University of Wollongong
Research Online
Faculty of Engineering and Information Sciences Papers
Faculty of Engineering and Information Sciences
2007
KK-theory and spectral flow in von Neumann
algebras
J Kaad
University of Copenhagen
R Nest
University of Copenhagen
Adam C. Rennie
University of Copenhagen, [email protected]
Publication Details
Kaad, J., Nest, R. & Rennie, A. C. (2007). KK-theory and spectral flow in von Neumann algebras. Journal of K-theory, 10 (2), 1-29.
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KK-theory and spectral flow in von Neumann algebras
Abstract
We present a definition of spectral flow for any norm closed ideal J in any von Neumann algebra N. Given a
path of selfadjoint operators in N which are invertible in N/J, the spectral flow produces a class in
Ko( J).Given a semifinite spectral triple (A, H, D) relative to (N, t) with A separable, we construct a class [D] ?
KK1(A, K(N)). For a unitary u ? A, the von Neumann spectral flow between D and u*Du is equal to the
Kasparov product [u]A[D], and is simply related to the numerical spectral flow, and a refined C*-spectral flow.
Keywords
von, spectral, theory, algebras, flow, kk, neumann
Disciplines
Engineering | Science and Technology Studies
Publication Details
Kaad, J., Nest, R. & Rennie, A. C. (2007). KK-theory and spectral flow in von Neumann algebras. Journal of
K-theory, 10 (2), 1-29.
This journal article is available at Research Online: http://ro.uow.edu.au/eispapers/477
arXiv:math/0701326v1 [math.OA] 11 Jan 2007
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) N, J 0 ) 0 0 " , + "$ .
1 , , N J + 2 2
/ KK ( . K ( ) $
/ 0 3 4 K ( K0 (B) B ⊂ J σ ( + ,
C ∗ ( + , 0 5 , C ∗ 1) KK ( 4 )
KK ( & / & 1 6 7 8 N (, % H J N 9 π : N → N/J 4 KK
S : H → H ker(S) N(S) ∈ L(H) S (S) (S) R(S) ∈ L(H) S N N(S) R(S) N p, q ∈ N S ∈ N p ∩ q ∈ N = N p, q ∈ L(H) (p)∩ (q) p∩q ∈ L(H)
Sp = pS Sq = qS Sp ∩ q = p ∩ qS S ∈ N S S
u∈N
S
u|S| = S
S ∗ = u∗ |S ∗ |
u∗ u = R(u∗ ) = R(S ∗ )
uu∗ = R(u) = R(S)
∗
∗
∗
1 − uu = N(u ) = N(S ) 1 − u∗ u = N(u) = N(S)
!" #
() *
K % %& % C ∗ %
' K % $ N/J [π(S)] ∈ K1(N/J) K1 (N/J) π(S)
S ∈ Mn (N) n ∈ N ∂[π(S)] = [N(S)] − [N(S ∗ )] ∈ K0 (J),
∂ : K1 (N/J) → K0 (J) K 8.3.1$
+ ,&
Mn (N) = Mn (C)⊗N %- . ⊕ni=1 H S Mn (N) u ∈ Mn (N) S -
u π(S) π(S) = π(u|S|) = π(u)π(S ∗ S)1/2 = π(u)
# & + !/ "0$ ∂[π(S)] = [1 − u∗ u] − [1 − uu∗ ] = [N(S)] − [N(S ∗ )]
' S : H1 → H2 S
1 & (* ' 2 - N
%
qNp
p, q ∈ N
& !!$ & - & & p, q N S ∈ qNp (qp)
T, R ∈ pNq π(T S) = π(p)
π(SR) = π(q)
π(T ) = π(T SR) = π(R) R = T T S '
S ∈ qNp N(S) ∩ p = N(S)p
N(S ∗ ) ∩ q = N(S ∗ )q
(1 − p)N(S) = (1 − p) = N(S)(1 − p) ⇒ pN (S) = N(S)p
N(S) ∩ p = N(S)p N(S ∗ ) q S ∈ qNp u ∈ N S u ∈ qNp p − u∗ u = N(S) − (1 − p) = N(S) ∩ p
q − uu∗ = N(S ∗ ) − (1 − q) = N(S ∗ ) ∩ q.
S ∈ qNp (q p) π(u∗u) = π(p) π(uu∗) = π(q)
u (q p) N(S) ∩ p, N (S ∗ ) ∩ q ∈ J
u qNp (1 − p)H ⊆ (S) = (u) (u) = (S) ⊆ qH (1 − p)N(S) = (1 − p) N(S) − (1 − p) = N(S) − N(S)(1 − p) = N(S)p = N(S) ∩ p N(S ∗ ) q S ∈ qNp (q p) ! T ∈ pNq S ∗ S ∈ pNp (p p) ! T T ∗ ∈ pNp π(S ∗ S)
! C ∗ ! π(p)N/Jπ(p) "!! π(SS ∗) ! C ∗ !
π(q)N/Jπ(q) #!! u ∈ qNp $ S ∈ qNp !$ $ π(S)π(S ∗S)−1/2 ∈
π(q)N/Jπ(p) !! π(u∗u) = π(S ∗ S)−1/2 π(S ∗ S)π(S ∗ S)−1/2 = π(p) π(uu∗) = π(S)π(S ∗ S)−1 π(S ∗ ) = π(q)
! !! $!! %
S ∈ qNp (qp) (qp) S (q )(S) = [N(S) ∩ p] − [N(S ∗ ) ∩ q]
K0 (J)
S ∈ qNp (q p) ! ! u ∈ qNp $ S !
(p, q, u) ! K ! % ! [S] := [p, q, u] ∈ K0 (N, N/J) !
K ! K K0 (N, N/J) ! K $ ! K0 (J)
&
' : K0(J) → K0 (N, N/J)
% ()* +% 4.3.7, ()* 4.3.8,
!! (q p) $ S ! $ ! ! [S] ∈ K0(N, N/J) - $ (q p) !! $!!
! !! !!
KK
S ∈ qNp (q p)
u ∈ qNp S −1[S] = q p (S)
K0 (J)
[S] ∈ K0 (N, N/J) [S] = [p, q, u] = [p − u∗ u, q − uu∗, 0] + [u∗ u, uu∗, u]
K (u∗u, uu∗, u) [S] = [p − u∗ u, q − uu∗, 0]
p − u∗u = N(S) ∩ p q − uu∗ = N(S ∗ ) ∩ q J −1[S] = [p − u∗u] − [q − uu∗] = q p (S)
S0 ∈ qNp S1 ∈ qNp (qp)
(q p)
S0 S1 (q
)
(S0 ) = (q) (S1 )
t → St ∈ qNp S0 S1 t → π(St )π(St∗St)−1/2 ∈ π(q)N/Jπ(p) ! π(p)N/Jπ(p)
t → vt ∈ qN/Jp (p, q, vt) K t ∈ [0, 1] "#
4.3.13$ u0 ∈ qNp u1 ∈ qNp S0 S1 π(u0 ) = π(v0 ) π(u1 ) = π(v1 ) ! [S0 ] = [p, q, u0] = [p, q, v0 ] = [p, q, v1] = [p, q, u1] = [S1 ]
K0(N, N/J) ! % (q p)(S0) = −1 [S0] = −1[S1 ] = (q p)(S1 )
S ∈ qNp (q p)
T ∈ rNq (rq)
T S (rp)
(r q)(T ) + (q p)(S) = (r p)(T S)
v ∈ rNq u ∈ qNp w ∈ rNp T S T S &
π(w) = π(T S)π(S ∗T ∗ T S)−1/2
= π(T )π(SS ∗T ∗ T )−1/2 π(S)
= π(T )π(T ∗T )−1/2 π(S)π(S ∗S)−1/2
= π(vu)
! [p, r, vu] = [p, r, w] K0(N, N/J)
[T ] [S] K0 (N, N/J) [T ] + [S] = [q, r, v] + [p, q, u] = [p, r, vu] = [p, r, w] = [T S]
K0 (N, N/J)
(rq) (T ) + (qp) (S) = −1 [T ] + −1 [S] = −1 [T S] = (rp) (T S)
N τ KN τ !
" #$ % &
KN #' % τ∗ : K0 (KN ) → R #( )!* +, ' # - . $/ $$ ,, ,0 ,(1
2 *!
0
0$
3!
pNq +$$1 4 !& p!q !* p = q = 1 T ∈ N J !* π(T ) N/J J F J Fsa
χ : R → R [0, ∞) χ(t) =
1
0
t ∈ [0, ∞)
t ∈ (−∞, 0)
+, 4.31 3 5 χ(T ) = χ π(T ) χ π(T ) 3 0 ∈/ π(T ) ε > 0 [−ε, ε] π(T ) f1 : R → R
⎧
t ∈ (−∞, −ε]
⎨ 0
t ∈ [−ε, 0]
f1 (t) = ε−1 t + 1
⎩
1
t ∈ [0, ∞)
f2 : R → R ⎧
t ∈ (−∞, 0]
⎨ 0
−1
t ∈ [0, ε]
f2 (t) = ε t
⎩ 1
t ∈ [ε, ∞)
T ∈ Fsa π
KK
f1 = χ = f2 (π(T )) f1 ≥ χ ≥ f2 (T )
χ π(T ) = f1 π(T ) = π f1 (T ) ≥ π χ(T ) ≥ π f2 (T ) = f2 π(T ) = χ π(T )
χ π(T ) = π χ(T ) t → Bt Fsa t → χ π(Bt) C ∗ N/J C ∗ A C ∗ U R Asa A {a ∈ Asa | (a) ⊆ U }
Asa
a ∈ Asa (a) ⊆ U (·, U c ) : C → [0, ∞[ (λ, U c ) = inf{|λ − μ| | μ ∈ U c }
λ ∈ C (a)
/ U c = U c λ ∈ (a) (λ, U c ) > 0 λ ∈
(a), U c
inf{|λ − μ| | λ ∈ (a), μ ∈ U c }
=
ε=
! b ∈ Asa
2
b − a <
ε
2
2
>0
λ ∈ (b)
Bε (λ) ∩ (a) = ∅
" Bε (λ) ε > 0 λ
μ ∈ Bε/4 (λ)
μ ∈
/ (a)
3ε
(μ − a)−1 −1 = sup{|μ − α|−1 | α ∈ (a)}−1 = μ, (a) ≥
4
ε ε
+ ≤ (μ − a)−1 −1
4 2
λ − b #$% & 17.3'
" λ ∈ (b) Bε (λ) ∩ (a) = ∅
( ε (b) ⊆ U (b) ⊆ U b ∈ Asa b − a < ε/2
(λ − b) − (μ − a) ≤ |λ − μ| + a − b <
t0 ∈ [0, 1]
) ε > 0 [−ε, ε] π(Bt0 )
!
π(Bt0 ) ⊆ (−∞, −ε) ∪ (ε, ∞)
( *
+ t → π(Bt ) δ > 0 π(Bt ) ⊆ (−∞, −ε) ∪ (ε, ∞)
t ∈ (t0 − δ, t0 + δ) ∩ [0, 1]
f
t ∈ (t0 − δ, t0 + δ) ∩ [0, 1]
χ π(Bt ) = f π(Bt )
f1
t → f π(Bt )
K0 (J)
t → Bt Fsa t → π χ(Bt ) = χ π(Bt )
0 = t0 < t1 < . . . < tn = 1 π χ(Bt ) − π χ(Bs ) < 1/2 t, s ∈ [ti−1 , ti ]
pi = χ(Bti ) {Bt} n
{Bt } =
(1 − pi ) ∩ pi−1 − (1 − pi−1 ) ∩ pi ∈ K0 (J)
i=1
!
" $ % pi pi−1 ∈ pi Npi−1 (pi &pi−1 )&'
) ) * ) t ∈ [0, 1]
{Bt }
# i ∈ {1, . . . , n}
(
(
" {Bt } (
{Ct } Bt − Ct ∈ J
(
p, q ∈ N π(p) − π(q) < 1
qp ∈ qNp (q p) ! " (1 − q) ∩ p ∈ J (1 − p) ∩ q ∈ J #
!
"
π(pqp) − π(p) ≤ π(p) − π(q) < 1
π(p)
π(pqp) π(pNp)+ T ∈ pNp π(T pqp) =
, "
π(qpq) − π(q) ≤ π(q) − π(q) < 1
π(qpq) π(qNq)+ R ∈ qNq π(qpqR) =
qp (q &p)&'
π(q)
)
{B } F
t
sa
0 = t0 < t1 < . . . < tn = 1 t, s ∈ [ti−1 , ti ]
π χ(Bt ) − π χ(Bs ) < 1/2
KK
i ∈ {1, . . . , n}
{Bt }
=
n
(pi pi−1 )
(pi pi−1 ) (pi pi−1 )
i=1
pi = χ(Bti )
{Bt }
i ∈ {0, . . . , n}
= (pn p0 ) (pn . . . p0 ) = [N(pn . . . p0 ) ∩ p0 ] − [N(p0 . . . pn ) ∩ pn ]
p, q, r
π(p) − π(q) < 1/2 ,
π(q) − π(r) < 1/2
N
π(r) − π(p) < 1/2
(rq) (rq) + (q p) (qp)
= (rp) (rp)
! (r q) (rq) + (q p) (qp) − (r p) (rp)
=0
(r r) (rqpr)
=0
π(rqpr) − π(r) ≤ π(qp) − π(r)
≤ π(qp) − π(q) + π(q) − π(r)
≤ π(p) − π(q) + π(q) − π(r)
<1
t ∈ [0, 1]
π (1 − t)rqpr + tr − π(r) = (1 − t)π(rqpr) − π(r) < (1 − t)
π (1 − t)rqpr + tr π(rNr) t ∈ [0, 1] t → (1 − t)rqpr + tr (r r) ! rqpr r " # $
0 = (rr) (r) = (rr) (rqpr)
% &'
"
{Bt }
{Ct }
J #
"
H : [0, 1] × [0, 1] → Fsa {Bt } {Ct } H H(t, 0) = Bt $ H(t, 1) = Ct t ∈ [0, 1] H(0, s) = B0 $
H(1, s) = B1 s ∈ [0, 1] % B0 = C0 B1 = C1 {Bt } = {Ct }
½
ζ : [0, 1] × [0, 1] → N/J ζ(t, s) = π χ H(t, s)
0 = t0 < t1 . . . < tn = 1 ,
0 = s0 < s1 . . . < sn = 1
[0, 1] × [0, 1] (t, s), (u, v) ∈ [ti−1, ti ] × [sj−1, sj ] ζ(t, s) − ζ(u, v) < 12
i, j ∈ {1, . . . , n} i, j ∈ {1, . . . , n}
J ! ! u → H (1 − u)ti−1 + uti , sj−1
H (ti−1 , sj−1), (ti, sj−1)
" # " $% H (ti−1, sj−1), (ti, sj−1) + H (ti, sj−1), (ti, sj )
+ H (ti , sj ), (ti−1 , sj ) + H (ti−1 , sj ), (ti−1 , sj−1) = 0
!
sfH (ti−1 , sj−1 ), (ti , sj−1) = −sfH (ti , sj−1), (ti−1 , sj−1)
# p, q ∈ N p − q < 1 &(p) ∩ '(q) = 0 = &(q) ∩ '(p)
J '(p q)(pq) = [(1 − p) ∩ q] − [(1 − q) ∩ p] = 0
1 − p + pqp N p − pqp ≤ p − q < 1
x &(q) ∩ '(p) 1 − p + pqp
(1 − p + pqp)x = 0
&(q) ∩ '(p)
x = 0
&(p) ∩ '(q) = 0 p q
= 0
Ct ∈ J
{Bt} {Ct} J !
Bt −
t ∈ [0, 1] '(p
0 q0 )
(p0 q0 ) = '(q1 p1 ) (q1 p1 ) = 0
p0 = χ(B0 ) p1 = χ(B1) q0 =
($))* χ(C0 )
χ(B0 ) − χ(C0 ) < 1
q1 = χ(C1 )
{Bt}
χ(C1 ) − χ(B1 ) < 1
=
{Ct} KK
0 = t0 < t1 < . . . < tn = 1 1
π χ(Bt ) − π χ(Bs ) <
4
1
π χ(Ct ) − π χ(Cs ) <
4
t, s ∈ [ti−1 , ti ] i ∈ {1, . . . , n}
Bt Ct i ∈ {0, . . . , n} (BC)i
Ct Bt (CB)i
J i
i
i
i
π (1 − t)Bti + tCti = π(Bti )
t ∈ [0, 1] i ∈ {1, . . . , n}
Cti−1 ←−−− Cti
⏐
⏐
⏐
(CB)i−1 (BC)i ⏐
Bti−1 −−−→ Bti
!
(BC)0 (BC)1 " #$ sf {Bt} = sf {Ct}
& (A, H, D) (N, J) #%
∗ A N H ! ! J ! !" D #! N '()
'%)
[D, a] !$! %(D) ! &! !! H a ∈ A
a(λ − D)−1 ∈ J λ ∈
/ R ! a ∈ A
' J ! N
A
* (A, H, D) J N
FD := D(1 + D 2 )−1/2 .
" t → At N + t → Dt := D + At
J 1
t → FDt = Dt (1 + Dt2 )− 2
½
t ∈ [0, 1]
J x → x(1 + x2 )−1/2 .
t, s ∈ [0, 1]
J FDt − FDs = Dt (1 + Dt2 )
− 21
− Ds (1 + Ds2 )
− 12
!
≤ At − As " J Bε ∈ N
t ∈ [0, 1]
#
2.7
0 < ε < 1/4
FDt − FD0 = Bε (1 + D02 )−(1/2−ε)
B ≤ C()At − A0 ε = 1/4 "
$ # % FDt − FD0 = B1/4 (1 + D02 )−1/4 .
√
x2
x
! " f (x) = 1 +
+
x2 + 4
2
2
(1 + D02 )−1 ≤ f (A0 )(1 + D 2 )−1 ∈ J,
f (A0 )
(1 + D02 )−1/4 ≤ f (A0 )1/4 (1 + D 2 )−1/4 ∈ J.
B1/4
N
π(FD0 ) = π(FDt ).
t ∈ [0, 1]
π(FDt )π(FDt ) = π Dt2 (1 + Dt2 )−1 = π (1 + Dt2 )(1 + Dt2 )−1 = π(1)
π(FDt )
t ∈ [0, 1]
! & '(
) {At}t∈[0,1] N (A, H, D) (N, J) J t → D+At
{Dt } := {FDt }
{At}t∈[0,1] N (N, J) p1 = χ(FD+A1 ) p0 = χ(FD+A0 ).
t → D + At D + A0 D + A1
(A, H, D)
{Dt } = {FDt } = [(1 − p1 ) ∩ p0 ] − [(1 − p0 ) ∩ p1 ]
= *(p1 p0 ) (p1 p0 ) ∈ K0 (J).
KK
π χ(FDt ) − π χ(FDs ) = χ π(FDt ) − χ π(FDs ) = 0
s, t ∈ [0, 1] {Dt } = {FD } = (1 − p1 ) ∩ p0 − (1 − p0 ) ∩ p1
t
(A, H, D) (N, J)
J σ (MA, FD ) ∈ E(A, J) MA :
A → L(J) A pF = F 2+1 [MA , pF ]1 ∈ KK 1 (A, J) ! u ∈ A D u∗Du [u]⊗ˆ A[MA, pF ]1 " "
D
# $ (A, H, D) (N, J) % J σ
& J '
J ( x, y = x∗ y J % A A N FD ∈ N FD ∈ L(J) ∗
MA : A → L(J) a ∈ A [FD , a] a(1 − FD2 ) a(FD − FD∗ ) J (MA , FD ) AJ ! FD = FD∗ $ a ∈ A ) J
aFD2 = a D 2 (1 + D 2 )−1 ∼ a D 2 (1 + D 2 )−1 + (1 + D 2 )−1 = a
% a(FD2 − 1) ∈ J a ∈ A
$ a, b ∈ A ! [FD , a]b = D (1 + D 2 )−1/2 , a b + [D, a](1 + D 2 )−1/2 b
* [D, a] ∈ N +#, 456- & [D, a](1 + D 2 )−1/2 b ∈ J
D (1 + D 2 )−1/2 , a b ∈ J
+., 8-
2 −1/2
(1 + D )
1
=
π
0
∞
λ−1/2 (1 + D 2 + λ)−1 dλ.
½
(1 + D2 + λ)−1 R(λ) 1
π
0
∞
λ−1/2 D R(λ), a b dλ
D (1 + D 2 )−1/2 , a b.
D R(λ), a b = DR(λ)[a, D 2 ]R(λ)b
= DR(λ)[a, D]DR(λ)b + DR(λ)D[a, D]R(λ)b.
1
!" R(λ) = (1 + D2 + λ)−1 ≤ 1+λ
1
#" DR(λ) = D(1 + D2 + λ)−1 ≤ 2√1+λ
$" D2R(λ) = D2(1 + D2 + λ)−1 ≤ 1
λ ∈ [0, ∞) 1
π
0
∞
λ
−1/2
1
D R(λ), a b dλ ≤ b[a, D]
π
1
D (1 + D 2 )−1/2 , a b =
π
0
∞
0
∞
λ
−1/2
1
1
+
4(1 + λ) 1 + λ
dλ < ∞.
λ−1/2 D R(λ), a b dλ
D R(λ), a b = DR(λ)ab − [D, a]R(λ)b − aDR(λ)b
= DR(λ)1/2 R(λ)1/2 ab − [D, a]R(λ)b − aDR(λ)1/2 R(λ)1/2 b ∈ J
λ ∈ [0, ∞) J D (1 + D 2 )−1/2 , a b ∈ J
[FD , a]b ∈ J a, b ∈ A % & [FD , a]b ∈ J a, b ∈ A =
A
' %( ) * ! +
,# K1 (A) - ' (A, H, D) .
(N, J) / u ∈ A t → Dt := (1 − t)D + tu∗ Du = D + t[u∗ , D]u
t → t[u∗, D]u ( N - Dt Dt → FD t
KK
(A, H, D) (N, J) p = χ(FD )
u ∈ A up − pu ∈ J FD FD = (2p − 1)|FD |.
FD π(FD ) = π(2p − 1)π(|FD |) = π(2p − 1)π(FD2 )1/2 = π(2p − 1).
2[u, p] − [u, FD ] = [u, (2p − 1) − FD ] ∈ J.
[u, FD ] ∈ J [u, p] ∈ J (A, H, D) (N, J)
u ∈ A t → Dt {Dt} = {FD } = ∂ π(pup) + π(1 − p)
= (p p) (pup)
t
p = χ(FD ) D u∗ Du (D, u∗Du)
{FD } =
t
#
(1 − u∗ pu) ∩ p − (1 − p) ∩ u∗ pu
χ(Fu∗ Du ) = χ(u∗ FD u) = u∗ χ(FD )u = u∗ pu
!
x ∈ "(u∗ pu) ∩ (p) ⇔ px = x
pux = 0 ⇔ x ∈ "(pup) ∩ (p)
x ∈ "(p) ∩ (u∗ pu) ⇔ px = 0 ux = pux
⇔ pu∗ pux = 0 ux = pux ⇔ ux ∈ "(pu∗ p) ∩ (p).
(1 − u∗ pu) ∩ p = N(pup) ∩ p
u (1 − p) ∩ u∗pu u∗ = N(pu∗ p) ∩ p
N(pup + 1 − p) = N(pup) ∩ p N(pu∗ p + 1 − p) = N(pu∗ p) ∩ p
% & # $
N(pup) ∩ p − N(pu∗ p) ∩ p = ∂ π(pup) + π(1 − p)
π(pup) + π(1 − p) # N/J pu − up ∈ J {FD } =
t
pF = F 2+1 D
(D, u∗Du) = ∂[π(pF upF + 1 − pF )]
$ % π(2p − 1)
= π(FD )
π(p) = π(pF )
½
D u∗Du # ∂ : K1 (N/J)
→ K0 (J)
∂J⊗K : K
1 C(J
⊗ K) → K0 (J ⊗ K) ∂J : K1 C(J) → K0 (J)
C ∗ B C(B) ! L(B)/B π : N → N/J πJ⊗K : L(J ⊗ K) → C(J ⊗ K)
"! πJ : L(J) → C(J)
J σ C ∗ A = A
[D] = [MA , pF ]1 KK 1 (A, J) (MA , FD ) ∈ E(A, J) pF = FD2+1 u ∈ A [u] K1 (A) (D, u∗Du) = ∂[π(pF upF + 1 − pF )] = [u]⊗ˆ A[D]
$ K1 (A) ∼
= K1 (A ⊗ K) KK 1 (A, J) ∼
=
KK 1 (A ⊗ K, J ⊗ K) % [u ⊗ e11 + e] [MA⊗K , pF ⊗ 1]1
K1(A ⊗ K) KK 1 (A ⊗ K, J ⊗ K) e11 & K e = 1 − 1 ⊗ e11 ' ()* 7.1.9+ (, 17.8.8+ -. ˆ A [D] = ∂J⊗K πJ⊗K pF ⊗ 1(u ⊗ e11 + e)pF ⊗ 1 + 1 − pF ⊗ 1
[u]⊗
= ∂J⊗K πJ⊗K (pF upF ) ⊗ e11 + pF ⊗ 1 − pF ⊗ e11 + 1 − pF ⊗ 1
= ∂J⊗K πJ⊗K (pF upF + 1 − pF ) ⊗ e11 + e
K0(J ⊗ K) / πJ (p2F − pF ) = 0 [MA, pF ]1 ∈ KK 1 (A, J) A 0 ∂J [πJ (pF upF + 1 − pF )] ∈ K0(J) K0 (J) K0 (J ⊗ K) (*1 "
4.2.4+ 2 ∂J [πJ (pF upF + 1 − pF )] = ∂[π(pF upF + 1 − pF )]
x ∈ N π(pF upF + 1 − pF ) ∈ N/J N J
N ⊆ L(J) x ∈ L(J) πJ (pF upF + 1 −
pF ) ∈ C(J) / (*1
3 4.8.10+ 4
C∗
5 ! σ J 6 6
σ K R KK
J
σ C ∗ B
KK (N, J)
L(B)
A=A
B⊆J
(A, H, D)
A
A
σ C ∗ (MA , FD ) ∈ E(A, B)
A B C ∗ MA : A →
B ! B "
x, y = x∗ y x, y ∈ B B F +1
# [MA , pF ]1 ∈ KK 1 (A, B) [DB ] pF = D2 ∂B : K1 C(B) → K0 (B) C(B) $% L(B)/B πB : L(B) → C(B) "
(A, H, D) B ⊆ J ⊆ N C ∗ B (D, u
∗
C
Du) = ∂B [πB (pF upF + 1 − pF )] ∈ K0 (B)
# & ∗
[DB ]
& ' (A, H, D) (N, J) B ⊆ J σ C ∗ (MA , FD ) ∈ E(A, B) MA : A → L(B) u ∈ A C ∗ D u∗ Du [DB ] ∈
KK 1 (A, B) [u] ∈ K1 (A) ˆ A [DB ] = ∂B [πB (pF upF + 1 − pF )] = B (D, u∗Du)
[u]⊗
# # (
C ∗ (MA , FD ) & AB # )
* B
∗
σ C ∗ (A, H, D) (N, J) B C L(H) FD [FD , a] b[FD , a]
FD b[FD , a] aϕ(D)
a, b ∈ A ϕ ∈ C0 (R) B ! J (MA , FD ) "
AB # B σ
A # C ∗ C0 (R) x → (i + x)−1 (i + D)−1 J aϕ(D) J ϕ ∈ C0 (R) , # ( [FD , a] ∈ J B J B ⊆ J - B σ + A
B
.
1−
FD2
2 −1
= (1 + D )
∈B
FD ϕ(D) ∈ B
FD
L(B)
½
ϕ ∈ C0 (R)
(MA , FD )
AB
C ∗ (A, H, D)
C ∗ !! (N, J)
u ∈ A
"
(D, u∗Du) = ∂[π(pF upF + 1 − pF )] ∈ K0 (J)
B
σ C ∗ J
# $
B
C ∗ (MA , FD ) AB "
(D, u∗Du) = ∂B [πB (pF upF + 1 − pF )] ∈ K0 (B)
% K0 (J)
i∗ : K0 (B) → K0 (J)
i : B → J
(A, H, D) (N, J) B
σ C ∗ J (MA, FD ) ∈ E(A, B) MA : A → L(B) B L(H) ∗
i : L(B) → L(H)
i(T )(bx) = (T b)x T ∈ L(B) b ∈ B x ∈ H i
B ⊆ L(H) L(B) N
(MA , FD ) AB A ' 1 − FD2 =
(1 + D 2 )−1 ∈ B % (1 + D 2 )−1 ∈ L(H) D 2 H %
B H i % ' ()*' 2.1+' " L(B) ,
∗
&
i : L(B) → L(H)
i(T )(bx) = (T b)x
S ∈ B y ∈ H T ∈ L(B)
T ∈ L(B)' b ∈ B
x∈H
& x ∈ H
x = by
b∈B
i(T )Sby = i(T )bSy = (T b)Sy = S(T b)y = Si(T )by
i(T )S = Si(T )
H
i(T ) ∈ B ⊆ N = N (A, H, D) (N, J) B σ C ∗ J (MA, FD ) AB KK
C ∗
i∗ : K0 (B) → K0 (J) u ∈ A (D, u
∗
Du) = i∗
B (D, u
∗
Du)
i:B→J
i : L(B) → N
i : C(B) → N/J
x ∈ L(B) πB (pF upF + 1 − pF ) ∈ C(B) i(x) ∈ N
π(pF upF + 1 − pF ) ∈ N/J !" # 4.8.10$ i∗ ∂B πB (pF upF + 1 − pF )
1 0
xx∗
x(1 − x∗ x)1/2
−
= i∗
0 0
x∗ (1 − xx∗ )1/2
1 − x∗ x
1/2 i(x)i(x)∗
1 0
i(x) 1 − i(x)∗ i(x)
=
−
1/2
0 0
i(x)∗ 1 − i(x)i(x)∗
1 − i(x)∗ i(x)
= ∂[π(pF upF + 1 − pF )]
% (A,
H, D) (N, J) ∗
B σ
C J (MA, FD ) AB u ∈ A D u∗Du ! [DB ] ∈ KK 1(A, B) [u] ∈ K1(A) u ∈ A
(D, u
&
∗
& & '
(
) * * +
. ˆ A [DB ])
Du) = i∗ ([u]⊗
N
!", '' '-$
+ / 0 τ 1 ∗ + F ⊆ N F + ∗ + + N F N % F F + x + λId x ∈ F λ ∈ C
FN ∗
N " p # τ (p) < ∞ $ !"- 2 "
3$ FN N FN τ
KN (A, H, D) (N, τ ) (A, H, D) (N, KN ) K0 (KN ) ! τ∗ : K0 (KN ) → R " τ #
" " $
$ 2.1
n ∈ N {x1 , . . . , xm } ⊆ Mn (FN ) p ∈ Mn (FN ) pxi = xi i ∈ {1, . . . , m} p {x1 , . . . , xm }
" %
{p1 , . . . , pm} FN & sup{p1 , . . . , pm } ≤
p1 + . . . + pm sup{p1 , . . . , pm } ∈ FN " i ∈ {1, . . . , m} pi ≤ sup{p1 , . . . , pm } ! sup{p1 , . . . , pm }pi = pi sup{p1 , . . . , pm } "
{p1 , . . . , pm } " FN FN #
" ! %
n ∈ N # " {x1 , . . . , xm} ⊆ Mn (FN ) ' %
kl
kl
p ∈ FN pxkl
i = xi " i ∈ {1, . . . , m} k, l ∈ {1, . . . n} xi # ! k l !(p, . . . p)xi = xi " i ∈ {1, . . . , m} n ∈ N ∗ FN x ∈ Mn (FN ) f x Mn (KN ) f (0) = 0 f (x) ∈ Mn (FN ) FN
C ∗ KN C ∗ K ( )
3 i : FN → KN i∗ : K0 (FN ) → K0 (KN )
* & " +, )
4 f (x) ∈ Mn (KN )
Mn (KN ) C ∗ -! " γ ! " x Mn (KN ) ! 0 1
f (x) =
2πi
γ
f (λ)(λ − x)−1 dλ
p " x λ " x $ (1 − p) = (1 − p)(x − λ)(x − λ)−1 = −λ(1 − p)(x − λ)−1 .
" λ = 0 1
(1 − p)(x − λ)−1 = − (1 − p)
λ
KK
1
(1 − p)f (x) =
f (λ)(1 − p)(λ − x)−1 dλ
2πi γ
−f (λ)
1
(1 − p)dλ
=
2πi γ λ
= (1 − p)f (0) = 0
pf (x) = f (x) Mn (FN ) Mn(KN ) p ∈ Mn(FN ) f (x) ∈ Mn(FN ) τ∗ : K0(KN ) → R τ∗ [x + λId] − [y + μId] = τn (x) − τn (y)
x + λId, y + μId ∈ Mn(FN+) [λ] = [μ] K0 (C) τn = τ ⊗ FN ⊗ Mn (C) = Mn (FN ) Mn (C)
τ̂ : FN+ → R τ̂ (x + λId) = τ (x) τ̂ τ̂ (u∗xu) = τ̂ (x)
u ∈ FN+ u = v + αId v ∈ FN α ∈ C αα = 1 v ∗ v + v ∗ α + vα = 0 = vv ∗ + v ∗ α + vα
(v ∗ + αId)(x + λId)(v + αId) = (v ∗ xv + v ∗ xα + v ∗ λv + v ∗ λα + αxv + x + αλv + λId)
= (v ∗ xv + v ∗ xα + αxv + x + λId)
τ̂ τ̂ (v ∗ + αId)(x + λId)(v + αId) = τ (v ∗ xv + v ∗ xα + αxv + x) = τ (x)
τ∗ : K0 (FN+) → R τ̂ ([x + λId] −
[y + μId]) = τn (x) − τn (y) (x + λId), (y + μId) ∈ Mn (FN+ ) !
K0 (FN ) " π∗ : K0 (FN+ ) → K0 (C) π : FN+ → C # $%
p Mn (KN ) p ∈ Mn(FN )
! Mn (FN ) Mn (KN ) e ∈ Mn (FN ) 1
e − p < 24
e < 2 e2 − e ≤ e(e − p) + (e − p)p + p − e <
1
4
e 1/2 ∈/ !(e) e ε > 0 !(e) ⊆ [0, 1/2 − ε] ∪ [1/2 + ε, 5/4]
f : R/{ 12 } → R f (t) =
0
1
t<
t>
1
2
1
2
(e)
Mn (FN )
f (0) = 0
f (e)
sup{|f (t) − t| | t ∈ (e)} ≤ sup{1/2 − ε, 1/4}
f (e) − e <
1
2
p − f (e) ≤ p − e + e − f (e) < 1
+
p f (e) ! u Mn (KN
) u f (e)u = p "#$ % 4.1.7& '
Mn (FN )
Mn (N)
∗
"(( (&
{Bt } N π(Bt) ∈
0 = t0 < t1 < . . . < tn = 1 [0, 1] N/KN t
i ∈ {1 . . . , n} π χ(Bt ) − π χ(Bs ) < 1/2
t, s ∈ [ti−1 , ti ] KN {Bt} n τ N(pi ) ∩ pi−1 − τ N(pi−1 ) ∩ pi
{Bt } =
i=1
pi = χ(Bti )
{B } t
{Bt } = τ N(pn . . . p0 ) ∩ p0 − τ N(p0 . . . pn ) ∩ pn
! {Bt} " #
τ∗ : K0 (KN ) → R
)
$ * + , - p ∈ KN
'
.
(A, H, D) (N, τ ) $
A = A A C ∗ % & {At} N t → D + At :=
Dt {Dt } := {FDt }
", - (( (&
/ 0
1 (A, H, D) (N, τ ) $
A = A A C ∗ % u ∈ A $
Dt = (1 − t)D + tu∗ Du = D + tu∗ [D, u] t → Dt {Dt } = τ ∂[π(pup + 1 − p)] = τ N(pup + 1 − p) − τ N(pu∗ p + 1 − p)
KK
τ∗ : K0 (KN ) → R p = χ(FD ) C ∗ B ⊆ KN [DB ] ∈ KK 1 (A, B) {Dt }
ˆ A [DB ])
= τ∗ i∗ ([u]⊗
i : B → KN [u] ∈ K1 (A) τ∗ : K0 (KN ) → R
KN
!
KK "
#
$
%&!' $
(
) * $
* KK "
) A B Z C AB ∗
ψ : A → L(E)! E !
" B ! V ∈ L(E) 2
(ψ, V )
+&,
+-,
+,
∗
ψ(a)(V 2 − 1) ∈ K(E)
ψ(a)(V − V ∗ ) ∈ K(E)
[V, ψ(a)] ∈ K(E)
a ∈ A AB E(A, B) (ψ, V ) ∈
E(A, B) a(V 2 − 1) = a(V − V ∗ ) = [V, a] = 0
E(A, B)
KK "
KK(A, B) ∼oh $
. $
∼u [ψ, V ] ∈ KK(A, B) % / 17.3.3'
$
$
(ψ, V ) ∈ E(A, B)
KK "
0 C1 C ∗ "
ˆ 1 ) C⊕C KK 1 (A, B) = KK(A, B ⊗C
ˆ
⊗ % 14.4'
1 C
∗
C ∗ "
A B KK "
*
"
2
% / 17.6.5' 3*
−1
(A, B ⊗ K) ∼
= KK 1 (A, B ⊗ K)
∗
ψ : A → L(B ⊗ K) p ∈
L(B ⊗K) # τ : a → π pψ(a)p ∈ C(B ⊗K) ˆ (2p − 1)⊗ε
ˆ ∈ E A, (B ⊗ K)⊗C
ˆ 1
A (B ⊗ K) ψ ⊗1,
ε = (1, −1) ∈ C1
ˆ : A → L (B ⊗ K)⊗C
ˆ 1
ψ ⊗1
ˆ (2p − 1)⊗ε]
ˆ ∈
[ψ, p] [ψ ⊗1,
ˆ
KK (A, B ⊗ K) = KK A, (B ⊗ K)⊗C ψ : A → L(B ⊗ K) p ∈ L(B ⊗ K)
ψ(a)(p − p) ∈ B ⊗ K
ψ(a)(p − p ) ∈ B ⊗ K
[p, ψ(a)] ∈ B ⊗ K
a ∈ A
A B D C A D B
σ ! KK "
ˆ : KK (D, A) × KK (A, B) → KK (D, B)
⊗
#
" K (A) = KK (C, A) KK (A, B) $ %
18&
' ( K K KK 1
1
1
2
∗
∗
i
A
j
1
1
A
i+j
C∗
1
KK 1 (C, A ⊗ K)
K1 (A ⊗ K)
[MC , p]1 → ∂[π(p)]
MC : C → L(A ⊗ K) π : L(A ⊗ K) →
C(A ⊗ K) ∂ : K0 C(A ⊗ K) → K1 (A ⊗ K) 17.5.7 $
&
)
∂ [u] ∈ K1 (A⊗K) q ≤ 1 [u] = [exp(2πiq)]
12.2.2 $* )
q ∈ L(A ⊗ K)
! KK(C, A ⊗ K)
K0 (A ⊗ K)
&
[MC , V ] → ∂[π(T )]
MC : C → L (A ⊗ K) ⊕ (A ⊗ K) V ∈ L (A ⊗ K) ⊕ (A ⊗ K) 0 T∗
V =
T 0
∂ : K1 C(A ⊗ K) → K0 (A ⊗ K) "
(A ⊗ K) ⊕
0 1
(A ⊗ K) γ =
17.5.5 1 0
$ ) &
A B D C A D B σ #
+
(ψ , V ) ∈ E(D, A) ψ : D → L(E ) (ψ , V ) ∈ E(A, B) ψ : A → L(E )
∗
1
1
1
2
2
2
2
1
KK
ˆ ψ2 E2 14.4 x ∈ E1
E = E1 ⊗
ˆ y ∈ E2 4.6
Tx ∈ L(E2 , E) Tx (y) = x⊗y
F ∈ L(E) V2
E1 x ∈ E1 ∂x Tx V2 − (−1)∂x F Tx ∈ K(E2 , E)
x E1 !
" # C ∗ $
D
% " &
18.10.1 x = (ψ1 , V1 ) ∈ E(D, A) V1 = V1∗ V1 ≤ 1
ˆ :
y = (ψ2 , V2) ∈ E(A, B) F V2
E1 E = E1⊗ˆ ψ2 E2 ψ = ψ1 ⊗1
A → L(E) ˆ + (1 − V12 )1/2 ⊗1
ˆ F
V = V1 ⊗1
ˆ ψ(a)] ∈ K(E) a ∈ A z = (ψ, V ) ∈ E(D, B) [V1⊗1,
x y [x]⊗ˆ A[y] = [z] KK(D, B)
' KK 1
& ' " ( A B C ∗
ˆ 1 , B ⊗ K)
ˆ 1 → KK(A⊗C
ϕ : KK 1 (A, B ⊗ K) = KK A, (B ⊗ K)⊗C
0
2p − 1
2p − 1
0
ˆ 1 → L (B ⊗ K) ⊕ (B ⊗ K)
∗
σ : A⊗C
0
−iψ(a)
σ(a, −a) =
iψ(a)
0
ϕ[ψ, p] = σ,
1
σ(a, a) =
ψ(a)
0
0
ψ(a)
(B ⊗ K) ⊕ (B ⊗ K) γ =
*
K KK 1
1
' 0 1
1 0
)
' *)
A B D C A D B
σ ∗
[x] KK 1 (D, A ⊗ K) ! [x] ˆ (2q − 1)⊗ε)
ˆ ∈ E D, (A ⊗ K)⊗C
ˆ 1
x = (ψ1 ⊗1,
ˆ 1 17.4.3 ˆ : D → L(E1 ) E1 = (A ⊗ K)⊗C
ψ1 ⊗1
∗
q = q q ≤ 1
ˆ 1, B ∼
[y] KK (A ⊗ K)⊗C
= KK 1 (A ⊗ K, B) [y] ˆ 1, B
y = (ψ2 , V2 ) ∈ E (A ⊗ K)⊗C
ˆ 1 → L(E2 )
ψ2 : (A ⊗ K)⊗C
ˆ ⊗1
ˆ F
E = E1 ⊗ˆ ψ E2 ψ = (ψ1 ⊗1)
2
∈ L(E) V2 E1 ˆ ⊗1
ˆ F ∈ L(E)
ˆ ⊗1
ˆ + sin(πq)⊗1
V = − cos(πq)⊗ε
ˆ ⊗1,
ˆ ψ(d) ∈ K(E)
cos(πq)⊗ε
d ∈ D (ψ, V ) DB x y [ψ, V ] = [x]⊗ˆ A⊗K [y]
!
d ∈ D
ψ1 (d)(q 2 − q) ∈ A ⊗ K A ⊗ K
∞
(πq)2k
+ ψ1 (d)
ψ1 (d) cos(πq) = ψ1 (d)
(−1)k
(2k)!
k=1
∞
π 2k
q + ψ1 (d)
(2k)!
k=1
= ψ1 (d) cos(π)q − q + 1
∼ ψ1 (d)
(−1)k
= −ψ1 (d)(2q − 1)
d ∈ D
x ˆ − cos(πq)⊗ε
ˆ ∈ E D, (A ⊗ K)⊗C
ˆ 1
(ψ1 ⊗1,
ˆ 1 KK D, (A ⊗ K)⊗C
! "# 1/2
ˆ
ˆ 2
= sin(πq)⊗1
1 − cos(πq)⊗ε
ˆ sin(πq)⊗1
q ≤ 1
q = q∗
(q) ⊆ [−1, 1]
$
A B " C ∗ " A B σ
[u] ∈ K1 (A ⊗ K) # [u] [MC , q]1 ∈
KK 1 (C, A ⊗ K) q = q ∗ q ≤ 1 $ [u] = [exp(2πiq)] "
u = exp(2πiq) y = [ψ, p]1 ∈ KK 1 (A ⊗ K, B ⊗ K) ψ : A ⊗ K → L(B ⊗ K) " [u]⊗ˆ A⊗K y % ψ & ∂ π pψ(u)p + (1 − p) ∈ K0 (B ⊗ K)
(A ⊗ K)+ ∂ : K1 C(B ⊗ K) → K0 (B ⊗ K)
KK
ˆ 1, B ⊗ K
ϕ : KK 1 (A ⊗ K, B ⊗ K) → KK (A ⊗ K)⊗C
y ϕy = ϕ[ψ, p]1 = [σ, V2 ]
V2 =
0
2p − 1
2p − 1
0
ˆ 1→L
σ : (A ⊗ K)⊗C
σ(a, −a) =
0
−iψ(a)
iψ(a)
0
(B ⊗ K) ⊕ (B ⊗ K)
ψ(a)
0
σ(a, a) =
0
ψ(a)
ˆ 1, B ⊗ K y KK (A ⊗ K)⊗C
ˆ 1
[MC, q]1 ∈ KK 1 (C, A ⊗ K) [x] ∈ KK C, (A ⊗ K)⊗C
ˆ ∈ E C, (A ⊗ K)⊗C
ˆ 1
x = MC , (2q − 1)⊗ε
ˆ 1
z = [u]⊗ˆ A⊗K y = [x]⊗ˆ A⊗K ϕy E1 = (A⊗K)⊗C
ˆ σ E2 (B ⊗ K) ⊕ (B ⊗ K) = E2
E2 = (B ⊗ K) ⊕ (B ⊗ K) E = E1 ⊗
0 1
γ = 1 0 ψ σ ˆ σ (B ⊗ K) ⊕ (B ⊗ K)
ˆ 1 ⊗
ˆ σ E2 = (A ⊗ K)⊗C
w : E1 ⊗
→ (B ⊗ K) ⊕ (B ⊗ K) = E2
ˆ σ x2 ) = σ(x1 )x2
w(x1 ⊗
x1 ∈ E1 , x2 ∈ E2
x ∈ E1 wTx = σ(x) wTx V2 − (−1)∂x V2 wTx = σ(x), V2 ∈ K(E2 )
! w∗V2w ∈ E V2
E1 " !
z
(MC , V ) ∗
ˆ
ˆ
ˆ
ˆ w V2 w ∈ L(E)
V = − cos(πq)⊗ε ⊗1 + sin(πq)⊗1 ⊗1
" (MC, V ) ∼u (MC , wV w∗) z = [MC , wV w ∗ ] ∈ KK(C, B ⊗ K)
ˆ + σ sin(πq)⊗1
ˆ V2
wV w ∗ = −σ cos(πq)⊗ε
0
−iψ
cos(πq)
ψ
sin(πq)
0
V2
=−
+
iψ cos(πq)
0
0
ψ sin(πq)
0
iψ cos(πq) + ψ sin(πq) (2p − 1)
=
−iψ cos(πq) + ψ sin(πq) (2p − 1)
0
∈ L (B ⊗ K) ⊕ (B ⊗ K)
ˆ 1 → L(B ⊗ K) ψ : L(A ⊗ K) → L(B ⊗ K) σ : L (A ⊗ K)⊗C
2.1
KK(C, B ⊗ K) ∼
= K0 (B ⊗ K) !
∂ π − iψ cos(πq) + ψ sin(πq) (2p − 1) ∈ K0 (B ⊗ K)
"
v = i exp(iπq) = i cos(πq) − sin(πq) # v L(A ⊗ K) 1 ∂ π − iψ cos(πq) + ψ sin(πq) (2p − 1) =
∂ π − iψ v cos(πq) + ψ v sin(πq) (2p − 1)
$
# %
π cos(πq) cos(πq) = π (1 − 2q)(1 − 2q) = π(1)
cos2(πq) − 1 ∈ A ⊗ K &% sin(πq) ≥ 0 q = q∗ q ≤ 1 sin(πq) = 1 −
1/2
cos2 (πq)
∈ A⊗K ' % v cos(πq) ∈ (A⊗K)+ ( ψ(a)p−pψ(a) ∈ B⊗K
a ∈ (A ⊗ K)+ π − iψ v cos(πq) + ψ v sin(πq) (2p − 1)
= π − ipψ v cos(πq) p − i(1 − p)ψ v cos(πq) (1 − p)
+ pψ v sin(πq) p − (1 − p)ψ v sin(πq) (1 − p)
2
= π pψ(−v )p + (1 − p)ψ v[−i cos(πq) − sin(πq)] (1 − p)
= π pψ(u)p + (1 − p)
#
z = ∂ π − iψ cos(πq) + ψ sin(πq) (2p − 1) = ∂ π pψ(u)p + (1 − p)
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