Moreover
dim (K
) ^ sup Jdim (P) P ! K nitely generated projective R-submoduleK .
Then with Denition 0.4, Theorem 0.6 carries over to R. One has essentially to copy
the part of the proof which begins with 2.4.
An easy example where these axioms are satised is the case where R is a principal
ideal domain and dim is the usual rank of a nitely generated free R-module. Then the
extended dimension for an R-module M is just the dimension of the F-vector space F M
R
for F the quotient eld of R. Notice that the case of a von Neumann algebra R ^ A is
harder since A is not noetherian in general.
In Denition 0.5 we have dened TM and PM only for nitely generated A-modules
M although the denition makes sense in general. The reason is that the following denition
for arbitrary A-modules seems to be more appropriate
(2.15)
TM : T JN ! M dim (N ) ^ 0K ;
(2.16)
PM : MTM .
One easily checks using Theorem 0.6 (4) that TM is the largest submodule of M with trivial
dimension and that these denitions (2.15) and (2.16) agree with Denition 0.5 if M is
nitely generated. One can show by example that they do not agree if one applies them
to arbitrary A-modules. In the case of a principal ideal domain R the torsion submodule
of an R-module M is just TM in the sense of denition (2.15). ;
3. Induction for group von Neumann algebras
Next we investigate how the dimension behaves under induction. Let i : " 1 ! be
an inclusion of groups. We claim that associated to i there is a ring homomorphism of the
group von Neumann algebras, also denoted by
(3.1)
i : N(") 1 N(!) .
L }ck, von Neumann algebras and L2-Betti numbers I
147
Recall that N(") is the same as the ring B (l2 ("), l2 ("))" of bounded "-equivariant
operators f : l2 (") 1 l2 ("). Notice that C! C" l2 (") can be viewed as a dense subspace
of l2 (!) and that f denes a C!-homomorphism id C" f : C! C" l2 (") 1 C! C" l2 (")
which is bounded with respect to the pre-Hilbert structure induced on C! C" l2 (") from
l2 (!). Hence id C" f extends to a !-equivariant bounded operator i ( f ) : l2(!) 1 l2(!).
Given an N(")-module M, dene the induction with i to be the N(!)-module
(3.2)
i (M) : N(!) M.
<
N(B)
Obviously i is a covariant functor from the category of N(")-modules to the category
<
of N(!)-modules, preserves direct sums and the properties nitely generated and projective
and sends N(") to N(!).
Theorem 3.3. Let i : " 1 ! be an injective group homomorphisms. Then:
(1) Induction with i is a faithfully at functor from the category of N(")-modules to
the category of N(!)-modules, i.e. a sequence of N(")-modules M 1 M 1 M is exact
0
1
2
at M if and only if the induced sequence of N(!)-modules i M 1 i M 1 i M is exact
1
< 0
< 1
< 2
at i M .
< 1
(2) For any N(")-module M we have:
dim
(M ) ^ dim
(i M ) .
N(B)
N(A) <
Proof.
The proof consists of the following steps.
Step 1. dim
(M ) ^ dim
(i (M )), provided M is a nitely generated projective
N(B)
N(A) <
N(")-module.
Let A ` M (n, n, N(")) be a matrix such that A ^ A<, A2 ^ A and the image of the
N(")-linear map A : N(")n 1 N(")n induced by right multiplication with A is N(")isomorphic to M. Let i (A) be the matrix in M (n, n, N(!)) obtained from A by applying
i to each entry. Then i (A) ^ i (A)< ^ i (A)2 ^ i (A) and the image of the N(!)-linear map
i (A) : N(!)n 1 N(!)n induced by right multiplication with i (A) is N(!)-isomorphic to
i M. Hence we get from Denition 1.6
<
dim
(M ) ^ tr (A) ;
N(B)
N(B)
(i (A)) .
dim
(i M ) ^ tr
N(A) <
N(A)
(i (a)) ^ tr (a) for a ` N("). This is an easy conseTherefore it suices to show tr
N(B)
N(A)
quence of the Denition 1.3 of the standard trace.
Step 2.
If M is nitely presented N(")-module, then
(i (M )) ;
dim
(M ) ^ dim
N(B)
N(A) <
TorN(B) (N(!), M ) ^ 0 .
1
148
L }ck, von Neumann algebras and L2-Betti numbers I
Since M is nitely presented, it splits as M ^ TM PM where PM is nitely generated
f
projective and there is an exact sequence 0 1 N(")n __B N(")n 1 TM 1 0 with f < ^ f
G21H, Theorem 1.2, Lemma 3.4. If we apply the right exact functor induction with i to it,
i f
<
N(!)n 1 i TM 1 0 with (i f )< ^ i f. Because
we get an exact sequence N(!)n __B
<
<
<
of Step 1, Additivity (see Theorem 0.6 (4)) and the denition of Tor it suices to show
that i f is injective. Let l be the functor introduced in Theorem 1.8 or G21H, section 2.
<
Then i (l ( f )) is l (i f ). Because l respects weak exactness (see Theorem 1.8 or G21H, Lemma
<
i f
<
N(")n 1 0 is weakly exact. Then one easily
2.3) l ( f ) has dense image since N(")n __B
checks that l (i f ) ^ i (l ( f )) has dense image since C! C" l2 (") is a dense subspace of
<
<
l2 (!). Since the kernel of a bounded operator of Hilbert spaces is the orthogonal complement of the image of its adjoint and l (i f ) is selfadjoint, l (i f ) is injective. Since l+1
<
<
respects exactness (see Theorem 1.8 or G21H, Lemma 2.3) i f is injective.
<
Step 3.
TorN(B) (N(!), M ) ^ 0 provided, M is a nitely generated N(")-module.
1
g
Choose an exact sequence 0 1 K __B P 1 M 1 0 such that P is a nitely generated
projective N(")-module. The associated long exact sequence of Tor-groups shows that
TorN(B) (N(!), M ) is trivial if and only if i g : i K 1 i P is injective. For each element
<
<
<
1
x in i K there is a nitely generated submodule K F! K such that x lies in the image of
<
the map i K F 1 i K induced by the inclusion. Hence it suices to show for any
<
<
nitely generated submodule K F! P that the inclusion induces an injection i K F 1 i P.
<
<
This follows since Step 2 applied to the nitely presented module PK shows
TorN(B) (N(!), PK ) ^ 0.
1
Step 4.
i is an exact functor.
<
By standard homological algebra we have to show that Tor1 (N(!), M ) ^ 0 is
N(B)
trivial for all N(")-modules M. Notice that M is the colimit of the directed system of its
nitely generated submodules (directed by inclusion) and that the functor Tor commutes
in both variables with colimits over directed systems G6H, Proposition VI. 1.3 on page 107.
Now the claim follows from Step 3.
Step 5. Let JM i ` IK be the directed system of nitely generated submodules of
i
the N(")-module M. Then
dim
(M ) ^ sup Jdim
(M ) i ` IK ;
N(B)
N(B) i
dim
(i M ) ^ sup Jdim
(i M ) i ` IK .
N(A) <
N(A) < i
Because of Step 4 we can view i M as a submodule of i M. Now apply Conality (see
< i
<
Theorem 0.6(4).
Step 6.
The second assertion of Theorem 3.3 is true.
Because of Step 5 it suices to prove the claim in the case that M is nitely generated
because any module is the colimit of the directed system of its nitely generated submodules.
g
Choose an exact sequence 0 1 K __B P 1 M 1 0 such that P is a nitely generated
projective N(")-module. Because of Step 4 and Additivity (see Theorem 0.6) we get
L }ck, von Neumann algebras and L2-Betti numbers I
149
dim
(M ) ^ dim
(P) ^ dim
(K ) ;
N(B)
N(B)
N(B)
dim
(i M ) ^ dim
(i P) ^ dim
(i K ) .
N(A) <
N(A) <
N(A) <
Because of Step 1 it remains to prove
dim
(K ) ^ dim
(i K ) .
N(B)
N(A) <
Because of Step 5 it suices to treat the case where K ! P is nitely generated. Since N(")
is semi-hereditary (see Theorem 0.6 (1)) K is nitely generated projective and the claim
follows from Step 1.
Step 7.
The rst assertion of Theorem 3.3 is true.
Because we know already from Step 4 that i is exact, it remains to prove for an
<
N(")-module M
i M^0 * M^0.
<
Suppose i M ^ 0. In order to show M ^ 0 we have to prove for any N(")-map
<
f : N(") 1 M that it is trivial. Let K be the kernel of f. Because i is exact by Step 4 and
<
i M ^ 0 by assumption, the inclusion induces an isomorphism i K 1 i N("). Since i K
<
<
<
<
is a nitely generated N(!)-module and i is exact by Step 4, there is a nitely generated
<
submodule K F! K such that the inclusion induces an isomorphism i K F 1 i N("). Let
<
<
N(")m 1 K F be an epimorphism. Let g : N(")m 1 N(") be the obvious composition.
Because i is exact by Step 4 the induced map i g : i N(")m 1 i N(") is surjective.
<
<
<
<
Hence it remains to prove that g itself is surjective because then K F^ N(") and the map
f : N(") 1 M is trivial. Since the functors l+1 and l of Theorem 1.8 are exact we have
to show for a "-equivariant bounded operator h : l2 (")m 1 l2 (") that h is surjective if
i (h) : l2 (!)m 1 l2 (!) is surjective. Let JE j ~ 0K be the spectral family of the positive
J
operator h h<. Then Ji (E ) j ~ 0K is the spectral family of the positive operator i (h) i (h)<.
J
Notice that h resp. i (h) is surjective if and only if E ^ 0 resp. i (E ) ^ 0 for some j ~0.
J
(imJ(l+1(E ))) ^ 0 resp.
Because E ^ 0 resp. i (E ) ^ 0 is equivalent to dim
J
J
J
N(B)
(im (l+1 (i (E )))) ^ 0 and im (l+1 (i (E ))) ^ i im (l+1(E )), the claim
follows from
dim
J
J
<
J
N(A)
Step 6. This nishes the proof of Theorem 3.3. ;
The proof of Theorem 3.3 would be obvious if we would know that N(!) viewed
as an N(")-module is projective. Notice that this is a stronger statement than proven in
Theorem 3.3. One would have to show that the higher Ext-groups instead of the Tor-groups
vanish to get this stronger statement. However, the proof for the Tor-groups does not go
through directly since the Ext-groups are not compatible with colimits.
Lemma 3.4. Let H ! ! be a subgroup. Then:
(1) dim (N(!) C! C G!HH) ^ H +1, where H +1 is dened to be zero if H is innite.
(2) N(!) C! C G!HH is trivial if and only if H is non-amenable.
(3) If ! is innite and V is a C!-module which is nite-dimensional over C, then
dim (N(!) C! V ) ^ 0 .
150
L }ck, von Neumann algebras and L2-Betti numbers I
Proof. (3) Since V is nitely generated as C!-module N(!) C! V is a nitely
generated N(!)-module. Because of Theorem 0.6 it suices to show that there is no
N(!)-homomorphism from N(!) C! V to N(!). This is equivalent to the claim that
there is no C!-homomorphism from V to N(!). Since the map N(!) 1 l2 (!) given by
evaluation at the unit element e ` ! ! l2 (!) is !-equivariant and injective it suices to
show that l2 (!) contains no !-invariant linear subspace W which is nite-dimensional as
complex vector space. Since any nite-dimensional topological vector space is complete,
W is a Hilbert N(!)-submodule. Let pr : l2 (!) 1 l2 (!) be an orthogonal !-equivariant
projection onto W. Then we get for any c ` !
(3.5)
dim(W ) ^ Vpr (c), cW .
Let Jv , v , . . . , v K be an orthonormal basis for the Hilbert subspace W ! l2(!). For c ` !
1 2
r
r
we write pr (c) ^ \ j (c) v . We get from pr (c) 2 ~ 1
i
i
i:1
(3.6)
j (c) ~ 1 .
i
Given e ~0, we can choose c (e) satisfying
(3.7)
Vv , c (e)W ~ r+1 e
i
for i ^ 1, 2, . . . , r .
Now (3.6) and (3.7) imply
(3.8)
Vpr (c (e)), c (e)W ~ e .
Since (3.8) holds for all e ~0, we conclude dim (W ) ^ 0 and hence W ^ 0 from equation
(3.5).
(1) and (2) If i : H 1 ! is the inclusion, then i (N(H ) C C) and
H
<
N(!) C! C G!HH are isomorphic as N(!)-modules. Because of Theorem 3.3 it remains
to treat the special case !^ H for the rst two assertions. The rst assertion follows from
the third for innite ! and is obvious for nite !. Next we prove the second assertion.
r
s+1
e
s`S
C! __B
C _B 0 is exact
Let S be a set of generators of !. Then C! ___B
s `S
where e ( \ j c) ^ \ j and r denotes right multiplication with u ` C!. We obtain an
C
C
u
C`A
C`A r
s+1
e
s`S
exact sequence N(!) ___B
N(!) __B N(!) C! C 1 0 since the tensor product
s`S
r
s+1
s`S
is right exact. Hence N(!) C! C is trivial if and only if N(!) ___B
N(!) is
s`S
surjective. This is equivalent to the existence of a nite subset T ! S such that
N(!)
s `T
r
s+1
s`T
___B
N(!)
is surjective. Let " ! ! be the subgroup generated by T. Then the map above
r
t+1
t`T
is induction with the inclusion of " ! ! applied to N(") ___B
N("). Hence we
t`T
151
L }ck, von Neumann algebras and L2-Betti numbers I
conclude from Theorem 3.3 (1) that N(!) C! C is trivial if N(") C" C is trivial for
some nitely generated subgroup " ! !. Since ! is amenable if and only if each of its
nitely generated subgroups is amenable G28H, Proposition 0.16 on page 14, we can assume
without loss of generality that ! is nitely generated, i.e. S is nite. We can also assume
that S is symmetric, i.e. s ` S implies s+1 ` S.
Because the functor l of Theorem 1.8 is exact, N(!) C! C is trivial if and only if
r
s+1
s`S
l2(!) is surjective. This is equivalent to the bijectivity of
the operator f : l2(!) ___B
s`S
the operator
1
id ^ \
r
1
S s
s`S
f f < : l2(!) _____B
l2(!) .
2 S 1
\
r
S s
s`S
___
_B
l2(!) is
It is bijective if and only if the spectral radius of the operator l2(!)
dierent from 1. Since this operator is convolution with a probability distribution whose
support contains S, namely
P : ! 1 G0, 1H, c )
S +1, c ` S ,
0,
c / S,
the spectral radius is 1 precisely if ! is amenable G16H. This nishes the proof of Lemma 3.4. ;
4. L 2-invariants for arbitrary A-spaces
In this section we extend the notion of L2-Betti numbers for regular coverings of
CW-complexes of nite type (i.e. with nite skeletons) with ! as group of deck transformations to (compactly generated) topological spaces with action of a (discrete) group !.
We will continue with using Notation 1.1.
Denition 4.1. Let X be a (left) !-space and V be a A-Z!-bimodule. Let HA (X; V )
p
be the singular homology of X with coeicients in V, i.e. the A-module given by the homology
of the A-chain complex V Z! C sing (X ), where C sing (X ) denotes the singular Z!-chain
<
<
complex of X. Dene the p-th L2-Betti number of X with coeicients in V by
b(2) (X; V ) : dim (HA (X; V )) ` G0, 8H ,
p
A p
and the p-th L2-Betti number of the group ! by
b(2) (!) : b(2) (E!; N(!)) .
p
p
;
Next we compare cellular and singular chain complexes and show that it does not
matter whether we use singular or cellular chain complexes in the case that X is a !-CWcomplex. For basic denitions and facts about !-CW-complexes we refer for instance to
G8H, sections II.1 and II.2, G20H, sections 1 and 2.
152
L }ck, von Neumann algebras and L2-Betti numbers I
Lemma 4.2. Let X be a !-CW-complex. Then there is a up to Z!-homotopy unique
and in X natural Z!-chain homotopy equivalence
f (X ) : C cell (X ) 1 C sing (X ) .
<
<
In particular we get for any A-Z!-bimodule V an in X and V natural isomorphism
H (V Z! C cell (X )) __B
H (V Z! C sing (X )) . ;
p
<
p
<
Proof. Obviously the second assertion follows from the rst assertion which is
proven as follows.
Let Y be a CW-complex with cellular Z-chain complex C cell and singular Z-chain
<
complex C sing. We dene a third (intermediate) Z-chain complex C inte (Y ) as the sub<
<
complex of C sing whose n-th chain module is the kernel of
<
cnsing
C sing (Y ) __B C sing (Y , Y ) .
C sing (Y ) __B
n+1 n
n+1 n n+1
n
n
There are an in Y natural inclusion and an in Y natural epimorphism of Z-chain complexes
(4.3)
i (Y ) : C inte (Y ) 1 C sing (Y ) ;
<
<
(4.4)
p (Y ) : C inte (Y ) 1 C cell (Y ) ;
<
<
which induce isomorphisms on homology G20H, page 263.
If ! acts freely on the !-CW-complex X, then C cell and C sing are free Z!-chain
<
<
complexes and we get a Z!-chain homotopy equivalence well-dened up to Z!-homotopy
from the fundamental theorem of homological algebra and the fact that the chain maps
(4.3) and (4.4) induce isomorphisms on homology. In the general case one has to go to
the orbit category Or (!) and apply module theory over this category instead of over Z!.
The orbit category Or (!) has as objects homogenous spaces and as morphisms
!-maps. The !-CW-complex X denes a contravariant functor
X : Or (!) 1 JCW^ COMPLEXESK,
!H ) map (!H, X )A^ X H.
Its composition with the functor C cell, C inte resp. C sing from the category of CW-complexes
<
<
<
to the category of chain complexes yields Z Or (!)-chain complexes, i.e. contravariant
functors
C cell (X ) : Or (!) 1 JZ ^ CHAIN-COMPLEXESK ;
<
C inte (X ) : Or (!) 1 JZ ^ CHAIN-COMPLEXESK ;
<
C sing (X ) : Or (!) 1 JZ ^ CHAIN-COMPLEXESK .
<
We obtain natural transformations from the natural chain maps (4.3) and (4.4)
(4.5)
i (X ) : C inte (X ) 1 C sing (X ) ;
<
<
L }ck, von Neumann algebras and L2-Betti numbers I
(4.6)
153
p (X ) : C inte (X ) 1 C cell (X ) ;
<
<
which induce isomorphisms on homology. Hence it suices to show that C cell (X ) and
<
then
C sing (X ) are free and hence projective in the sense of G20H, Denition 9.17 because
<
we obtain a homotopy equivalence of Z Or(!)-chain complexes from C cell (X ) to C sing (X )
<
<
G20H, Lemma 11.3 whose evaluation at !1 is the desired Z!-chain homotopy equivalence.
The proofs that these two chain complexes are free are simple versions of the arguments
in G20H, Lemma 13.2. Notice that in G20H the !-CW-complex is required to be proper, but
this condition is needed there only because there ! is assumed to be a Lie group and
universal coverings are built in, and can be dropped in the discrete case. ;
Remark 4.7. Originally the L2-Betti numbers of a regular covering M
1 M of a
closed Riemannian manifold with group of deck transformations ! were dened by Atiyah
G2H in terms of the heat kernel as explained in (0.2) in the introduction. It follows from
the L2-Hodge-deRham theorem G10H that this analytic denition agrees with the combinatorial denition of b(2) (X
) in terms of the associated cellular L2-chain complex and the
p
von Neumann dimension of nitely generated Hilbert N(!)-modules for a triangulation
X of M. Because of Lemma 4.2 this combinatorial denition agrees with the Denition 4.1.
Analogously to the case of L2-Betti numbers we will extend the notion of NovikovShubin invariants for regular coverings of compact Riemannian manifolds to arbitrary
!-spaces and prove that they are positive for the universal covering of an aspherical closed
manifold with elementary-amenable fundamental group in another paper. ;
The next results are well-known in the case where X is a regular covering of a
CW-complex of nite type. We call a map g : Y 1 Z homologically n-connected for
n ~1 if the map induced on singular homology with complex coeicients
g : H sing (Y; C) 1 H sing (Y; C) is bijective for k~ n and surjective for k ^ n. The map g
< k
k
is called a weak homology equivalence if it is n-connected for all n ~ 1.
Lemma 4.8. Let f : X 1 Y be a !-map and let V be a A-Z!-bimodule.
(1) Suppose for n ~ 1 that for each subgroup H ! ! the induced map f H : X H 1 Y H
is homologically n-connected. Then the map induced by f
f : HA (X; V ) 1 HA (Y; V )
< p
p
is bijective for p~ n and surjective for p ^ n and we get
b(2) (X; V ) ^ b(2) (Y; V )
p
p
b(2) (X; V ) ~ b(2) (Y; V )
p
p
for p ~ n ;
for p ^ n .
(2) Suppose such that for each subgroup H ! ! the induced map f H : X H 1 Y H is a
weak homology equivalence. Then for all p ~ 0 the map induced by f
is bijective and we get
f : H A (X; V ) 1 H A (Y; V )
< p
p
154
L }ck, von Neumann algebras and L2-Betti numbers I
b(2) (X; V ) ^ b(2) (Y; V ) .
p
p
;
Proof. We give only the proof of the second assertion, the one of the rst assertion
is an elementary modication. The map f induces a homotopy equivalence of Z Or(!)-chain
complexes C sing ( f ) : C sing (X ) 1 C sing (Y ) in the notation of the proof of Lemma 4.2
<
<
<
since the singular Z Or(!)-chain
complexes
of X and Y are free in the sense of G20H,
Denition 9.17 (see G20H, Lemma 11.3). Its evaluation
at !1 is a Z!-chain equivalence.
Hence f induces a chain equivalence
V Z! C sing ( f ) : V Z! C sing (X ) 1 V Z! C sing (Y )
<
<
<
and Lemma 4.8 follows. ;
We get as a direct consequence from Theorem 3.3
Theorem 4.9. Let i : " 1 ! be an inclusion of groups and let X be a "-space. Then
H A (!^ X; N(!)) ^ i HB (X; N(")) ;
p
B
< p
b(2) (!^ X; N(!)) ^ b(2) (X; N(")) . ;
p
B
p
Theorem 4.10. Let X be a path-connected !-space. Then:
(1) There is an isomorphism of N(!)-modules HA (X; N(!)) N(!) Z! C .
0
(2) b(2) (X; N(!)) ^ ! +1, where ! +1 is dened to be zero if the order ! of ! is
0
innite.
(3) HA (X; N(!)) is trivial if and only if ! is non-amenable.
0
Proof. The rst assertion follows from the fact that C sing (X ) 1 C sing (X ) 1 Z 1 0
1
0
is an exact sequence of Z!-modules and the tensor product is right exact. The other two
assertions follow from Lemma 3.4. ;
Remark 4.11. Let M 1 M be the universal covering of a closed Riemannian manifold
with fundamental group n. Brooks G5H has shown that the analytic Laplace operator "
0
on M in dimension zero has zero not in its spectrum if and only if n is non-amenable.
Now " has zero not in its spectrum if and only if H N (M, N(n)) is trivial because of G21H,
0
0
paragraph after Denition 3.11, Theorem 6.1, and the fact that the analytic and combinatorial spectral density function are dilatationally equivalent G11H. Hence Theorem 4.10
generalizes the result of Brooks. Notice that both Brook's and our proof use G16H. Compare
also with the result G15H, Corollary III.2.4 on page 188, that a group ! is non-amenable
if and only if H1 (!, l2(!)) is Hausdor. ;
Remark 4.12. Next we compare our approach with the one in G7H, section 2. We
begin with the case of a countable simplicial complex X with free simplicial !-action. Then
for any exhaustion X ! X ! X ! ! X by !-equivariant simplicial subcomplexes for
0
1
2
which X! is compact, the p-th L2-Betti number in the sense and notation of G7H, 2.8 on
page 198 is given by
L }ck, von Neumann algebras and L2-Betti numbers I
155
i<
j,k
b(2) (X : !) ^ lim lim dim
(im (H
p (X : !)) __B
H
p (X : !)) ,
p
N(A)
(2) k
(2) j
jU' kU'
where i
j,k
: X 1 X is the inclusion for j~ k. We get from G21H, Lemma 1.3, and Lemma 4.2
j
k
i<
j,k
(im (H
p (X : !)) __B
H
p (X : !))
dim
(2) k
(2) j
N(A)
(i )
j,k < H
^ dim
(im (H
A (X ; N(!))) __B
A (X ; N(!))) .
N(A)
p j
p k
Hence we conclude from Theorem 2.9 that the denitions in G7H, 2.8 on page 198, and in
4.1 agree:
(4.13)
b(2) (X : !) ^ b(2) (X; N(!)) .
p
p
If ! is countable and X is a countable simplicial complex with simplicial !-action, then
by G7H, Proposition 2.2 on page 198, and by (4.13)
(4.14)
b(2) (X : !) ^ b(2) (E!^X : !) ;
p
p
(4.15)
b(2) (X : !) ^ b(2) (E!^X; N(!)) .
p
p
Cheeger and Gromov G7H, Section 2, dene L2-cohomology and L2-Betti numbers
of a !-space X by considering the category whose objects are !-maps f : Y 1 X for a
simplicial complex Y with cocompact free simplicial !-action and then using inverse limits
to extend the classic notions for nite free !-CW-complexes such as Y to X. Our approach
avoids the technical diiculties concerning inverse limits and is closer to standard notions,
the only non-standard part is the verication of the properties of the extended dimension
function (Theorem 0.6 and Theorem 3.3).
5. Amenable groups
In this section we investigate amenable groups. For information about amenable
groups we refer for instance to G28H. The main technical result of this section is the next
lemma whose proof uses ideas of the proof of G7H, Lemma 3.1 on page 203.
Theorem 5.1. Let ! be amenable and M be a C!-module. Then
(TorCA(N(!), M )) ^ 0 for p ~ 1 ,
dim
p
N(A)
where we consider N(!) as an N(!)-C!-bimodule.
Proof.
Step 1. If M is a nitely presented C!-module, then
dim
(TorCA (N(!), M)) ^ 0.
N(A)
1
Choose a nite presentation
156
L }ck, von Neumann algebras and L2-Betti numbers I
n
m
f
p
C! __B C! __B M __B 0 .
i:1
i:1
For an element u ^ \ j c in l2 (!) dene its support
C
C
supp (u) : Jc ` ! j @ 0K ! !.
C
Let B ` M (m, n, C!) be the matrix describing f, i.e. the component f : C! 1 C! is given
i,j
by right multiplication with b . Dene the nite subset S by
i,j
S : Jc c or c+1 ` T supp (b )K .
i,j
i,j
m
n
Let f (2) : l2 (!) 1 l2 (!) be the bounded !-equivariant operator induced by f.
i:1
i:1
m
Denote by K the !-invariant linear subspace of l2 (!) which is the image of the kernel of
i:1
m
m
f under the canonical inclusion k : C! 1 l2 (!). Next we show for the closure K
of K
i:1
i:1
(5.2)
K
^ ker ( f (2)) .
m
m
Let pr : l2 (!) 1 l2 (!) be the orthogonal projection onto the closed !-invariant
i:1
i:1
subspace K
R 3 ker ( f (2)). The von Neumann dimension of im (pr) is zero if and only if pr
itself is zero. Hence (5.2) will follow if we can prove
(5.3)
tr
(pr) ^ 0 .
N(A)
Let e ~0 be given. Since ! is amenable, there is a nite non-empty subset A ! ! satisfying
G3H, Theorem F. 6.8 on page 308,
(5.4)
A
S
~e,
A
where A is dened by Ja ` A there is s ` S with as / AK. Dene
S
" : Jc ` ! c ` A or cs ` A for some s ` SK ^ A 4 ( T ( A) s) .
S
S
S
S
s`S
Let pr : l2 (!) 1 l2 (!) be the projection sending \ j c to \ j c. Dene
A
C
C
C`A
C`A
pr : l2 (!) 1 l2 (!) analogously. Next we show for s ` S and u ` l2 (!)
B
L }ck, von Neumann algebras and L2-Betti numbers I
(5.5)
157
pr r (u) ^ r pr (u) , if pr (u) ^ 0 ,
A s
s
A
B
where r : l2 (!) 1 l2 (!) is right multiplication with s. Since s ` S implies s+1 ` S, we get
s
the following equality of subsets of !:
Jc ` ! cs ` A, c / "K ^ Jc ` ! c ` A, c / "K .
Now (5.5) follows from the following calculation for u ^
\ j c ` l2 (!):
C
C ` A, C / B
\ j cs
C
Cs `A, C / B
^ \ j cs
C
C `A, C / B
^ ( \ j c) s
C
C `A, C / B
^ r pr (u) .
s
A
pr r (u) ^
A s
We have dened S such that each entry in the matrix B describing f is a linear combination
of elements in S. Hence (5.5) implies
n
m
( pr ) f (2) (u) ^ f (2) ( pr ) (u) , if pr (u ) ^ 0 for i ^ 1, 2, . . . , m .
A
A
A i
j:1
i:1
m
m
Notice that the image of pr lies in C!. We conclude
A
i:1
i:1
m
pr (u) ` K , if u ` ker ( f (2)), pr (u ) ^ 0 for i ^ 1, 2, . . . , m .
A
B i
i:1
This shows
m
m
(pr pr ) (ker ( f (2)) 3 ker (pr )) ^ 0 .
B
A
i:1
i:1
Since ker (pr ) has complex codimension " in l2 (!) and " ~ ( S ^1) A , we conB
S
clude for the complex dimension dimC of complex vector spaces
(5.6)
m
dimC ((pr pr ) (ker ( f (2)))) ~ m ( S ^1) A .
A
S
i:1
Since pr pr is an endomorphism of Hilbert spaces with nite-dimensional image, it is
A
trace-class and its trace trC (pr pr ) is dened. We get
A
(5.7)
m
trC (pr pr )
A
i:1
tr (pr) ~
N(A)
A
158
L }ck, von Neumann algebras and L2-Betti numbers I
from the following computation for e ` ! ! l2 (!) the unit element
m
tr (pr) ^ \ Vpr (e), eW
N(A)
i,i
i:1
1 m
^
\ A Vpr (e), eW
i,i
A
i:1
1 m
\ \ Vpr (c), cW
^
i,i
A
i:1 C ` A
1 m
\ \ Vpr pr (c), cW
^
i,i
A
A
i:1 C ` A
1 m
\ \ Vpr pr (c), cW
^
i,i
A
A
i:1 C ` A
1 m
\ trC (pr pr )
^
i,i
A
A
i:1
1
m
^
trC (pr ( \ pr )) .
A
A
i:1
If H is a Hilbert space and f : H 1 H is a bounded operator with nite-dimensional image,
then trC ( f ) ~ f dimC ( f (im ( f ))). Since the image of pr is contained in ker ( f (2)) and
pr and pr have operator norm 1, we conclude
A
(5.8)
m
m
trC (pr pr ) ~ dimC ((pr pr ) (ker ( f (2)))) .
A
A
i:1
i:1
Equations (5.4), (5.6), (5.7) and (5.8) imply
tr (pr) ~ m ( S ^1) e .
N(A)
Since this holds for all e ~0, we get (5.3) and hence (5.2) is true.
m
m
m
Let pr : l2 (!) 1 l2 (!) be the projection onto K
. Let i : ker ( f ) 1 C! be
K
i:1
i:1
i:1
the inclusion. It induces a map
m
m
id C! i
N(!) C! C! __B
N(!) .
j : N(!) C! ker ( f ) ___B
i:1
i:1
Next we want to show
(5.9)
im (l+1(pr )) ^ im ( j) .
K
Let x ` ker ( f ). Then
(5.10)
(id ^ l+1(pr )) j (1 x) ^ (id ^ prK
) k i (x) ,
K
L }ck, von Neumann algebras and L2-Betti numbers I
159
m
m
where k : C! 1 l2 (!) is the inclusion. Since (id ^ pr ) is trivial on K we get
K
i:1
i:1
(id ^ pr ) k i ^ 0. Now we conclude from (5.10) that im ( j) ! ker (id ^ l+1(pr )) and
K prove
K ( j) ! im (l+1(pr )) holds. This shows im ( j) ! im (l+1(pr )). It remains to
hence im
K
m K
m
for any N(!)-map g : N(!) 1 N(!) with im ( j) ! ker (g) that g l+1(pr ) is trivial.
K
i:1
i:1
Obviously K! ker (l (g)). Since ker (l (g)) is a closed subspace, we get K
! ker (l (g)). We
conclude l (g) pr ^ 0 and hence g l+1(pr ) ^ 0. This nishes the proof of (5.9).
K
K
Since l+1 preserves exactness by Theorem 1.8 and id C! f ^ l+1( f (2)), we conclude
from (5.2) and (5.9) that the sequence
m
m
id C! i
id C! f
N(!) C! C! ___B
N(!) C! C!
N(!) C! ker ( f ) ___B
i:1
i:1
is weakly exact. Continuity of the dimension function (see Theorem 0.6 (4)) implies
(ker (id C! f )im (id C! i)) ^ 0 .
dim
N(A)
Since TorC! (N(!), M) ^ ker (id C! f )im (id C! i) holds, Step 1 follows.
1
Step 2.
If M is a C!-module, then dim
(TorC! (N(!), M)) ^ 0.
N(A)
1
Obviously M is the union of its nitely generated submodules. Any nitely generated
module M is a colimit over a directed system of nitely presented modules, namely, choose
an epimorphism from a nitely generated free module F to M with kernel K. Since K is
the union of its nitely generated submodules, M is the colimit of the directed system FL
where L runs over the nitely generated submodules of K. The functor Tor commutes in
both variables with colimits over directed system G6H, Proposition VI.1.3 on page 107. Now
the claim follows from Step 1 and Theorem 2.9.
Step 3.
(TorC! (N(!), M)) ^ 0 for all p ~ 1.
If M is a C!-module, then dim
p
N(A)
We use induction over p ~ 1. The induction begin is already done in Step 2. Choose
an exact sequence 0 1 N 1 F 1 M 1 0 of N(!)-modules such that F is free. Then we
obtain an isomorphism TorC! (N(!), M) TorC! (N(!), N) and the induction step
p
p+1
follows. This nishes the proof of Theorem 5.1. ;
Theorem 5.11. Let ! be an amenable group and X a !-space. Then
b(2) (X; N(!)) ^ dim
(N(!) C! H sing (X; C))
p
N(A)
p
where H sing (X; C) is the C!-module given by the singular homology of X with complex
p
coeicients. In particular b(2) (X; N(!)) depends only on the C!-module H sing (X; C).
p
p
Proof.
We have to show for a C!-chain complex C with C ^ 0 for p~ 0
<
p
160
L }ck, von Neumann algebras and L2-Betti numbers I
(5.12)
dim (H (N(!) C! C )) ^ dim (N(!) C! H (C )) .
p
<
p <
We begin with the case where C is projective. Then there is a universal coeicient spectral
<
sequence converging to H (N(!) C! C ) G30H, Theorem 5.6.4 on page 143, whose
<
p;q
(see Theorem 0.6 (4))
E 2-term is E2 ^ TorC! (N(!), H (C )). Now Additivity of dim
q <
N(A)
p,q
p
together with Theorem 5.1 imply (5.12) if C is projective.
<
Next we prove (5.12) in the case where C is acyclic. One reduces the claim to
<
two-dimensional C and then checks this special case using long exact Tor-sequences,
<
Additivity (see Theorem 0.6 (4)) and Theorem 5.1.
In the general case one chooses a projective C!-chain complex P together with a
<
C!-chain map f : P 1 C which induces an isomorphism on homology. Since the mapping
< <
<
cylinder is C!-chain homotopy equivalent to C , the mapping cone of f is acyclic and
<
<
hence (5.12) is true for P and the mapping cone, we get (5.12) for C from Additivity
<
<
(see Theorem 0.6 (4)). This nishes the proof of Theorem 5.11. ;
We obtain as an immediate corollary from Theorem 4.10 and Theorem 5.11 (cf. G7H,
Theorem 0.2 on page 191)
Corollary 5.13.
If ! is innite amenable, then for all p ~ 0
b(2) (!) ^ 0 .
p
;
Remark 5.14. It is likely that Theorem 5.1 characterizes amenable groups. Namely,
if ! contains a free group F of rank two, then ! is non-amenable and Theorem 5.1 becomes
false because Theorem 3.3 implies
dim
(TorC!(N(!), C G!F H)) ^ dim (TorCF (N(!), C))
N(A)
N(A)
1
1
^ dim
(N(!) TorCF (N(F ), C))
N(A)
N(F)
1
^ dim
(TorCF (N(F ), C))
N(F)
1
^ b(2) (F )
1
^^ s (BF )
^ 1.
;
Remark 5.15. In view of Theorem 5.1 the question arises when N(!) is at over C!.
Except for virtually cyclic groups, i.e. groups which are nite or contain an innite cyclic
subgroup of nite index, we know no examples of nitely presented groups ! such that
N(!) is at over C!. If N(!) is at over C!, then N(") is at over C" for any subgroup
" ! ! by Theorem 3.3 (1). Moreover, N(!) is at over C! if and only if N(") is at
over C" for any nitely generated subgroup " ! !. This follows from Theorem 3.3 (1)
and the facts that the functor Tor commutes in both variables with colimit over directed
systems G6H, Proposition VI.1.3 on page 107, any C!-module is the colimit of its nitely
generated submodules, any nitely generated C!-module is the colimit of a directed system
of nitely presented C!-modules and any nitely presented C!-submodule is obtained by
L }ck, von Neumann algebras and L2-Betti numbers I
161
induction from a nitely presented C"-module for a nitely generated subgroup " ! !. It
is not hard to check that N(!) is at over C!, if N(") is at over C" for some subgroup
" ! ! of nite index and that N(Z) is at over CZ using the fact that CG is semi-simple
for nite G and CZ is a principal ideal domain. In particular N(!) is at over C! if !
is virtually cyclic.
Now suppose that N(!) is at over C!. If B! is a CW-complex of nite type,
then b(2) (E!; N(!)) ^ 0 for p ~ 1 and the p-th Novikov-Shubin invariant satises
p
a (B!) ^8; for p ~ 2. This implies for instance that ! does not contain a subgroup
p
which is isomorphic to Z Z (Remark 5.14) or Z^Z G17H, Proposition 39 on page 494.
If ! is non-amenable and B! is a nite CW-complex, then B! is a counterexample to the
zero-in-the-spectrum-conjecture G18H, G22H, section 11. ;
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Fachbereich Mathematik und Informatik, Westf{lische Wilhelms-Universit{t M}nster, Einsteinstr. 62,
D-48149 M}nster
e-mail: lueckXmath.uni-muenster.de
Eingegangen 17. Juli 1997, in revidierter Fassung 4. September 1997