Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
Spectral triples on Cuntz-Pimsner algebras
Magnus Goffeng
joint work with Bram Mesland and Adam Rennie
Chalmers University of Technology and University of Gothenburg
2017–03–10
“KK-theory, Gauge Theory and Topological Phases”,
Lorentz center, Leiden
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
Setting the stage...
Noncommutative geometry is...
Noncommutative differential geometry: spectral triple (A, H, D)
Noncommutative differential topology: index theory for Fredholm modules
(bounded cycles (H, F ) and/or unbounded cycles (H, D))
Noncommutative topology: Kasparov’s KK -theory
Noncommutative spectral geometry: using the laplacian D 2 from a
spectral triple
Finite dimensionality: Weyl laws in Fredholm modules
(i + D)−1 ∈ Lp (H)
p
in the unbounded setting, or
[F , a] ∈ L (H) ∀a ∈ A
Magnus Goffeng
in the bounded setting
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
On a manifold
The prototypical spectral triple of a manifold
From a closed manifold M and a Dirac operator D acting on a Clifford bundle
S → M, one constructs the prototypical example of a spectral triple:
(C ∞ (M), L2 (M, S), D)
The associated bounded transform FD := D(1 + D 2 )−1/2 is an elliptic
pseudo-differential operator of order 0.
In this setting, adding the adjective noncommutative proves to be a more or less
“bijective” operation...
Bounded vs. Unbounded: FD and D are elliptic pseudo-differential operators
with (almost) the same principal symbol providing similar information (FD
determines D up to conformal change of metric and bounded perturbation).
Differential topology: HH∗ (C ∞ (M)) coincides with the vector fields on M and
HP∗ (C ∞ (M)) with de Rham cohomology.
Finite dimensionality: The Weyl law guarantees that
Ψ−1 (M, S) ⊆ ∩p>dim(M) Lp (L2 (M, S)) and
(i + D)−1 , [FD , a] ∈ Ψ−1 (M, S).
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The machine for spectral triples on Cuntz-Pimsner algebras
A – unital C ∗ -algebra
E – a finitely generated projective A-bimodule
OE – the associated Cuntz-Pimsner algebra
The bivariant cycle
If E satisfies additional assumptions, there is a distinguished unbounded
(OE , A)-Kasparov cycle (ΞE , DE ) playing a special role in KK1 (OE , A).
This provides a new testing ground for a large class of nonprototypical
noncommutative geometries!
Spectral triples on OE
If (A, H, D) is a spectral triple for A, for which ΞE admits a suitable
connection ∇Ξ , when we obtain a spectral triple for OE from the operator
D E ⊗ 1H + 1 ⊗ ∇ D
on
ΞE ⊗A H.
Standard problem: equicontinuity.
Possible solutions: twisting (??) or logarithmic dampening (??)
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
Torsion invariants
A special case of a Cuntz-Pimsner algebra is ON – the universal C ∗ -algebra
generated by N isometries S1 , . . . , SN satisfying
N
X
Sj Sj∗ = 1,
so in particular
Si∗ Sj = δij .
j=1
∗
For any C -algebra B, the groups KK∗ (ON+1 , B) and KK∗ (B, ON+1 ) are
N-torsion and there is an index pairing
KK∗ (ON+1 , B) × KK∗ (B, ON+1 ) → KK∗ (ON+1 , ON+1 ) Magnus Goffeng
Z/N Z.
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
1
Introduction
2
Cuntz-Pimsner algebras
3
Spectral triples on ON
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The Cuntz-Pimsner algebra of a module
Standing assumptions
A – unital C ∗ -algebra,
E = A EA – finitely generated projective bi-Hilbertian A-bimodule, with left inner
product A h·, ·i and right inner product h·, ·iA .
The Toeplitz algebra associated with E
⊗A k
FE := ⊕∞
= A ⊕ E ⊕ E ⊗2 ⊕ · · ·
k=0 E
Tξ η := ξ ⊗ η, ξ ∈ E
⊗k
, η ∈ FE ,
– The Fock module of E .
– The Toeplitz operator associated with ξ.
TE := the C ∗ -sub algebra of End∗A (FE ) generated by {Tξ : ξ ∈ E }.
One computes that for ξ ∈ E , Tξ∗ (η1 ⊗ · · · ηk ) = hξ, ηiA η2 ⊗ · · · ⊗ ηk and Tξ∗ (a) = 0.
The Cuntz-Pimsner algebra of E
The Cuntz-Pimsner algebra OE is defined from the short exact sequence
0→
KA (FE ) → TE → OE → 0.
This is called the defining extension, its class in KK1 (OE , A) is denoted by ∂E
(whenever it exists). We write
Sξ Sη∗ := Tξ Tη∗ mod
KA ∈ OE ,
Magnus Goffeng
⊗k .
for ξ, η ∈ ∪∞
k=0 E
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The algebra ON
We consider A =
C and E = CN with the standard inner products h·, ·iC and C h·, ·i.
The Fock space
k
Upon setting VN := ∪∞
k=0 {1, . . . , N} , we can (and will!) identify
∞
M
FCN :=
( N )k ≡ `2 (VN ), via eµ1 ⊗ eµ2 ⊗ · · · eµk 7→ δµ for µ = µ1 µ2 · · · µk .
C
k=0
The Toeplitz operators
To simplify notation, we write Ti := Tei and Tµ := Tµ1 · · · Tµk , so
Tµ∗ Tν = heµ1 ⊗ · · · ⊗ eµk , eν1 ⊗ · · · ⊗ eνk iC 1TE ,
µ = µ 1 · · · µ k , ν = ν1 · · · νk .
So Ti is an isometry. Moreover
N
X
Ti Ti∗ = 1 − δ∅ ⊗ δ∅∗ = 1
Tµ δν = δµν and
mod
K(`2 (VN )).
i=1
The case N = 1
C
N
L
k
2
When N = 1, FC = ∞
k=0 ( ) ≡ ` ( ) and T1 = the unilateral shift. The defining
extension for O1 = OC = C (S 1 ) is the Toeplitz extension
0→
K(`2 (N)) → TC → C (S 1 ) → 0.
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
Further structures on the Cuntz-Pimsner algebra
For ξ ∈ E ⊗k , we write |ξ| := k
Conditional expectations (Rennie-Robertson-Sims)
Under a technical assumption on the Jones-Watatani indices of (E ⊗k )k∈N ,
there exists a unital conditional expecation
Φ∞ : OE → A,
Φ∞ (Sξ Sη∗ ) = δ|µ|,|ν|A hµ, q|ν| νi,
for a positive adjointable operator qk : E ⊗k → E ⊗k .
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
Further structures on the Cuntz-Pimsner algebra, continued
We can equipp OE with an A-valued right inner product
ha, biA := Φ∞ (a∗ b),
and we let ΞE denote the Hilbert C ∗ -module completion.
Decomposing the module ΞE
Under a technical assumption on (qk )k∈ N , the A-module ΞE decomposes into
an orthogonal direct sum of finitely generated projective right A-modules
M
M
ΞE =
Ξn,k ,
Z k≥max(0,−n)
n∈
such that
1
2
Ξn,0 = E ⊗n for n ≥ 0 and
for ξ ∈ E and any k ≥ max(0, −n) ,
(
Ξn+1,k ,
Sξ Ξn,k ⊆
Ξn+1,k ⊕ Ξn+1,k+1 ,
Magnus Goffeng
for n + k > 0, and
for n + k = 0.
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The L2 -space of ON
For A =
qk = N
C, E = C
−k
,
and
N
and ON , the expectation Φ∞ : ON →
Φ∞ (Sµ Sν∗ )
= δµ,ν N
−|ν|
=
φ(Sµ Sν∗ )
C is computable.
– the KMS-state on ON .
We write L2 (ON ) := ΞCN . The C ∗ -algebra generated by
{Sµ Sµ∗ : µ ∈ VN } ⊆ ON is abelian with spectrum
ΩN := {1, . . . , N}N+ = {x = x1 x2 · · · : xi ∈ {1, . . . , N}}.
The KMS-state comes from a measure µΩ on ΩN defined from
µΩ (Cν ) = N −|ν| ,
where
Cν := {x ∈ ΩN : xi = νi , i = 1, . . . , |ν|}.
The Hilbert space Ξn,k for ON
The Hilbert space Ξn,k has dimension N n+2k and admits a basis
(eµ,ν )|ν|=k,|µ|=n+k where
|ν|/2
Sµ Sν∗ ,
t(µ) 6= t(ν),
N
eµ,ν :=
q
N
N |ν|/2
Sµ Sν∗ − N −1 Sµ Sν∗ , t(µ) = t(ν) 6= ∅.
N−1
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
A bivariant cycle on OE
Using Ξn,0 = E ⊗n , we can identify the Fock module FE with a complementable
submodule in ΞE , and let PF : ΞE → FE denote the projection.
For ξ ∈ E ,
Tξ − PF Sξ PF = 0 ∈
K
A (FE ).
Toeplitz operators on ΞE
The pair (ΞE , 2PF − 1) is an (OE , A)-Kasparov module which represents
∂E ∈ KK1 (OE , A) coming from the defining extension
0→
K
The inclusion mapping
OE = TE /
K
A (FE )
→ TE → OE → 0.
A (FE )
,→ End∗A (FE )/
Proof.
K
A (FE ),
coincides with the a priori linear mapping
OE → End∗A (FE )/ A (FE ), a 7→ PF aPF mod
K
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K
A (FE ).
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
A unbounded bivariant cycle on OE
We can define self-adjoint and regular operators c and κ densely on ΞE by
declaring
c|Ξn,k := n and κ|Ξn,k = k.
Note c + κ ≥ 0 and κ ≥ 0.
An unbounded representative of ∂E
The operator
DE := (2PF − 1)|c| − κ : ΞE 99K ΞE ,
is self-adjoint and regular. Moreover, (ΞE , DE ) is an unbounded Kasparov
(OE , A)-cycle representing ∂E .
Proof
DE preserves Ξn,k . Moreover, DE ≥ 0 on Ξn,k if and only if k = 0. Thus
χ(DE ) = 2PF − 1, χ(t) = (t + 1/2)|t + 1/2|−1 .
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The spectral triple on ON
On L2 (ON ), ceµ,ν = (|µ| − |ν|)eµ,ν and κeµ,ν = |ν|eµ,ν . The operator DN := DCN
takes the form
(
|µ|eµ,∅ ,
if ν = ∅
DN eµ,ν :=
−|µ|eµ,ν , if ν 6= ∅.
The case N = 1 and O1 = C (S 1 )
ΞC ∼
= L2 (S 1 ), FC ∼
= H 2 (S 1 ), PF = the Szegö projection and D = −i∂θ .
Summability properties of DN
2
The spectral triple (L2 (ON ), DN ) is θ-summable, that is, e−tDN ∈ L1 (L2 (ON )) for all
t > 0.
The Fredholm module (L2 (ON ), DN |DN |−1 ) is p-summable for all p > 0.
Proof of summability property
DN |DN |−1 = 2PF − 1 (possibly up to a rank one operator) and
Si 7→ Ti = PF Si PF
defines a ∗-homomorphism
algebraic nonsense.
mod ∩p>0 Lp (L2 (ON )),
C∗ [S1 , . . . , SN ] → B(L2 (ON ))/ ∩p>0 Lp (L2 (ON )). Now:
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The groupoid model
There is an explicit isomorphism ON ∼
= C ∗ (GN ) where the etale groupoid
GN ⇒ ΩN is defined from
GN := {(x, n, y ) ∈ ΩN ×
Z×Ω
: ∃k s.t. xj+n = yj ∀j ≥ k}.
N
The groupoid structure comes from
rG (x, n, y ) = x, dG (x, n, y ) = y ,
and
(x, n, y )(y , m, z) = (x, n + m, z).
Under ON ∼
= C (GN ),
∗
L2 (ON ) = L2 (GN , µG ),
where µG := dG∗ µΩ .
Indeed, φ corresponds to µG via
Z
φ(f ) =
f (g )dµG ,
f ∈ Cc (GN ).
G
We let Cc∞ (GN ) denote the algebra of compactly supported locally constant
functions.
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
The geometric Dirac operator
Some integral operators on GN
For a suitable extended metric ρG , we define the “log-Laplacian”
Z
f (g ) − f (h)
Tf (g ) :=
dµG (h).
log(N)
G ρG (g , h)
We also define
Z
Pf (g ) :=
kP (g , h)f (h)dµG (h),
G
where kP is the characteristic function of the set
∪µ∈VN {((x, n, y ), (z, n, w )) : n = |µ|, x, z ∈ Cµ , xn+j = yj , zn+j = wj ∀j}.
Finally, cf (x, n, y ) := nf (x, n, y ).
Spectral triples
It holds that P = PF and T − κ is “small”. The operator
Dgeo := (2P − 1)|c| − T
is self-adjoint and fits into a spectral triple (Cc∞ (GN ), L2 (GN , µG ), Dgeo ) which is
θ-summable and χ[0,∞) (Dgeo ) − P is of finite rank.
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
Introduction
Cuntz-Pimsner algebras
Spectral triples on ON
Thanks
Thanks for your attention!
Some references:
1
M. Goffeng, B. Mesland, Spectral triples and finite summability on
Cuntz-Krieger algebras, Doc. Math., 20 (2015), 89–170.
2
A. Rennie, D. Robertson, A. Sims, The extension class and KMS states for
Cuntz–Pimsner algebras of some bi-Hilbertian bimodules, to appear in the
Journal of Topology and Analysis.
3
M. Goffeng, B. Mesland, A. Rennie, Shift tail equivalence and an
unbounded representative of the Cuntz-Pimsner extension, to appear in
Ergod. Th. Dyn. Sys.
4
F. Arici, A. Rennie, The Cuntz-Pimsner extension and mapping cone exact
sequences, arXiv:1605.08593.
5
M. Goffeng, B. Mesland, Spectral triples on ON , conference proceedings
from the MATRIX-program ”Refining C ∗ -algebraic invariants for dynamics
using KK -theory” in Creswick, Australia, 2016
Magnus Goffeng
Spectral triples on Cuntz-Pimsner algebras
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