Dao of the principal symbol
Dmitriy Zanin
(written in collaboration with Edward McDonald and Fedor Sukochev)
University of New South Wales
29.03.17
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Principal symbol mapping
Traditional definition of principal symbol
Let P be pseudo-differential operator, that is
ˆ
(Px)(t) =
p(t, u)e iht,ui (Fx)(u)du.
Rd
Here, p is some good function on Rd × Rd .
If
X
p=
pn ,
n≥0
where pn (t, u) is the homogeneous function of u of order −n, then the
function p0 is called a principal symbol of a pseudo-differential operator P.
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Principal symbol mapping
Why principal symbol is important?
The most important feature of the principal symbol mapping is that
p0 (PQ) = p0 (P)p0 (Q)
for sufficiently good operators P and Q.
So, principal symbol is a homomorphism.
Question
Which algebra of operators is the natural domain for the principal symbol?
Answer
We construct a principal symbol as a continuous ∗−homomorphism of
C ∗ −algebras.
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Principal symbol mapping
Domain of a principal symbol
Let π1 : L∞ (Rd ) → B(L2 (Rd )) and π2 : L∞ (Sd−1 ) → B(L2 (Rd )) be the
unital ∗−representations given by the formulae
π1 (f ) = Mf ,
π2 (g ) = g (
∇
1
).
(−∆) 2
Let Π be the C ∗ −subalgebra in B(L2 (Rd )) generated by the algebras
π1 (L∞ (Rd )) and π2 (L∞ (Sd−1 )).
¯ ∞ (Sd−1 ) be the weak closure of
In what follows, we denote by L∞ (Rd )⊗L
d
the algebraic tensor product L∞ (R ) ⊗ L∞ (Sd−1 ) with respect to the
¯ ∞ (Sd−1 ) with the algebra
representation π1 ⊗ π2 . We identify L∞ (Rd )⊗L
d
d−1
L∞ (R × S
).
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Principal symbol mapping
Statement of the main result
Our first main result is proved in [DAO 1].
Theorem
There exists a unique norm-continuous ∗−homomorphism
¯ ∞ (Sd−1 ) such that
symb : Π → L∞ (Rd )⊗L
f ∈ L∞ (Rd ),
symb(π1 (f )) = f ⊗ 1,
symb(π2 (g )) = 1 ⊗ g ,
g ∈ L∞ (Sd−1 ).
We call symb a principal symbol mapping.
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Principal symbol mapping
Why not more general results? I
Question
Why consider symb on a C ∗ −algebra Π and not on its von Neumann
closure?
Observe that the closure of Π in weak operator topology is B(L2 (Rd )).
Indeed, let A be in the commutant of Π. Since π1 (L∞ (Rd )) is the maximal
abelian subalgebra in B(L2 (Rd )), it follows that A ∈ π1 (L∞ (Rd )). Since A
also commutes with π2 (L∞ (Sd−1 )), it follows that A is scalar. Thus,
commutant of Π is C. By von Neumann bicommutant theorem, the weak
closure of Π is B(L2 (Rd )).
Observe that the subspace [B(L2 (Rd )), B(L2 (Rd ))] is B(L2 (Rd )). In
particular, B(L2 (Rd )) does not admit a non-trivial character. In particular,
there cannot exist a continuous ∗−homomorphism
symb : B(L2 (Rd )) → L∞ (Rd × Sd−1 ).
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Principal symbol mapping
Why not more general result? II
Question
Why consider only homogeneous functions of the gradient?
Let π3 (h) = h(∇) for every h ∈ L∞ (Rd ). Let Σ be the C ∗ −subalgebra in
B(L2 (Rd )) generated by the algebras π1 (L∞ (Rd )) and π3 (L∞ (Rd )).
Suppose there exists a continuous ∗−homomorphism π : Σ → L∞ (R2d ).
We have
khk2∞ = kh ⊗ hk∞ ≤ kπkkπ1 (h)π3 (h)k∞ ≤
≤ kπk∞ kπ1 (h)π3 (h)k2 = cd kπk∞ khk22 .
This inequality is impossible — take h = χ(0,)d for sufficiently small > 0.
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Principal symbol mapping
Outline of the proof. I
Let P be the ∗−subalgebra in B(L2 (Rd )) generated by the algebras
π1 (L∞ (Rd )) and π2 (L∞ (Sd−1 )). Every element of P can be written as
follows
p
n Y
X
x=
π1 (fk,l )π2 (gk,l ).
k=1 l=1
The key idea is that
σm · x · σ 1 →
p
n Y
X
m
fk,l (0)π2 (gk,l )
k=1 l=1
as m → ∞ in strong operator topology. Here,
(σm ξ)(t) = ξ(
Dmitriy Zanin (University of New South Wales)
t
),
m
t ∈ Rd .
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Principal symbol mapping
Outline of the proof. II
Let x ∈ P be as in the previous slide. We have
k
p
n Y
X
fk,l (0) · gk,l k∞ ≤ kxk∞ .
k=1 l=1
Applying the shift by t ∈ Rd , we obtain that
k
p
n Y
X
fk,l (t) · gk,l k∞ ≤ kxk∞ .
k=1 l=1
Taking the supremum over t ∈ Rd , we obtain
k
p
n Y
X
fk,l ⊗ gk,l k∞ ≤ kxk∞ .
k=1 l=1
Thus, for x ∈ P, we have
ksymb(x)k∞ ≤ kxk∞ .
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Traces on L1,∞
General notations
Fix throughout a separable infinite dimensional Hilbert space H. We let
B(H) denote the algebra of all bounded operators on H. For a compact
operator T on H, let µ(k, T ) denote k−th largest singular value (these are
the eigenvalues of |T |). The sequence µ(T ) = {µ(k, T )}k≥0 is referred to
as to the singular value sequence of the operator T . The standard trace on
B(H) is denoted by Tr.
Fix an orthonormal basis in H (the particular choice of a basis is
inessential). We identify the algebra l∞ of bounded sequences with the
subalgebra of all diagonal operators with respect to the chosen basis. For
a given sequence α ∈ l∞ , we denote the corresponding diagonal operator
by diag(α).
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Traces on L1,∞
Principal ideals Lp,∞
Let Lp,∞ be the principal ideal in B(l2 ) generated by the element
A0 = diag({(k + 1)
− p1
}k≥0 ). Equivalently,
1
Lp,∞ = {A : sup(k + 1) p µ(k, A) < ∞}.
k≥0
In Noncommutative Geometry, a compact operator A is called an
infinitesimal of order p1 if
µ(k, A) = O((k + 1)
− p1
),
k ∈ Z+ .
In other words, Lp,∞ is the set of all infinitesimals of order p1 .
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Traces on L1,∞
Traces on L1,∞
Definition
A linear functional ϕ : L1,∞ → C is called trace if ϕ(AB) = ϕ(BA) for
every A ∈ L1,∞ and for every B ∈ B(H).
There exists a plethora of traces on L1,∞ . The most famous ones are
Dixmier traces.
Definition (Dixmier)
If ω is a free ultrafilter on Z+ , then the functional
n
A → lim
n→ω
X
1
µ(k, A),
log(n + 2)
0 ≤ A ∈ L1,∞
k=0
is finite and additive on the positive cone of L1,∞ . Thus, it uniquely
extends to a unitarily invariant linear functional on L1,∞ . The latter is
called Dixmier trace and is denoted by trω .
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Traces on L1,∞
Basic properties of traces
1
Every Dixmier trace is positive
2
Every positive trace is continuous with respect to the natural
quasi-norm on L1,∞
3
Every continuous trace is a linear combination of 4 positive traces.
4
There are positive traces which are not Dixmier traces
5
There exist discontinuous traces
6
There are 22 positive traces
N
More information on the traces is available in [LSZ].
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Connes Trace Formula
Original Connes Trace Formula
Let M be a d−dimensional compact Riemannian manifold and let P be a
(uniform classical) pseudo-differential operator of order 0 on M. Let ∆M
be the Laplace-Beltrami operator. The assertion below is due to Connes
[Connes-action].
Theorem
For every P as above and for every Dixmier trace trω , we have
d
trω (P(1 − ∆M )− 2 ) = ResW (P).
Here, ResW (P) is the integral of principal symbol of P (we do not define
it here) over co-sphere bundle.
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Connes Trace Formula
Connes Trace Formula in terms of symb mapping
We say that T ∈ Π is compactly based if there is a function φ with
compact support such that T π1 (φ) = T . Our version of Connes Trace
Theorem (for Rd ) established in [DAO 1] reads as follows.
Theorem
If T ∈ Π is compactly based, then
− d2
ϕ(T (1 − ∆)
ˆ
)=
symb(T )
Rd ×Sd−1
for every continuous normalised trace ϕ on L1,∞ .
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Connes Trace Formula
How our result extends the original one?
Let P be a (uniform classical) compactly based pseudo-differential
operator on Rd and let p be its principal symbol. Let
ˆ
u
(Tx)(t) =
p(t, )e iht,ui (Fx)(u)du.
|u|
Rd
We have that T ∈ Π and symb(T ) = p.
Moreover, T − P ∈ Ld,∞ and also
d
d
T (1 − ∆)− 2 − P(1 − ∆)− 2 ∈ Ld,∞ × L1,∞ ⊂ L1 .
Since ϕ vanishes on L1 , it follows from our version of Connes Trace
Formula that
ˆ
ˆ
d
d
ϕ(P(1 − ∆)− 2 ) = ϕ(T (1 − ∆)− 2 ) =
symb(T ) =
Rd ×Sd−1
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p.
Rd ×Sd−1
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Further perspectives
Noncommutative (Moyal) plane: von Neumann algebra
Let {Us }s∈Rd be a family of unitary operators satisfying the condition
i
Us1 +s2 = e 2 hs1 ,θs2 i Us1 Us2 ,
s1 , s2 ∈ Rd .
Let us construct a canonical representation π of a non-commutative plane
on the Hilbert space L2 (Rd ). We set
i
(π(Us )ξ)(t) = e − 2 hs,θti ξ(t − s),
ξ ∈ L2 (Rd ).
We now define a von Neumann algebra of the non-commutative plane by
setting
wot
L∞ (Rdθ ) = span(π(Us ), s ∈ Rd ) .
d
The key fact is that L∞ (Rdθ ) is canonically isomorphic to B(L2 (R 2 )).
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Further perspectives
Noncommutative (Moyal) plane: smooth subalgebra
We now define a smooth algebra of a non-commutative plane. Let Dk ,
1 ≤ k ≤ d, be multiplication operators
(Dk ξ)(t) = tk ξ(t),
ξ ∈ L2 (Rd ).
Note that, for x ∈ L∞ (Rdθ ),
[Dk , x] ∈ B(L2 (Rd )) =⇒ [Dk , x] ∈ L∞ (Rdθ ).
This allows a definition of a Sobolev space
W m,p (Rdθ ) = {x ∈ L∞ (Rdθ ) : ∂ k (x) ∈ Lp (Rdθ ),
ord(k) ≤ m}.
Here, partial derivatives of order n are defined by the usual formula
∂ k (x) = [Dk1 , [Dk2 , [· · · , [Dkn , x]]]],
k = (k1 , · · · , kn ).
According to the Leibniz rule, every Sobolev space is an algebra. Each
Sobolev space is weakly dense in L∞ (Rdθ ).
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Further perspectives
Noncommutative (Moyal) plane: C ∗ −algebra, Dirac
operator
d
Recall that L∞ (Rdθ ) is canonically isomorphic to B(L2 (R 2 )). The
norm-closure of every Sobolev space is the ideal of all compact operators
d
in B(L2 (R 2 )). This is the algebra of continuous functions on the Moyal
plane; it is denoted by C (Rdθ ).
Dirac operator on the Moyal plane is defined by the formula
D=
d
X
γk ⊗ Dk ,
k=1
where {γk }dk=1 are Pauli matrices. As usual, we denote ∇ = (D1 , · · · , Dd )
P
and dk=1 Dk2 by −∆.
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Further perspectives
Symbol for the noncommutative plane
Let π4 be the representation of the algebra L∞ (Sd−1 ) on L2 (Rd ) defined
by the formula π4 (g ) = g ( ∇ 1 ).
(−∆) 2
Let A be the C ∗ −algebra generated by C (Rdθ ) and by π4 (L∞ (Sd−1 )). Our
third main result proved in [DAO 2] reads as follows.
Theorem
There exists a unique continuous ∗−homomorphism
¯ ∞ (Sd−1 ) such that
symb : A → L∞ (Rdθ )⊗L
symb(x) = x ⊗ 1,
symb(π4 (g )) = 1 ⊗ g ,
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x ∈ C (Rdθ ),
g ∈ L∞ (Sd−1 ).
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Further perspectives
Connes Trace Formula for the noncommutative plane
We say that an element T ∈ A is compactly based if there exists
x ∈ W m,1 (Rdθ )) (for sufficiently large m) such that Tx = T . This
d
condition guarantees (see [Cwikel]) that T (1 − ∆)− 2 ∈ L1,∞ . We can now
state the Connes Trace Formula as established in [DAO2].
Theorem
If T ∈ A is compactly based, then
− d2
ϕ(T (1 − ∆)
ˆ
) = (τθ ⊗
)(symb(T ))
for every normalised continuous trace ϕ on L1,∞ .
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Further perspectives
Where else we can define a principal symbol?
A natural principal symbol mapping (a continuous C ∗ −algebra
homomorphism) can be defined in the following spectral triples
1
commutative torus Td (DONE)
2
noncommutative torus Tdθ (DONE)
3
4
5
commutative sphere Sd−1 (DONE for d = 4, WORK IN PROGRESS
for other d)
noncommutative sphere Sd−1
(DONE for d = 4, WORK IN
θ
PROGRESS for other d)
quantum groups SUq (d) (WORK IN PROGRESS)
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Further perspectives
References
[Connes-action] Connes A. The action functional in noncommutative
geometry.
[NCG] Connes A. Noncommutative Geometry.
[mps] Gayral V., Gracia-Bondia J., Iochum B., Schücker T., Varilly J.
Moyal planes are spectral triples.
[LSZ] Lord S., Sukochev F., Zanin D. Singular traces. Theory and
applications.
[Cwikel] Levitina G., Sukochev F., Zanin D. Cwikel estimates revisited.
[DAO 2] McDonald E., Sukochev F., Zanin D. Dao of the principal
symbol. II
[DAO 1] Sukochev F., Zanin D. Dao of the principal symbol. I
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Further perspectives
Happy 70-th anniversary!
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