Intrinsic Chern-Connes Characters for Crossed Products by $\mathbb Z^d$

0.23 MB·English

by  Emil Prodan

INTRINSIC CHERN-CONNES CHARACTERS FOR CROSSED
PRODUCTS BY Zd
5
1
EMILPRODAN
0
2
n Abstract. By imbedding A⋊ξZd into the C∗-algebra of adjointable oper-
a ators over the standard Hilbert A-module HA, we define Fredholm modules
J and Chern-Connes characters that are intrinsic to the C∗-dynamical system
9 (A,ξ,Zd). We introduce a T-index for the generalized Fredholm operators
1 over HA and develop ageneralized Fedosov principleand formula. We prove
anindexformulaforthepairbingofthecharacterswithK0(A⋊ξZd)andcon-
] clude that the pairing is in the image of K0(A) under the trace T. A local
A
indexformulaenables newapplicationsincondensedmatter physics.
O b
.
h
t
a INTRODUCTION
m
Consider a classical dynamical system (Ω,ξ), where ξ =(ξ ,...,ξ ) is a system
[ 1 d
of d-commuting homeomorphisms (d = even). Let C(Ω),ξ,Zd be its dual C∗-

2 dynamicalsystem and C(Ω)⋊ Zd the canonicallyas(cid:0)sociatedcros(cid:1)sed-product. Let
ξ
v π be the standard representation of the crossed product on ℓ2(Zd):
9 ω
7
(0.1) π φ q ψ = φ (ξ ω)ψ ,
4 (cid:16) ω(cid:16) q q · (cid:17) (cid:17)x q q x x−q
3 P P
d
0 and D =γ (X +x ) be the (shifted-) Dirac operator on C22 ℓ2(Zd). Then
. d x0 ⊗ 0 ⊗
1 C22 ℓ2(Zd),π ,F ,γ withF =sign(D )isanaturalevenFredholmmodule
0 o(cid:0)ver C⊗(Ω)⋊ Zdωandx0its(cid:1)Chern-Cxo0nnes charaxc0ter, defined as the cohomology class
5 ξ
of the cyclic cocycle:
1
v: (0.2) τd(a0,...ad)=1/2 Tr γFx0[Fx0,πω(a0)]...[Fx0,πω(ad)] ,
i (cid:8) (cid:9)
X pairs well and integrally with the K0-group [12]:
r (0.3) K (C(Ω)⋊ Zd) [p] τ (p,...p)=Index π+(p)F π−(p) Z.
a 0 ξ ∋ 0 → d ω x0 ω ∈
(cid:8) (cid:9)
In [3, 32], under certain optimal conditions, the local formula:
(0.4) τ (a ,...a )=Λ ( 1)ρ a ∂ a ...∂ a , Λ = (2π√−1)d2
d 0 d d X − T{ 0 ρ1 1 ρd d} (cid:16) d (d/2)! (cid:17)
ρ∈Sd
was proven by elementary means. Above, S is the group of permutations and
d
(∂, ) is the noncommutative differential calculus over C(Ω)⋊ Zd.
ξ
T
The identity 6.19 played a key role in this computation. In d= 2, this identity
is due to Connes [12], who used it to compute the 2-dimensional Chern characters
1991 Mathematics Subject Classification. 46L87,19K35,19K56,19L64.
Key words and phrases. Chern-Connes character, discrete crossed products, local index
formula.
ThisworkwassupportedbytheU.S.NSFgrantDMR-1056168.
1
2 EMILPRODAN
oftheconvolutionalgebrasC∞(R2)andC∞(SL(2,R)). Thelocalindexformulaof
c c
[3, 32] can be seen as a particularization of the generic Connes-Moscovici formula
[14] and its later extensions [19, 10, 9, 11, 8]. A recent related work is [1], where
local index formulas for Rieffel deformed crossed products are derived. Among
these works, only [8] seems to cover the general settings of [3, 32] where the index
formulas were shown to hold over certain noncommutative Sobolev spaces. Notice
alsothat0.4givesalocalformulaforthecharacteritselfandnotjustofthepairing.
There is some interest from the condensed matter physics community in the
resultssummarizedby0.1-0.4becauseallnon-interactingquantumlatticemodelsof
homogeneousmaterials can be generatedas representationsof crossedproducts by
Zd. The righthandside of 0.4 relates to the transportcoefficients of real materials,
in particular, it has certain relevance for topological insulators [31]. While [3,
32] explained some outstanding properties of these materials in the presence of
strong disorder and magnetic fields, the approach is limited to the single particle
theory of solids, where the electron-electron interaction is treated as a mean-field
correction. In fact, even before considering the electron-electron interaction, one
needs to address the following shortcomings:
(a) The formalism works only for crossed products of commutative algebras.
(b) The K-groups of many crossed-products used in condensed matter can be
fully resolved by the top and the lower Chern numbers. [32] produced a non-
commutative theory only for the top Chern number.
(c) The pairing of the characters with the K-groups is always integral, hence not
relevantforthesequencesoffractionaltopologicalphases,suchasthefractional
Chern insulators [27].
Removing these deficiencies while maintaining the elementary character of the cal-
culation was the main motivation for the present work.
TherootoftheproblemistherepresentationoftheFredholmmodulesonHilbert
spacesandanaturalcureisprovidedbyKK-theory. Whileexploringthispath,we
discoveredthatthecrossed-product ⋊ ZdcanbecanonicallyimbeddedintheC∗-
ξ
A
algebra of adjointable operators over the standard Hilbert -module . A Dirac
A
operator over can be naturally defined from the actioAn of Zd afHter tensoring
A
H
with an appropriate Clifford algebra. Furthermore, if posses a continuous trace,
A
thenthis tracecanbe naturallypromotedto a lowersemicountinuoustrace over
T
the imbedding algebra. The definitions of the intrinsic Fredholm modules, of the
“
notion of summability and of the intrinsic Chern-Connes characters are then fairly
straightforward(see Defs. 5.2, 5.3 and 6.1, respectively).
Oneoutcomeoftheapproach(seeTh.6.3)isthattheoperatorreplacingtheone
appearinginside the Index in 0.3is now a generalizedFredholm operatorover .
A
Assuch,Mingo’sindex[24]forC∗-modulesprovidesaKK-mapfromK ( ⋊HZd)
0 ξ
to K ( ). One could imagine the possibility of the crossedproduct ⋊ ZAd being
0 ξ
A A
imbeddedinthealgebraofadjointableoperatorsover ofanotherC∗-algebra,in
B
H
whichcaseone will perhaps obtaina KK-mapinto K ( ). This couldbe a useful
0
B
toolforthecomputationoftheK -groups. However,whatwefindinterestingabout
0
our construction is that it is intrinsic, in the sense that the entire construction is
natural and relying entirely on data from the C∗-dynamical system ( ,ξ,Zd).
A
InanalogywiththeBreuer-Fredholmindex forvonNeumannalgebras[5,6]and
the later generalizations [26], and especially [28], we define a numerical -index
T
“
INTRINSIC CHERN-CONNES CHARACTERS 3
for the multiplier algebra, by applying the trace on the Mindex of generalized
T
Fredholm operators over . This numerical index takes values in the image of
A “
H
K ( ) under the trace andthe structure of this sub-groupof R can be far more
0
compAlex than Z. For eTxample, if is the irrational rotational algebra , then
“ θ
this image is atleast(Z+θZ) [0,1A][34]. This addressespoint(c) above.AIn order
∩
to compute the -index,we developageneralizedFedosovprinciple anda Fedosov
T
formula (see Th. 4.6), which enables us to prove an index formula for the pairing
“
of the intrinsic Chern-Connes characters with K ( ⋊ Zd). Hence, this pairing is
0 ξ
A
in the image of K ( ) under the trace .
0
A T
Lastly,usingthesameelementarymethodsasin[32],wederivealocalformulafor
“
the Chern-Connes cocycle, similar to 0.4 (see Th. 6.5). The theory now covers the
lower Chern numbers, though only in the regime of weak disorder. An application
for disordered topological insulators in 3 space-dimensions is provided in the last
Chapter, where an index formula for the so called weak topological invariants is
derived and predictions about their possible values are made.
1. PRELIMINARIES
Let( ,ξ,Zd)beaC∗-dynamicaland ⋊ Zd itscanonicalcrossed-product. The
ξ
A A
C∗-algebra is assumed separable and to posses a unit and a faithful, continuous
A
and ξ-invariant trace . This is the only input we need for the definition and
A
T
characterizationof the intrinsic Chern-Connes characters.
Throughout our presentation, we follow closely the notation from Davidson’s
monograph[15]. In particular, the core algebra Zd will be represented by formal
A
finite sequences:
(1.1) a= a q, a , q Zd,
q q
· ∈A ∈
X
q
togetherwiththe standardalgebraicoperations. The algebra isimbeddedinthe
A
crossed-productas:
(1.2) a a=a 0 ⋊ Zd,
ξ
A∋ → · ∈A
and the additive group as:
(1.3) Zd q u =1 q ⋊ Zd.
q ξ
∋ → · ∈A
The first imbedding is isometric and the second one is in the group of unitaries of
⋊ Zd.
ξ
A
The Fourier calculus is defined by the group of automorphisms ρ on
λ λ∈Td
⋊ Zd [15], which act on the core algebra as: (cid:8) (cid:9)
ξ
A
(1.4) Td λ ρ (a)= λqa q, λq =λq1...λqd.
∋ → λ q· 1 d
X
q
The Fourier coefficients of a ⋊ Zd are defined by the Riemann integral:
ξ
∈A
(1.5) Φ (a)= dµ(λ) ρ (au−1)= dµ(λ) λ−qρ (a)u−1,
q Z λ q Z λ q
Td Td
whereµistheHaarmeasureonTd. TheFouriercoefficientswillbeseenaselements
of .
A
4 EMILPRODAN
Proposition 1.1 ([15], pg. 223). The Cesa`ro sums:
N N d
q
(1.6) a = ... 1 | j| Φ (a) q
N Å − N +1ã q ·
X X Y
q1=−N qd=−Nj=1
converge in norm to a ⋊ Zd as N .
ξ
∈A →∞
Hence, generic elements from ⋊ Zd can be representated as a Fourier series:
ξ
A
(1.7) a= Φ (a) q,
q
·
qX∈Zd
where the infinite sum must be interpreted via 1.1.
TheFouriercalculusgeneratesacanonicalfaithfulandcontinuoustraceon ⋊
ξ
Zd [15]: A
(1.8) a = A Φ0(a) .
T{ } T { }
It also generates a set of un-bounded derivations, ∂ = (∂ ,...,∂ ), through the
1 d
generators of the d-parameter group of automorphisms {ρλ}λ∈Td. The derivations
act as:
(1.9) ∂a=ı qΦ (a) q, (ı=√ 1).
q
· −
qX∈Zd
Together, (∂, ) define the non-commutative calculus over ⋊ Zd.
ξ
GenericallyT, the cocycles can be defined only on a pre C∗A-sub-algebra of ⋊Zd
A
[13],whichcanbegeneratedbyvariousmeans[37,33]. Belowwedescribeonesuch
sub-algebra which we find particularly convenient for the calculations to follow.
Proposition 1.2. Consider the set ( ⋊ Zd) :
ξ loc
A
(1.10) a ⋊ Zd α<1, s.t. sup α−|q| Φ (a) < .
ξ q
n ∈A | ∃ q (cid:0) k k(cid:1) ∞o
Then:
(i) The set ( ⋊ Zd) can be equivalently characterized as:
ξ loc
A
(1.11) a ⋊ Zd ρ (a) analytic of λ in a strip around Td .
ξ λ
n ∈A | o
(ii) When equipped with the algebraic operations, ( ⋊ Zd) becomes a dense
ξ loc
sub-algebra of ⋊ Zd. A
ξ
A
(iii) This sub-algebra is stable under the holomorphic functional calculus.
Proof. (i) Note that ρ is entire on Zd. If b Zd and a ⋊ Zd with ρ (a)
λ ξ α
analyticofαinafinite striparoundATd,thenρ∈(Aab)canbe∈anAalyticallycontinued
α
in the same strip via ρ (ab) = ρ (a)ρ (b). Likewise, if λ Td, then ρ (a) can
α α α αλ
∈
be analytically continued in α via:
(1.12) ρ (a)=ρ ρ (a)=ρ ρ (a).
αλ λ α α λ
◦ ◦
The last equality holds because the analytic continuations are unique [4]. Then:
(1.13) Φ ρ (a) = dµ(λ) ρ ρ (a)u =αq dµ(λ) ρ ρ (au ) .
q α Z λ α −q Z α λ −q
(cid:0) (cid:1) Td (cid:0) (cid:1) Td (cid:0) (cid:1)
INTRINSIC CHERN-CONNES CHARACTERS 5
Wecanexchangeρ andtheintegralbyusingthedominatedconvergencetheorem,
α
to conclude:
(1.14) Φ ρ (a) =αqρ Φ (a) =αqΦ (a).
q α α q q
(cid:0) (cid:1) (cid:0) (cid:1)
This identity is valid for any α in a strip around Td, and this strip is independent
of q. Taking α =α−qi/|q| with α close-enoughto 1, we obtain:
i
| |
(1.15) Φ (a) α|q| ρ (a) , q Zd.
q α
k k≤ k k ∀ ∈
Now assume a ( ⋊ Zd) . We can apply ρ on the Ces`aro sums:
ξ loc α
∈ A
N N d
q
(1.16) ρα(aN)= ... Å1− N|+j|1ãαqΦq(a)·q,
X X Y
q1=−N qd=−Nj=1
and the righthand-side and its derivatives with respect to α can be seen to be
absolutely norm-convergent as N , for α in a thin-enough strip around Td.
→ ∞
The statement then follows.
(ii) ( ⋊ Zd) is closed under the algebraic operations, which can be seen
ξ loc
A
directly from representation 1.11. Indeed, if ρ (a) and ρ (b) are analytic in a
α α
striparoundTd, then ρ (ab) canbe analyticallycontinuedoverthe same stripvia
α
ρ (ab)=ρ (a)ρ (b). ( ⋊ Zd) is dense in ⋊ Zd because it contains Zd.
α α α ξ loc ξ
A A A
(iii) Consider a ( ⋊ Zd) which is invertible in ⋊ Zd. We need to show
ξ loc ξ
that its inverse belo∈ngAs to ( ⋊ Zd) . Since the latterAis dense in ⋊ Zd, there
ξ loc ξ
isbfrom( ⋊ Zd) suchtAhat ab 1 <1. Thenab=1 (1 abA)isinvertible,
ξ loc
A k − k − −
hence b is invertible and a−1 = b(ab)−1. Taking r = 1 ab, the problem is
reducedtoshowingthat(1 r)−1 belongsto( ⋊ Zd) . Re−callthat r <1and
ξ loc
r ( ⋊ Zd) . Now, ρ (−r) is analytic of αAin a finite strip aroundTkdkhence, by
ξ loc α
ta∈kinAg α sufficiently close to the unit circle, ρ (r) < 1 and ∞ ρ (r) n is
| | k α k n=0 α
convergesinnormtogetherwithits∂ -derivatives. WeconcludetPhatρ (cid:0)(1 r(cid:1))−1
α α
−
is analytic in a strip around Td. (cid:0) (cid:3)(cid:1)
( ⋊ Zd) is a Fr`echetalgebra and ( ⋊ Zd) belongs to the domain of any
ξ loc ξ loc
A A
simple or higher derivation ∂n. The stability under the holomorphic functional
calculus ensures that the K -groups of ( ⋊ Zd) and ⋊ Zd coincide [13].
0 ξ loc ξ
A A
Besides, as we shall see, the canonical Fredholm module associated to the crossed
product ⋊ Zd is automatically (d+1)-summable. These facts made us believe
ξ
that ( ⋊A Zd) is the natural domain for the intrinsic Chern-Connes characters.
ξ loc
A
2. THE REPRESENTATION
In this Chapter we show that the C∗-algebra of adjointable operators over the
standardHilbert -module can serve as a naturalimbedding algebra for ⋊
A ξ
Zd. This will provAide a natuHralconnection between the K-theoriesof ⋊ ZdAand
ξ
A
of , as formulated within the framework of the standard Hilbert -module (see
A A
for example Chapter III in [38]).
Viewing as a right Hilbert -module, is defined as the tensor product
A
A A H
of Hilbert C∗-modules, with being an ordinary separable Hilbert space
A⊗H H
[16]. The inner product on is a φb ψ = φψ a∗b. The C∗-algebra of
A
adjointablecompactoperatorHs([20],hDe⊗f.4|)o⊗veri ihsi|soimorphicto K,where
A
KisthealgebraofcompactoperatorsoverseparabHleHilbertspaces. ThAe⊗C∗-algebra
ofadjointableoperators([20],Def.3)over is isomorphicto ( K)(cf.[20],
A
H M A⊗
6 EMILPRODAN
Th. 1), the multiplier algebra or double centralizer [7] of K. From a standard
A⊗
propertyofthetensorproductsandtheirmultiplieralgebras([38],CorollaryT.6.3),
we have a chain of algebra inclusions:
(2.1) K B ( K),
A⊗ ⊂A⊗ ⊂M A⊗
where B is the algebra of bounded operators over separable Hilbert spaces and the
tensorproductin BiswiththespatialC∗-norm. Thisobservationisimportant
forusbecausetheAcr⊗ossed-productalgebracanbenaturallyimbeddedin Band
2.1 will provide an imbedding in ( K), which is what we formallyAne⊗ed.
In the presentcontext, it is conMvenAien⊗tto make the choice =ℓ2(Zd), in which
H
case the elements of can be uniquely expressed as:
A
H
(2.2) (b )= b δ ℓ2(Z2),
x x x
⊗ ∈A⊗
xX∈Zd
with δ being the canonicalorthonormalbasis in ℓ2(Zd). Similarly, the elements
x
of { K}and B take the form:
A⊗ A⊗
(2.3) aˆ = a E ,
xy x,y
⊗
X
x,y
with E being the system of matrix-units over ℓ2(Zd), E δ = δ δ . Let
x,y x,y z yz x
us also mention the ideal of finite-rank elements, algebraically generated by the
rank-one operators:
(2.4) θ(bx) (c ) =(a ) (b )(c ) , (a ), (b ), (c ) .
(ax) x x h x | x i x x x ∈HA
(cid:0) (cid:1)
In our settings:
(2.5) θ(bx) = a b∗ E .
(ax) x y⊗ x,y
x,Xy∈Zd
The ideal of finite-rank operators is dense in K.
A⊗
Proposition 2.1. The following map
(2.6) ⋊ Zd a πˆ(a)= ξ Φ (a) E B,
ξ x q x,x−q
A ∋ → ⊗ ∈A⊗
x,Xq∈Zd
is well defined. It provides a faithful morphism of C∗-algebras and a faithful imbed-
ding of ⋊ Zd in ( K).
ξ
A M A⊗
Proof. Ourfirsttaskistoshowthatπˆisreallyinto B,i.e. thatπˆ(a)isabounded
operator when represented on some Hilbert spaceA⊗ ℓ2(Zd). For a Zd, this
K⊗ ∈ A
can be accomplished by elementary means using a covariant representation of the
dynamicalsystem( ,ξ,Zd). Furthermore,foraandbfrom Zd,onecanexplicitly
A A
verify that (note the similarity between 2.6 and 0.1):
(2.7) πˆ(a)πˆ(b)=πˆ(ab) and πˆ(a∗)=πˆ(a)∗.
The map is obviously faithful. At this point we established that πˆ is a faithful
-representation of Zd inside the C∗-algebra B. By the very definition of
∗ ⋊ Zd, this represAentation must extend over thAe⊗whole crossed-product. (cid:3)
ξ
A
INTRINSIC CHERN-CONNES CHARACTERS 7
3. THE CANONICAL TRACE AND ITS SPECIFIC CLASSES OF
ELEMENTS
The trace on and the ordinarytrace on B provide a trace = Tr on
A A
B. This tTrace caAn be characterized more directly as follows. T T ⊗
“
A⊗
Proposition 3.1. Let B be the positive cone of B. The functional:
+
A⊗ A⊗
(3.1) B aˆ aˆ = a [0, ],
+ A xx
A⊗ ∋ →T{ } T { }∈ ∞
“ xX∈Zd
is a faithful, lower semicontinuous trace.
We recall the following standard consequences of being a trace:
T
The set of trace-class elements: “
•
(3.2) =span aˆ B aˆ <
1 +
S { ∈A⊗ | T{ } ∞}
is a two-sided ideal in B. “
A⊗
The set of Hilbert-Schmidt elements:
•
(3.3) = aˆ B aˆ∗aˆ <
2
S { ∈A⊗ | T{ } ∞}
is a two-sided ideal in B and “= .
1 2 2
A⊗ S S ·S
The trace is cyclic:
•
(3.4) aˆbˆ = bˆaˆ
T{ } T{ }
for all aˆ and ˆb “ B. “
A
∈S ∈A⊗
The trace is invariant to conjugation by unitaries:
•
(3.5) uˆ∗aˆuˆ = aˆ
T{ } T{ }
for all aˆ and unita“ry uˆ “B.
A
∈S ∈A⊗
The following characterization of the trace-class elements is essential for the
generalized Fedosov principle elaborated in the next Chapter.
Proposition 3.2. = S , where S is the ideal of trace-class operators in B.
1 1 1
S A⊗
Proof. Given that is the extension of Tr, S automatically belongs to
A 1
T T ⊗ A⊗
the domain of . W“e need to show the reverse inclusion, which follows if we can
show that T S where S is the ideal of Hilbert-Schmidt operators in B.
2 “ 2 2
S ⊂ A⊗
Since is separable and is faithful and continuous, can be completed to a
A
A T A
separable Hilbert space when equipped with the inner product ab = (a∗b).
A
h | i T
Let e be an orthonormal basis of this Hilbert space. Then any element in
k
{B c}an∈bAe uniquely represented as e T . If such element belongs to ,
A⊗ k k⊗ k S2
then P
∗
(3.6) e T e T = Tr T∗T < ,
Tn(cid:16)X k⊗ k(cid:17) X k⊗ ko X { k k} ∞
“ k k k
hence e T S . (cid:3)
k k⊗ k ∈A⊗ 2
P
Itwillalsobeimportanttorealizethattheidealoffinite-rankoperatorsbelongs
to the domain of the trace. Indeed:
(3.7) θ(bx) = (a )(b ) ∗ .
Tn (ax)o TA h x | x i
(cid:8) (cid:9)
“
Then it is obvious that any finite combination of θ’s is in .
1
S
8 EMILPRODAN
4. GENERALIZED FREDHOLM INDEX AND THE FEDOSOV
FORMULA
Within the framework of standard modules, K ( ) is defined as ([38], pg. 264):
0
A
(4.1) K ( )= [pˆ] [qˆ] pˆ, qˆprojectors in K ,
0
A { − | A⊗ }
where[]indicatesthe classesunder the Murray-von-Neumannequivalencerelation
in ( K). We recall the index map for Hilbert C∗-modules, developed over
M A⊗
a stretch of works by Kasparov [20, 21], Miˇsˇcenco and Fomenko [25], Pimsner,
Popa, and Voiculescu [29] and assembled in the final form by Mingo [24]. Se also
the pedagogical exposition in [38] which calls this index the Mindex. We will do
the same here. In short, one defines the generalized Calkin algebra as the corona
( K)/ K, and the class of generalized Fredholm elements as:
M A⊗ A⊗
(4.2) = fˆ ( K) p(fˆ) invertible in ( K)/ K ,
A
F { ∈M A⊗ | M A⊗ A⊗ }
where p is the quotient map ( K) ( K)/ K.
( K)isnotavon-NeuMmaAnn⊗algeb→raM,heAnc⊗ethepAol⊗ardecompositioncannot
M A⊗
be assumedautomatically.1 Nevertheless,any fˆ admits acompactperturba-
A
tion gˆ (i.e. fˆ gˆ K) which does accept∈aFpolar decomposition gˆ = wˆ gˆ,
− ∈ A⊗ | |
with wˆ a partial isometry (see [24], Proposition1.7). This partial isometry defines
two compact projectors:
(4.3) ker gˆ=1 wˆwˆ∗ K
− ∈A⊗
and
(4.4) ker gˆ∗ =1 wˆ∗wˆ K,
− ∈A⊗
which at their turn define an element of the K ( )-group:
0
A
(4.5) [ker gˆ] [ker gˆ∗] K ( ).
0
− ∈ A
This element of K ( ) is insensitive of the compact perturbation gˆ used in the
0
A
construction, hence it is truly determined by the Fredholm element fˆ. Mingo’s
index map over is then defined as [24]:
A
H
(4.6) Mindex fˆ =[kerfˆ] [kerfˆ∗] K ( ),
0
{ } − ∈ A
whereitisunderstoodthatpossible(irrelevant)compactperturbationsareusedto
define the kernels. The Mindex is invariant to norm-continuous deformations of fˆ
and two Fredholm operators have the same Mindex precisely when they are in the
samepath-componentof . Thegroupofthehomotopyclassesofthegeneralized
A
F
Fredholm elements characterizes completely the K -group:
0
Theorem 4.1 ([24]). K ( ) [ ].
0 A
A ≃ F
OurgoalforthisChapteristodefineanumericalindexwhichiscomputationally
moreadvantageous. Westartwithastatementwhichisstandardforoperatorsover
ordinary Hilbert spaces:
Proposition 4.2. Any projector in K is finite-rank.
A⊗
1According to an argument due to W. J. Phillips, the polar decomposition is equivalent to
askingthatfˆhasclosedrange(see[38],Th.15.3.8).
INTRINSIC CHERN-CONNES CHARACTERS 9
Proof. We will take advantage of the fact that the set of finite-rank elements is an
ideal in ( K) and that this ideal is dense in K. Let
M A⊗ A⊗
(4.7) pˆ = 1 E ,
N x,x
⊗
X
|x|<N
be the standard approximation of identity for K. Then, for any projector
e K, eˆpˆ converges to eˆ in ( K) aAs⊗N . Consequently, for N
N
∈ A⊗ M A⊗ → ∞
large enough:
(4.8) sˆ=1 eˆ(1 pˆ )
N
− −
is invertible in ( K) and eˆsˆ= eˆpˆ , hence a finite rank operator. Therefore
N
eˆ=(eˆpˆ )sˆ−1 iMs finAit⊗e-rank. (cid:3)
N
This detail is important for our development because it showsthat the compact
projectors belong to the domain of the trace . As such, the pairing of with
T T
K ( ) is straightforward:
0 “ “
A
Proposition 4.3. The map:
(4.9) K ( ) [pˆ] [qˆ] pˆ qˆ R.
0
A ∋ − →T{ }−T{ }∈
is a group morphism. “ “
Definition 4.4. The generalized Fredholm index, which we call the -index, is
T
defined as:
“
(4.10) fˆ Index fˆ = kerfˆ kerfˆ∗ .
A
F ∋ → { } T{ }−T{ }
“ “
The following characterization of the -index is a direct consequence of the
T
properties of the Mindex mentioned earlier.
“
Proposition 4.5. The -index defines a map : R which is a locally
A
constant homomorphismTof semigroups (multiplicaTtiveFfor → and additive for R).
“ “ A
F
As for the Breuer-Fredholmindex [11], the injectivity modulo path components
present in 4.1 may be lost and the information about K ( ) group, that can be
0
A
extracted using the -index, can be limited. One recalls that this is not the case
T
for the classic Fredholm index. However, there is an advantage for using both
“
the -index and the Mindex, the former being algorithmically computable as the
T
following generalization of the Fedosov’s work [17] shows.
“
Theorem 4.6 (Fedosovprinciple andformula). Let fˆ ( K) with fˆ 1.
∈M A⊗ k k≤
Then:
(i) If there exists a natural number n such that:
(4.11) (1 fˆ∗fˆ)n and (1 fˆfˆ∗)n ,
1
− − ∈S
then fˆ .
A
∈F
(ii) If (1) holds, then the -index can be computed via:
T
(4.12) Index fˆ = “ (1 fˆ∗fˆ)n (1 fˆfˆ∗)n .
{ } T{ − }−T{ − }
Proof. (i) Assume 4.11 true. T“hen 3.2 assures th“at (1 fˆ∗fˆ)n belongs to S ,
1
− A⊗
hence compact. Taking n′ > n of the form n′ = 2k, we can apply successively the
squarerooton(1 fˆ∗fˆ)n′ toaccess1 fˆ∗fˆ. Thelatterisnecessarilycompactdue
− −
10 EMILPRODAN
to the following generic argument. Suppose aˆ K is positive. Then its square
∈A⊗
root is defined as a positive element by the continuous functional calculus and:
(4.13) √aˆ = limaˆ ǫ+√aˆ −1,
ǫց0
(cid:0) (cid:1)
where the limit is in ( K). Since K is an ideal, the operators inside the
limit belong to KManAd,⊗since KAis⊗closed, the limit is in K. Similarly,
1 fˆfˆ∗ KA.⊗Therefore, p(fˆA) is⊗invertible in ( K)/ AK⊗.
− ∈A⊗ M A⊗ A⊗
(ii) Assume for the beginning that fˆaccepts a polar decomposition fˆ= wˆ fˆ,
| |
with wˆ =wˆwˆ∗wˆ and wˆ∗ =wˆ∗wˆwˆ∗. We write:
(4.14) 1 fˆfˆ∗ =1 wˆwˆ∗+wˆ(1 fˆ2)wˆ∗,
− − −| |
and note that:
(4.15) (1 wˆwˆ∗) wˆ(1 fˆ2)wˆ∗ =0
− −| |
(cid:0) (cid:1)
and
(4.16) wˆ(1 fˆ2)wˆ∗ (1 wˆwˆ∗)=0.
−| | −
(cid:0) (cid:1)
When combined with the elementary fact wˆ∗wˆ fˆ = fˆ, these lead to:
| | | |
(4.17) (1 fˆfˆ∗)n =1 wˆwˆ∗+wˆ(1 fˆ2)nwˆ∗.
− − −| |
Using the cyclic property of the trace:
(4.18) wˆ(1 fˆ2)nwˆ∗ = wˆ∗wˆ(1 fˆ2)n
T{ −| | } T{ −| | }
and, since wˆ∗wˆ fˆ =“fˆ, it follows that: “
| | | |
(4.19) wˆ∗wˆ(1 fˆ2)n =wˆ∗wˆ 1+(1 fˆ2)n.
−| | − −| |
This together with 4.12 gives:
(4.20) (1 fˆ∗fˆ)n (1 fˆfˆ∗)n = 1 wˆ∗wˆ 1 wˆwˆ∗ .
T{ − }−T{ − } T{ − }−T{ − }
The affirma“tion is then prov“en for the particu“lar case when“fˆ accepts a polar
decomposition.
For the generic case, note that we already established that fˆis unitary modulo
K. Then we can apply Lemma 7.4 of [29] which, as noted in [24], extends
A⊗
to the present context. This Lemma assures us that fˆ(1 pˆ ) accepts the polar
N
−
decomposition, with pˆ defined in 4.7 and N large enough. Then fˆ(1 pˆ ) is a
N N
−
compact perturbation of fˆwith a polar decomposition, and since 4.11 still applies
for fˆ(1 pˆ ):
N
−
(4.21) Index fˆ = (1 (1 pˆ )fˆ∗fˆ(1 pˆ ))n (1 fˆ(1 pˆ )fˆ∗)n .
N N N
{ } T{ − − − }−T{ − − }
Using the cyclic pro“perty of the trace, one can check d“irectly that all the terms
containingpˆ cancelidentically,hencethislastequationcanbereducedto4.12. (cid:3)
N
The above statements can be generalized in the following way. Suppose there
areprojectionspˆandpˆ′ suchthat pˆ′fˆ=fˆpˆ=fˆandpˆ′ =sˆpˆsˆwith sˆa symmetry
(i.e. sˆself-adjoint and sˆ2 =1). Let:
(4.22) f˜=(1 pˆ)sˆ+fˆ.
−
ByapplyingthegeneralizedFedosovprincipleonf˜,welearnthatf˜ provided:
A
∈F
(4.23) (pˆ′ fˆ∗fˆ)n and (pˆ fˆfˆ∗)n .
1
− − ∈S