Lefschetz Numbers and Geometry of Operators in W*-modules

16 Pages·0.16 MB·English

by  Michael Frank

Lefschetz Numbers and Geometry of Operators in
W*-modules
5
9 M. Frank E. Troitsky
9
1
February 7, 2008
n
a
J

4
2
1 Introduction

1
v
The main goal of the present paper is to generalize the results of [18, 19] in the following
7
0 way: To be able to define K (A) C-valued Lefschetz numbers of the first type of an
0
0 endomorphism V on a C*-elliptic⊗complex one usually assumes that V = T for some
1 g
0 representation Tg of a compact group G on the C*-elliptic complex. We try to refuse this
5 restriction in the present paper. The price to pay for this is twofold:
9
(i) We have to define Lefschetz numbers valued in some larger group as K (A) C.
/ 0
n ⊗
(ii) We have to deal with W*-algebras instead of general unital C*-algebras.
a
- To obtain these results we have got a number of by-product facts on the theory of Hilbert
t
c W*-andC*-modulesandonboundedmoduleoperatorsonthemwhichareofindependent
n
u interest.
f
: The present paper is organized as follows: In 2 we prove the necessary facts on Hilbert
v §
i W*-modules andtheir bounded module mappings extending results of W.L. Paschke [14],
X
J.-F. Havet [5] and the first author [3]. In 3 we define Lefschetz numbers of two types
r §
a and show the main properties of them. In 4 we discuss the C*-case and obstructions to
§
refine the main results of 3.
§
Our standard references for the theory of Hilbert C*-modules are the papers [14, 15, 2,
9, 3, 10, 11] and the book of E. C. Lance [8]. The topological considerations are based
on the publications [12, 13, 17, 18, 19, 11]. We are going to continue the investigations
presented therein.
1
2 Hilbert W*-modules and module mappings
We want to show some more very nice properties of Hilbert W*-modules which often do
not appear in the general C*-case. This partial class of Hilbert C*-modules was brought
to the attention of the public by W. L. Paschke in his classical paper [14], and they
are of use in many cases. The facts below can be reproved for the class of monotone
complete C*-algebras carrying out much technical work, cf. [4], but not for larger classes
of C*-algebras, in general. However, since we are going to understand the structure of
general Hilbert C*-modules and their C*-duals better it suffices to treat the W*-case, and
we can avoid these technicalities. Let us start with a property generalizing the (double)
annihilator property of arbitrary subsets of W*-algebras.
Lemma 1 Let A be a W*-algebra and , .,. be a Hilbert A-module. For every subset
{M h i}
the bi-orthogonal set ⊥⊥ is a Hilbert A-submodule and a direct summand
S ⊆ M S ⊆ M
of , as well as the orthogonal complement ⊥.
M S
Proof: Thepropertyof ⊥⊥ tobeaHilbertA-submoduleisobviousbythedefinition
S ⊆ M
of orthogonal complements. Since the A-dual Banach A-module ′ of is a self-
M M
dual Hilbert A-module by [14, Th. 3.2] one can consider the Hilbert A-submodule
N
of ′ consisting of the direct sum of ⊥⊥ ֒ ′ and of the Hilbert A-module of all
M S → M
A-linear bounded mappings from to A vanishing on ⊥⊥. The second summand is
M S
the orthogonal complement of ⊥⊥ with respect to ′ by construction and hence, it
S M
is a self-dual Hilbert A-submodule and direct summand of by [3, Th. 3.2,Th. 2.8].
N
Consequently, the canonical embedding of ⊥⊥ into is a direct summand of , and
S N N
because of the submodule inclusion ⊥⊥ ֒ it is a direct summand of , too.
S ⊆ M → N M •
Example 1 below shows that situations different to that described at Lemma 1 can appear
e. g. for Hilbert C*-modules over the C*-algebra A = C([0,1]).
Lemma 2 Let A be a W*-algebra, , .,. be a Hilbert A-module and φ be a bounded
{M h i}
module operator on it. Then the kernel Ker(φ) of φ is a direct summand of and has
M
the property Ker(φ) = Ker(φ)⊥⊥.
Proof: By [14, Prop. 3.6] every bounded module operator φ on continues to a bounded
M
module operatoronits A-dualHilbert A-module ′. The kernel of theextended operator
M
is a direct summand of ′ because of the completeness of its unit ball with respect to
M
the τ -convergence induced by the functionals f( .,y ) : f A ,y ′ there, (cf. [3,
2 ∗,1
{ h i ∈ ∈ M}
Th. 3.2]). Consequently, the kernel of φ inside has to coincide with its bi-orthogonal
M
complement in , and by Lemma 1 it is a direct summand.
M •
2
Example 1 Note, that the kernel of bounded A-linear operators on Hilbert A-modules
over arbitrary C*-algebras A is not a direct summand, in general. For example, consider
the C*-algebra A = C([0,1]) of all continuous functions on the interval [0,1] as a Hilbert
A-module over itself equipped with the standard inner product a,b = ab∗. Define the
A
h i
mapping φ by the formula φ (f) = g f for a fixed function
g g
·
2x+1 : x 1/2
g(x) = − ≤
0 : x 1/2
(
≥
and for every f A. Then Ker(φ ) equals to the Hilbert A-submodule and (left) ideal
g
∈
f A : f(x) = 0 for x [0,1/2] , being not a direct summand of A, but nevertheless,
{ ∈ ∈ }
coinciding with the bi-orthogonal complement of itself with respect to A.
Corollary 1 LetA be a W*-algebra, and be two Hilbert A-modules andφ :
M N M → N
be a bounded A-linear mapping. Then the kernel Ker(φ) of φ is a direct summand of
M
and has the property Ker(φ) = Ker(φ)⊥⊥.
Proof: Consider the Hilbert A-module formed as the direct sum = equipped
K K M⊕N
with the A-valued inner product .,. + .,. . The mapping φ can be identified with
M N
h i h i
a bounded A-linear mapping φ′ on acting on the direct summand as φ and on the
K M
direct summand as the zero operator. Since the kernel of φ′ is a direct summand of
N K
containing by Lemma 2 its orthogonal complement is a direct summand of . The
N M
desired result turns out.
•
Now we are in the position to give a description of the inner structure of arbitrary Hilbert
W*-modules generalizing an analogous statement for self-dual Hilbert W*-modules by
W. L. Paschke ([14, Th. 3.12]).
Proposition 1 Let A be a W*-algebra and , .,. be a left Hilbert A-module. Then
{M h i}
is the closure of a direct orthogonal sum of a family D : α I of norm-closed
α
M { ∈ }
left ideals D A, where the closure of this direct sum is predetermined by the given
α
⊆
on A-valued inner product .,. and the A-valued inner products on the ideals are the
M h i
standard A-valued inner product on A. Moreover, for every bounded A-linear mapping
r : A there is a net x : β J of elements of for which the limit
β
M → { ∈ } M
. lim y,x
A β
k k −β∈Jh i
exists for every y and equals r(y).
∈ M
3
Proof: Fix an arbitrary bounded A-linear mapping r : A. The kernel of r is a direct
M →
summand of by Corollary 1. Consider its orthogonal complement. Since r can be
M
continued to an bounded A-linear mapping r( ) = .,x on the A-dual (self-dual) Hilbert
r
· h i
A-module ′ of (x Ker(r)⊥ ′) and since the orthogonal complement of the
r
M M ∈ ⊆ M
kernel of r inside ′ is a direct summand isomorphic to Ap, .,. for some projection
M { h i}
p A by the structural theorem [14, Th. 3.12] for self-dual Hilbert W*-modules the
∈
orthogonal complement of the kernel of r with respect to is isomorphic to the Hilbert
M
A-module I, .,. for some norm-closed left ideal I Ap of A, where the left-strict
A
{ h i } ⊆
closure of the left ideal I is the w*-closed ideal Ap of A. Now, r can be identified with
the element x Ap, and x Ap is the left-strict limit of a net x : β J of elements
r r β
∈ ∈ { ∈ }
of I , cf. [16, 3.12].
∩M §
Finally, bytransfinit induction onehastodecompose into asumof pairwiseorthogonal
M
directsummands oftypeKer(r)⊥ forboundedA-linearfunctionalsr on ,whereKer(r)⊥
M
is always isomorphic to a left norm-closed ideal I of A with the standard A-valued inner
product on it.
•
We go on to investigate the image of bounded module mappings between Hilbert W*-
modules. In general, many quite non-regular things can happen as the example below
shows, but embeddings of self-dual Hilbert W*-modules into other Hilbert W*-modules
canbeshowntobemappingsontodirectsummands incontrasttothesituationforgeneral
Hilbert C*-modules.
Example 2 Let Abe the set of all bounded linear operatorsB(H) on a separable Hilbert
space H with basis e : i N . Denote by k the operator k(e ) = λ e for a sequence
i i i i
{ ∈ }
λ : i N of non-zero positive real numbers converging to zero. Then the mapping
i
{ ∈ }
φ : A A , φ : a a k
k k
→ → ·
isaboundedA-linearmappingontheleftprojectiveHilbertA-moduleA. Buttheimageis
not a direct summand of this A-module and is not even Hilbert because direct summands
of A are of the form Ap for some projection p of A, and 1 k should equal p. The image
A
·
of φ is a subset of the set of all compact operators on H. Note, that the mapping φ is
k k
not injective.
The following proposition was proved for arbitrary C*-algebras A, countably generated
HilbertA-modules , withoutself-dualityrestrictionandaninjectiveboundedmodule
M N
mapping α : with norm-dense range by H. Lin [10, Th. 2.2]. We present another
M → N
variant for a similar situation in the W*-case.
Proposition 2 Let A be a W*-algebra, be a self-dual Hilbert A-module and , .,.
M {N h i}
be another Hilbert A-module. Suppose, there exists an injective bounded module mapping
4
α : with the range property α( )⊥⊥ = . Then the operator α(α∗α)−1/2 is a
M → N M N
bounded module isomorphism of and . In particular, they are isomorphic as Hilbert
M N
A-modules.
Proof: The mapping α possesses an adjoint bounded module mapping α∗ :
N → M
because of the self-duality of , cf. [14, Prop. 3.4]. Since α∗α is a positive element of
M
the C*-algebra End ( ) of all bounded (adjointable) module mappings on the Hilbert
A
M
A-module the square root of it, (α∗α)1/2, is well-defined by the series
M
n (α∗α) k
(α∗α)1/2 = . lim (α∗α) 1/2 id λ id
k k−n→∞k k  M − k M − (α∗α) ! 
kX=1 k k
 
with coefficients λ taken from the Taylor series at zero of the complex-valued function
k
{ }
f(x) = √1 x on the interval [0,1]. Moreover, because
−
(α∗α)1/2(x),(α∗α)1/2(x) = α(x),α(x)
h i h i
andbecauseoftheinjectivity ofαthemapping(α∗α)1/2 hastrivialkernel. Atthecontrary
one can only say that the range of (α∗α)1/2 is τ -dense in , (cf. [3]). Indeed, for every
1
M
A-linear bounded functional r( ) = .,y on the self-dual Hilbert A-module mapping
· h i M
the range of (α∗α)1/2 to zero one has
0 = (α∗α)1/2(x),y = x,(α∗α)1/2(y)
h i h i
for every x . Hence, y = 0 since (α∗α)1/2 is injective and x was arbitrarily
∈ M ∈ M
chosen.
Now, consider the mapping α(α∗α)−1/2 where it is defined on . Since (α∗α)1/2 has both
M
τ -dense range and trivial kernel by the assumptions on α its inverse unbounded module
1
operator (α∗α)−1/2 is τ -densely defined. One obtains
1
α(α∗α)−1/2(x),α(α∗α)−1/2(y) = x,y
h i h i
for every x,y from the (τ -dense) area of definition of (α∗α)−1/2. Consequently, the opera-
1
tor α(α∗α)−1/2 continues to a bounded isometric module operator on by τ -continuity.
1
M
The range of it is τ -closed (i.e., a self-dual direct summand of ) and hence, equals
1
N N
by assumption. Finally, since the range of (α∗α)−1/2 is norm-closed and τ -dense in
1
M
and since is self-dual the mapping α is a (non-isometric, in general) Hilbert A-module
M
isomorphism itself.
•
Corollary 2 Let A be a W*-algebra, be a self-dual Hilbert A-module and , .,. be
M {N h i}
another Hilbert A-module. Every injective module mapping from into is a Hilbert
M N
A-module isomorphism of and of a direct summand of .
M N
5
For our application in 3 we need the following partial result:
§
Corollary 3 Let A be a W*-algebra, and be countably generated Hilbert A-modules
M N
and F : be a Fredholm operator (see [13]). Then KerF and (ImF)⊥ are
M → N
projective finitely generated A-submodules, and IndF = [KerF] [(ImF)⊥] inside K (A).
0
−
Proof: We denote by ˆ the direct orthogonal sum of two Hilbert A-modules, whereas
⊕
denotes the direct topological sum of two Hilbert A-submodules of a given Hilbert A-
⊕
module, where orthogonality of the two components is not required. Let = ˆ ,
0 1
M M ⊕M
= be the decompositions from the definition of A-Fredholm operator:
0 1
N N ⊕N
F 0
F = 0 : ˆ ,
0 F1 ! M0 ⊕M1 → N0 ⊕N1
F : = , F : , and are the projective finitely generated modules.
0 0 ∼ 0 1 1 1 1 1
M N M → N M N
Let x = x +x , x and x , and F(x) = 0, so 0 = F (x )+F (x ) .
0 1 0 0 1 1 0 0 1 1 0 1
∈ M ∈ M ∈ N ⊕N
Thus F (x ) = 0, F (x ) = 0, so x = 0 and x . Thus KerF = KerF .
0 0 1 1 0 1 1 1
∈ M ⊂ M
By Lemma 2 KerF is a projective finitely generated A-module and has an orthogonal
complement. So, by Corollary 2,
F 0 0
0
F = 0 F′ 0 : ˆ ′ KerF F( ′) ˆ (ImF)⊥
 1  M0 ⊕M1 ⊕ → N0 ⊕ M1 ⊕
0 0 0 (cid:16) (cid:17)
 
 
and IndF = [KerF] [(ImF)⊥].
− •
The following example shows that the situations may be quite different for general Hilbert
C*-modules and injective mappings between them:
Example 3 Consider the C*-algebra A = C([0,1]) of all continuous functions on the
interval [0,1] as a self-dual Hilbert A-module over itself equipped with the standard A-
valued inner product a,b = ab∗. The mapping φ : f(x) x f(x), (x [0,1]), is an
A
h i → · ∈
injective bounded module mapping. Its range has trivial orthogonal complement, but it
is not closed in norm and, consequently, not a direct summand of A. Nevertheless, the
bi-orthogonal complement of the range of φ with respect to A equals A.
Lemma 3 Let A be a W*-algebra. Let and be self-dual Hilbert A-submodules of a
P Q
Hilbert A-module . Then is a self-dual Hilbert A-module and direct summand of
M P∩Q
. Moreover, + is a self-dual Hilbert A-submodule.
M P Q ⊆ M
If is projective and finitely generated then the intersection is projective and
P P ∩ Q
finitely generated, too. If both and are projective and finitely generated then the sum
P Q
+ is also.
P Q
6
Proof. Let p : = ⊥ ⊥ be the canonical orthogonal projection existing
M P ⊕ P → P
by [3, Th. 2.8], (cf. [2] for the projective case). Let p = p : ⊥. Since is a
Q
Q → P Q
self-dual Hilbert A-module p admits an adjoint operator and Kerp is a direct
Q Q
⊆ Q
summand by Lemma 2. Consequently, it is a self-dual Hilbert A-submodule of .
Q ⊆ M
But Kerp = . To derive the second assertion one has to apply the fact again that
Q
P ∩Q
every self-dual Hilbert A-submodule is a direct summand, cf. [3].
If is projective and finitely generated then every direct summand of it is projective and
P
finitely generated, what shows the additional remarks.
•
3 Lefschetz numbers
Throughout this section A denotes a W*-algebra. This restriction enables us to apply the
results of the previuos section being valid only in the W*-case, in general.
Let U be a unitary operator in the projective finitely generated Hilbert A-module .
P
Then
U = eiϕdP(ϕ), (1)
ZS1
where P(ϕ) is the projection valued measure valued in the W*-algebra of all bounded
(adjointable) module operators on , and the integral converges with respect to the
P
norm. So we have a bounded and measurable function
L( ,U) : S1 K (A), ϕ [dP(ϕ)], (2)
0
P → 7→
This function is bounded in the sense that there exists a projection which is greater then
all values with respect to the partial order in the space of projections. Let us denote the
set of such functions by K (A) .
0 S
Let us note that the Lefschetz numbers for compact group action considered in [19] can
be thought of as evaluated (for unitary representation) in the subspace of finitely valued
(simple) functions:
Simple(S1,K (A)) K (A) .
0 0 S
⊂
Suppose, = An. In the case of L( ,U) Simple(S1,K (A)) associate with the integral
0
P P ∈
(1)
eiϕdP(ϕ) = eiϕkP( )
k
ZS1 k E
X
the following class of the cyclic homology HC (M(n,A)):
2l
P( ) ... P( ) eiϕk.
k k
E ⊗ ⊗ E ·
k
X
7
Passing to the limit we get the following element
T˜U = eiϕd(P ... P)(ϕ) HC (M(n,A)).
2l
ZS1 ⊗ ⊗ ∈
Then we define
T(U) = Trn(T˜U) HC (A),
∗ 2l
∈
where Trn is the trace in cyclic homology.
∗
Lemma 4 ( [19, Lemma 6.1])
Let J : = Am = An be an isomorphism, U : , U : be
M N
M → N M → M N → N
A-unitary operators and JU = U J. Then
M N
T(U ) = T(U ).
M N
Similar techniques can be developed for a projective finitely generated A-module in-
N
stead of An. For this purpose we take = q(An), where q denotes the orthogonal
N
projection from An onto its direct orthogonal summand . Then we set
N
U 1 : An = (1 q)An (1 q)An = An,
∼ ∼
⊕ N ⊕ − → N ⊕ −
T˜U = eiϕd(qPq ... qPq)(ϕ).
ZS1 ⊗ ⊗
The correctness is an immediate consequence of the Lemma 4.
Let us consider an A-elliptic complex (E,d) and its unitary endomorphism U. The results
of 1 (cf. Prop. 2, Lemma 3, Lemma 3) and the standard Hodge theory argument help us
§
to prove the following lemma.
Lemma 5 For the A-Fredholm operator
∗
F = d+d : Γ( ) Γ( ),
ev od
E → E
we have
def
Ker(F ) = H ( ) = H ( ),
|Γ(Eev) ev E ⊕ 2i E
def
Ker(F ) = H ( ) = H ( ),
|Γ(Eod) od E ⊕ 2i+1 E
where H ( ) is the orthogonal complement to Imd Kerd Γ( ) and H ( ) are
m m m
E ⊂ ⊂ E E
projective U-invariant Hilbert A-modules.
8
Proof. For u Γ(E ) while
2i 2i
∈
∗
(d+d )(u +u +u +...) = 0
0 2 4
we have
∗ ∗
du +d u = 0, du +d u = 0,...
0 2 2 4
Together with the equality
(du,d∗v) = (d2u,v) = 0
one obtains
∗ ∗
du = 0, du = 0,... ;d u = 0, d u = 0,....
0 2 2 4
what implies u Ker(d+d∗). On the other hand for v Imd, v = dv we have
2i 2 2 1
∈ ∈
∗
(v ,u ) = (dv ,u ) = (v ,d u ) = 0.
2 2 1 2 1 2
Thus u H (E). Conversely, let u = u + u + ..., u H (E), i.e. du = 0, (i =
2i 2i 0 2 2i 2i 2i
∈ ∈
0,1,2,...), and for any v E we have
2i−1 2i−1
∈
∗
(dv ,u ) = 0, (v ,d u ) = 0,
2i−1 2i 2i−1 2i
so d∗u = 0. Thus u Ker(d + d∗). The invariance and projectivity follow from the
2i
∈
proved identification and Corollary 3.
•
Definition 1 We define the Lefschetz number L as
1
L ( ,U) = ( 1)iT(U H ( )) K (A) .
1 i 0 S
E − | E ∈
i
X
Definition 2 We define the Lefschetz number L as
2l
L ( ,U) = ( 1)iT(U H ( )) HC (A).
2l i 2l
E − | E ∈
i
X
After all the following theorem is evident:
Theorem 1 Let the Chern character Ch be defined as in [1, 6, 7]. Then
L ( ,U) = (Ch0 ) (L ( ,U))(ϕ)dϕ.
2l E ZS1 2l ∗ 1 E
Remark 1 Insituations, whentheendomorphismV oftheellipticC*-complexrepresents
as an element of a represented there amenable group G acting on the C*-complex then
the A-valued inner products can be chosen G-invariant, what gives us the unitarity of V
(see [11]). However, there is another obstruction demanding new approaches which will
be shown at Example 4 below.
9
4 Obstructions in the C*-case and related topics
The aim of this chapter is to show some obstructions arising in the general Hilbert C*-
module theory for more general C*-algebras than W*-algebras which cause the made
restriction of the investigations in section three. The results underline the outstanding
properties of Hilbert W*-modules. To handle the general C*-case we often need a basic
construction introduced by W.L.Paschke andH. Lin. It gives alink between theW*-case
and the general C*-case.
Remark 2 (cf.[9, Def. 1.3], [14, 4])
§
Let , .,. be a left pre-Hilbert A-module over a fixed C*-algebra A. The algebraic
{M h i}
tensor product A∗∗ becomes a left A∗∗-module defining the action of A∗∗ on its
⊗ M
elementary tensors by the formula ab h = a(b h) for a,b A∗∗, h . Now , setting
⊗ ⊗ ∈ ∈ M
a h , b g = a h ,g b
i i j j i i j j
 ⊗ ⊗  h i
i j i,j
X X X
 
on finite sums of elementary tensors one obtains a degenerate A∗∗-valued inner pre-
product. Factorizing A∗∗ by N = z A∗∗ : [z,z] = 0 one obtains a
⊗ M { ∈ ⊗ M }
pre-Hilbert A∗∗-module denoted by # in the sequel. It contains as a A-submodule.
M M
If is Hilbert then # is Hilbert, and vice versa. The transfer of the self-duality is
M M
more difficult. If is self-dual then # is self-dual, too. But,
M M
Problem. Suppose, the underlying C*-algebra A is unital. Whether the
property of # to be self-dual implies that was already self-dual?
M M
Other standard properties like e.g. C*-reflexivity can not be transferred. But every boun-
ded A-linear operator T on has a unique extension to a bounded A∗∗-linear operator
M
on # preserving the operator norm, (cf. [9, Def. 1.3]).
M
Proposition 3 Let A be a C*-algebra, and be two Hilbert A-modules and φ :
M N
be a bounded A-linear mapping. Then the kernel Ker(φ) of φ coincides with its
M → N
bi-orthogonal complement inside . In general, it is not a direct summand.
M
Proof: Let us assume, Ker(φ) = Ker(φ)⊥⊥ with respect to the A-valued inner product
6
of . Form the direct sum = . The mapping φ extends to a bounded A-linear
M L M⊕N
mapping ψ on setting
L
φ(x) : x
ψ(x) = ∈ M .
0 : x
(
∈ N
10