Cartan MASAs and Exact Sequences of Inverse Semigroupsadonsig1/NIFAS/1410-Fuller.pdf · Cartan...
-
Upload
vuongtuyen -
Category
Documents
-
view
227 -
download
0
Transcript of Cartan MASAs and Exact Sequences of Inverse Semigroupsadonsig1/NIFAS/1410-Fuller.pdf · Cartan...
Cartan MASAs and Exact Sequences of InverseSemigroups
Adam H. Fuller (University of Nebraska - Lincoln)joint work with Allan P. Donsig and David R. Pitts
NIFAS Nov. 2014, Des Moines, Iowa
Cartan MASAs
Let M be a von Neumann algebra. A maximal abelian subalgebra(MASA) D in M is a Cartan MASA if
1 the unitaries U ∈M such that UDU∗ = U∗DU = D span aweak-∗ dense subset in M;
2 there is a normal, faithful conditional expectation E : M→D.
Alternatively
1 the partial isometries V ∈M such that VDV ∗, V ∗DV ⊆ Dspan a weak-∗ dense subset in M;
2 there is a normal, faithful conditional expectation E : M→D.
We will call the pair (M,D) a Cartan pair. We call the normalizingpartial isometries groupoid normalizers, written GM(D).
Cartan MASAs
Let M be a von Neumann algebra. A maximal abelian subalgebra(MASA) D in M is a Cartan MASA if
1 the unitaries U ∈M such that UDU∗ = U∗DU = D span aweak-∗ dense subset in M;
2 there is a normal, faithful conditional expectation E : M→D.
Alternatively
1 the partial isometries V ∈M such that VDV ∗, V ∗DV ⊆ Dspan a weak-∗ dense subset in M;
2 there is a normal, faithful conditional expectation E : M→D.
We will call the pair (M,D) a Cartan pair. We call the normalizingpartial isometries groupoid normalizers, written GM(D).
Cartan MASAs
Let M be a von Neumann algebra. A maximal abelian subalgebra(MASA) D in M is a Cartan MASA if
1 the unitaries U ∈M such that UDU∗ = U∗DU = D span aweak-∗ dense subset in M;
2 there is a normal, faithful conditional expectation E : M→D.
Alternatively
1 the partial isometries V ∈M such that VDV ∗, V ∗DV ⊆ Dspan a weak-∗ dense subset in M;
2 there is a normal, faithful conditional expectation E : M→D.
We will call the pair (M,D) a Cartan pair. We call the normalizingpartial isometries groupoid normalizers, written GM(D).
Examples of Cartan Pairs
Example
Let Mn be the n × n complex matrices, and let Dn be the diagonaln × n matrices. Then (Mn,Dn) is a Cartan pair:
1 the matrix units normalize Dn and generate Mn;
2 The mapE : [aij ] 7→ diag[a11, . . . , ann]
gives a faithful normal conditional expectation.
Example
Let D = L∞(T) and let α be an action of Z on T by irrationalrotation. Then L∞(T) is a Cartan MASA in L∞(T) oα Z.
Examples of Cartan Pairs
Example
Let Mn be the n × n complex matrices, and let Dn be the diagonaln × n matrices. Then (Mn,Dn) is a Cartan pair:
1 the matrix units normalize Dn and generate Mn;
2 The mapE : [aij ] 7→ diag[a11, . . . , ann]
gives a faithful normal conditional expectation.
Example
Let D = L∞(T) and let α be an action of Z on T by irrationalrotation. Then L∞(T) is a Cartan MASA in L∞(T) oα Z.
Examples of Cartan Pairs
Example
Let
G =
{(a b0 1
): a, b ∈ R, a 6= 0
},
and let
H =
{(1 b0 1
): b ∈ R
}.
Then H is a normal subgroup of G and L(H) is Cartan MASA inL(G ).
Feldman & Moore approach
Feldman and Moore (1977) explored Cartan pairs (M,D) whereM∗ is separable and D = L∞(X , µ). They showed:
1 there is a measurable equivalence relation R on X withcountable equivalence classes and a 2-cocycle σ on R s.t.
M'M(R, σ) and D ' A(R, σ),
where M(R, σ) are “functions on R” and A(R, σ) are the“functions” supported on diag. {(x , x) : x ∈ X};
2 every sep. acting pair (M,D) arises this way.
A simple example
Consider the Cartan pair (M3,D3). Let G = GM3(D3). E.g., anelement of G could look like
V =
0 λ 0µ 0 00 0 γ
,with λ, µ, γ ∈ T.Let P = G ∩ Dn. And let S = G/P. So elements of S are of theform
S =
0 1 01 0 00 0 1
.From (Mn,Dn) we have 3 semigroups: P, G and S.
A simple example: continued
Conversely, starting with S , we can construct P: P is all thecontinuous functions from the idempotents of S into T. From Sand P we can construct G , since every element of G is the productof an element in S and an element in P. From G we can construct(Mn,Dn) as the span of G .
Our Objective: Give an alternative approach using algebraicrather than measure theoretic tools which
conceptually simpler;
applies to the non-separably acting case.
Inverse Semigroups
A semigroup S is an inverse semigroup if for each s ∈ S there is aunique “inverse” element s† such that
ss†s = s and s†ss† = s†.
We denote the idempotents in an inverse semigroup S by E(S).The idempotents form an abelian semigroup. For any elements ∈ S , ss† ∈ E(S).
An inverse semigroup S has a natural partial order defined by
s ≤ t if and only if s = te
for some idempotent e ∈ E(S).
Inverse Semigroups
A semigroup S is an inverse semigroup if for each s ∈ S there is aunique “inverse” element s† such that
ss†s = s and s†ss† = s†.
We denote the idempotents in an inverse semigroup S by E(S).The idempotents form an abelian semigroup. For any elements ∈ S , ss† ∈ E(S).An inverse semigroup S has a natural partial order defined by
s ≤ t if and only if s = te
for some idempotent e ∈ E(S).
Matrix example
Example
Consider the Cartan pair (Mn,Dn) again. Again, let
G = GMn(Dn)
= {partial isometries V ∈ Mn : VDnV∗ ⊆ Dn, V
∗DnV ⊆ Dn}.
Then G is an inverse semigroup:
if V ,W ∈ G then
(VW )Dn(VW )∗ = V (WDnW∗)V ∗ ⊆ Dn,
so VW ∈ G ;
the “inverse” of V is V ∗;
the idempotents are the projections in Dn;
V ≤W if V = WP for some projection P ∈ Dn.
Matrix example
Example
Consider the Cartan pair (Mn,Dn) again. Again, let
G = GMn(Dn)
= {partial isometries V ∈ Mn : VDnV∗ ⊆ Dn, V
∗DnV ⊆ Dn}.
Then G is an inverse semigroup:
if V ,W ∈ G then
(VW )Dn(VW )∗ = V (WDnW∗)V ∗ ⊆ Dn,
so VW ∈ G ;
the “inverse” of V is V ∗;
the idempotents are the projections in Dn;
V ≤W if V = WP for some projection P ∈ Dn.
Matrix example
Example
Consider the Cartan pair (Mn,Dn) again. Again, let
G = GMn(Dn)
= {partial isometries V ∈ Mn : VDnV∗ ⊆ Dn, V
∗DnV ⊆ Dn}.
Then G is an inverse semigroup:
if V ,W ∈ G then
(VW )Dn(VW )∗ = V (WDnW∗)V ∗ ⊆ Dn,
so VW ∈ G ;
the “inverse” of V is V ∗;
the idempotents are the projections in Dn;
V ≤W if V = WP for some projection P ∈ Dn.
Matrix example
Example
Consider the Cartan pair (Mn,Dn) again. Again, let
G = GMn(Dn)
= {partial isometries V ∈ Mn : VDnV∗ ⊆ Dn, V
∗DnV ⊆ Dn}.
Then G is an inverse semigroup:
if V ,W ∈ G then
(VW )Dn(VW )∗ = V (WDnW∗)V ∗ ⊆ Dn,
so VW ∈ G ;
the “inverse” of V is V ∗;
the idempotents are the projections in Dn;
V ≤W if V = WP for some projection P ∈ Dn.
Bigger Matrix example
More generally...
Example
Let (M,D) be a Cartan pair. Then the groupoid normalizersGM(D) form an inverse semigroup.
if V ,W ∈ GM(D) then
(VW )D(VW )∗ = V (WDW ∗)V ∗ ⊆ D,
so VW ∈ GM(D);
the “inverse” of V is V ∗;
the idempotents are the projections in D;
V ≤W if V = WP for some projection P ∈ D.
Extensions of Inverse Semigroups
Let S and P be inverse semigroups. And let
π : P → S ,
be a surjective homomorphism such that π|E(P) is an isomorphismfrom E(P) to E(S).An idempotent separating extension of S by P is an inversesemigroup G with
P �� ι // G
q // // S
and
ι is an injective homomorphism;
q is a surjective homomorphism;
q(g) ∈ E(S) if and only if g = ι(p) for some p ∈ P;
q ◦ ι = π.
Note that E(P) ∼= E(G ) ∼= E(S).
The Munn Congruence
Let G be an inverse semigroup. Define an equivalence relation (theMunn congruence) ∼ on G by
s ∼ t if ses† = tet† for all e ∈ E(G ).
If s ∼ t and u ∼ v thensu ∼ tv .
Thus S = G/ ∼ is an inverse semigroup.Let P = {v ∈ G : v ∼ e for some e ∈ E(G )}. Then P is an inversesemigroup.And G is an extension of S by P:
P ↪→ G → S .
The Munn Congruence
Let G be an inverse semigroup. Define an equivalence relation (theMunn congruence) ∼ on G by
s ∼ t if ses† = tet† for all e ∈ E(G ).
If s ∼ t and u ∼ v thensu ∼ tv .
Thus S = G/ ∼ is an inverse semigroup.
Let P = {v ∈ G : v ∼ e for some e ∈ E(G )}. Then P is an inversesemigroup.And G is an extension of S by P:
P ↪→ G → S .
The Munn Congruence
Let G be an inverse semigroup. Define an equivalence relation (theMunn congruence) ∼ on G by
s ∼ t if ses† = tet† for all e ∈ E(G ).
If s ∼ t and u ∼ v thensu ∼ tv .
Thus S = G/ ∼ is an inverse semigroup.Let P = {v ∈ G : v ∼ e for some e ∈ E(G )}. Then P is an inversesemigroup.And G is an extension of S by P:
P ↪→ G → S .
From Cartan Pairs to Extensions of Inverse Semigroups
Let (M,D) be a Cartan pair. Let
G = GM(D)
= {v ∈M a partial isometry : vDv∗ ⊆ D and v∗Dv ⊆ D}.
Let S = G/ ∼, where ∼ is the Munn congruence on G and let
P = {V ∈ G : V ∼ P, P ∈ Proj(D)}.
Definition
We call the extensionP ↪→ G → S ,
the extension associated to the Cartan pair (M,D).
From Cartan Pairs to Extensions of Inverse Semigroups
Let (M,D) be a Cartan pair. Let
G = GM(D)
= {v ∈M a partial isometry : vDv∗ ⊆ D and v∗Dv ⊆ D}.
Let S = G/ ∼, where ∼ is the Munn congruence on G and let
P = {V ∈ G : V ∼ P, P ∈ Proj(D)}.
Definition
We call the extensionP ↪→ G → S ,
the extension associated to the Cartan pair (M,D).
Properties of associated extensions
Let (M,D) be a Cartan pair, and let
P ↪→ G → S ,
be the associated extension.Then P = GM(D) ∩D, i.e. P is simply the partial isometries in D.
The inverse semigroup S has the following properties
1 S is fundamental: E(S) is maximal abelian in S ;
2 E(S) is a hyperstonean boolean algebra, i.e. the idempotentsare the projection lattice of an abelian W ∗-algebra;
3 S is a meet semilattice under the natural partial order on S ;
4 for every pairwise orthogonal family F ⊆ S ,∨F exists in S .
5 S contains 1 and 0.
Definition
An inverse semigroup S , satisfying the conditions above is called aCartan inverse monoid.
Properties of associated extensions
Let (M,D) be a Cartan pair, and let
P ↪→ G → S ,
be the associated extension.Then P = GM(D) ∩D, i.e. P is simply the partial isometries in D.The inverse semigroup S has the following properties
1 S is fundamental: E(S) is maximal abelian in S ;
2 E(S) is a hyperstonean boolean algebra, i.e. the idempotentsare the projection lattice of an abelian W ∗-algebra;
3 S is a meet semilattice under the natural partial order on S ;
4 for every pairwise orthogonal family F ⊆ S ,∨F exists in S .
5 S contains 1 and 0.
Definition
An inverse semigroup S , satisfying the conditions above is called aCartan inverse monoid.
Properties of associated extensions
Let (M,D) be a Cartan pair, and let
P ↪→ G → S ,
be the associated extension.Then P = GM(D) ∩D, i.e. P is simply the partial isometries in D.The inverse semigroup S has the following properties
1 S is fundamental: E(S) is maximal abelian in S ;
2 E(S) is a hyperstonean boolean algebra, i.e. the idempotentsare the projection lattice of an abelian W ∗-algebra;
3 S is a meet semilattice under the natural partial order on S ;
4 for every pairwise orthogonal family F ⊆ S ,∨F exists in S .
5 S contains 1 and 0.
Definition
An inverse semigroup S , satisfying the conditions above is called aCartan inverse monoid.
Matrix example
Example
In the matrix example (Mn,Dn), the semigroups P, G and S arethe semigroups discussed earlier:
1 G is the partial isometries V such thatVDnV
∗, V ∗DnV ⊆ Dn;
2 P is the partial isometries in Dn;
3 S is the matrices in G with only 0 and 1 entries.
Equivalent Extensions of Cartan Inverse monoid
Let α : S1 → S2 be an isomorphism of Cartan inverse monoids.Then E(Si ) is the lattice of projections for a W ∗-algebra,
Di = C (E(Si )). The isomorphism α induces an isomorphism αfrom D1 to D2.
Definition
Let S1 and S2 be isomorphic Cartan inverse monoids. Let Pi bethe partial isometries in Di . Extensions Gi of Si by Pi areequivalent if there is an isomorphism α : G1 → G2 such that
P1ι1−−−−→ G1
q1−−−−→ S1
α
y α
y α
yP2
ι2−−−−→ G2q2−−−−→ S2.
commutes.
Equivalent Extensions of Cartan Inverse monoid
Let α : S1 → S2 be an isomorphism of Cartan inverse monoids.Then E(Si ) is the lattice of projections for a W ∗-algebra,
Di = C (E(Si )). The isomorphism α induces an isomorphism αfrom D1 to D2.
Definition
Let S1 and S2 be isomorphic Cartan inverse monoids. Let Pi bethe partial isometries in Di . Extensions Gi of Si by Pi areequivalent if there is an isomorphism α : G1 → G2 such that
P1ι1−−−−→ G1
q1−−−−→ S1
α
y α
y α
yP2
ι2−−−−→ G2q2−−−−→ S2.
commutes.
More on Extensions of Inverse Monoids
It was shown by Laush (1975) that there is one-to-onecorrespondence between extensions of S by P and the secondcohomology group H2(S ,P).It is also shown that every extension of S by P is determined bycocycle function σ : S × S → P.
Uniqueness of Extension
Theorem
Let (M1,D1) and (M2,D2) be two Cartan pairs with associatedextensions
Pi ↪→ Gi → Si
for i = 1, 2.There is a normal isomorphism θ : M1 →M2 such θ(D1) = D2 ifand only if the two associated extensions are equivalent.
Going in the other direction
Let S be a Cartan inverse monoid. Let D = C (E(S)), and let P bethe partial isometries in D. Given an extension
P ↪→ G → S
we want to construct a Cartan pair (M,D) with associatedextension (equivalent to) P ↪→ G → S .
A D-valued Reproducing kernel space
Let j be an order-preserving map, j : S → G such that j ◦ q = id.That is j(s) ≤ j(t) when s ≤ t and j : E(S)→ E(G ) is anisomorphism.
Define a mapK : S × S → D
by K (s, t) = j(s†t ∧ 1).The idempotent s†t ∧ 1 is the minimal idempotent e such that
se = te = s ∧ t.
Thus K (s, t) is the idempotent in G defining j(s) ∧ j(t).The map K is positive: that is for c1, . . . , ck ∈ C ands1, . . . , sk ∈ S ∑
i ,j
cicjK (si , sj) ≥ 0.
A D-valued Reproducing kernel space
Let j be an order-preserving map, j : S → G such that j ◦ q = id.That is j(s) ≤ j(t) when s ≤ t and j : E(S)→ E(G ) is anisomorphism.Define a map
K : S × S → D
by K (s, t) = j(s†t ∧ 1).
The idempotent s†t ∧ 1 is the minimal idempotent e such that
se = te = s ∧ t.
Thus K (s, t) is the idempotent in G defining j(s) ∧ j(t).The map K is positive: that is for c1, . . . , ck ∈ C ands1, . . . , sk ∈ S ∑
i ,j
cicjK (si , sj) ≥ 0.
A D-valued Reproducing kernel space
Let j be an order-preserving map, j : S → G such that j ◦ q = id.That is j(s) ≤ j(t) when s ≤ t and j : E(S)→ E(G ) is anisomorphism.Define a map
K : S × S → D
by K (s, t) = j(s†t ∧ 1).The idempotent s†t ∧ 1 is the minimal idempotent e such that
se = te = s ∧ t.
Thus K (s, t) is the idempotent in G defining j(s) ∧ j(t).
The map K is positive: that is for c1, . . . , ck ∈ C ands1, . . . , sk ∈ S ∑
i ,j
cicjK (si , sj) ≥ 0.
A D-valued Reproducing kernel space
Let j be an order-preserving map, j : S → G such that j ◦ q = id.That is j(s) ≤ j(t) when s ≤ t and j : E(S)→ E(G ) is anisomorphism.Define a map
K : S × S → D
by K (s, t) = j(s†t ∧ 1).The idempotent s†t ∧ 1 is the minimal idempotent e such that
se = te = s ∧ t.
Thus K (s, t) is the idempotent in G defining j(s) ∧ j(t).The map K is positive: that is for c1, . . . , ck ∈ C ands1, . . . , sk ∈ S ∑
i ,j
cicjK (si , sj) ≥ 0.
A D-valued Reproducing kernel space
For each s ∈ S define a “kernel-map” ks : S → D by
ks(t) = K (t, s).
Let A0 = span{ks : s ∈ S}. The positivity of K shows that the
〈∑
ciksi ,∑
djktj 〉 =∑i ,j
cidjK (si , tj)
defines a D-valued inner product on A0. Let A be completion ofA0.Thus A is a reproducing kernel Hilbert D-module of functions fromS into D.
A left representation of G
For g ∈ G define an adjointable operator λ(g) on A by
λ(g)ks = kq(g)sσ(g , s),
where σ : G × S → P is a “cocycle-like” function (related to thecocycles of Lausch). This is determined by the equation
gj(s) = j(q(g)s)σ(g , s),
i.e. elements of the form gj(s) can be factored into the product ofan element in j(S) by an element in P.
The mapping
λ : G → L(A)
is a representation of G by partial isometries.
A left representation of G
For g ∈ G define an adjointable operator λ(g) on A by
λ(g)ks = kq(g)sσ(g , s),
where σ : G × S → P is a “cocycle-like” function (related to thecocycles of Lausch). This is determined by the equation
gj(s) = j(q(g)s)σ(g , s),
i.e. elements of the form gj(s) can be factored into the product ofan element in j(S) by an element in P. The mapping
λ : G → L(A)
is a representation of G by partial isometries.
A left representation of G on a Hilbert space
Let π be a faithful representation of D on a Hilbert space H. Wecan form a Hilbert space A⊗π H by completing A⊗H withrespect to the inner product
〈a⊗ h, b ⊗ k〉 := 〈h, π(〈a, b〉)k〉.
Then π determines a faithful representation π of L(A) on theHilbert space A⊗π H by
π(T )(a⊗ h) = (Ta)⊗ h.
Thus, we have a faithful representation of G on the hilbert spaceA⊗π H by
λπ : g 7→ π(λ(g)).
A left representation of G on a Hilbert space
Let π be a faithful representation of D on a Hilbert space H. Wecan form a Hilbert space A⊗π H by completing A⊗H withrespect to the inner product
〈a⊗ h, b ⊗ k〉 := 〈h, π(〈a, b〉)k〉.
Then π determines a faithful representation π of L(A) on theHilbert space A⊗π H by
π(T )(a⊗ h) = (Ta)⊗ h.
Thus, we have a faithful representation of G on the hilbert spaceA⊗π H by
λπ : g 7→ π(λ(g)).
A left representation of G on a Hilbert space
Let π be a faithful representation of D on a Hilbert space H. Wecan form a Hilbert space A⊗π H by completing A⊗H withrespect to the inner product
〈a⊗ h, b ⊗ k〉 := 〈h, π(〈a, b〉)k〉.
Then π determines a faithful representation π of L(A) on theHilbert space A⊗π H by
π(T )(a⊗ h) = (Ta)⊗ h.
Thus, we have a faithful representation of G on the hilbert spaceA⊗π H by
λπ : g 7→ π(λ(g)).
Creating Cartan pairs
Let Mq = λ(G )′′, and Dq = λ(E(S))′′. Then (Mq,Dq) is aCartan pair such that
1 The pair (Mq,Dq) is independent of choice of j and π;
2 Dq is isomorphic to D = C (E(S));
3 The conditional expectation E : Mq → Dq is induced fromthe map
S → E(S)
s 7→ s ∧ 1.
4 The extension associated to (Mq,Dq) is equivalent to
P ↪→ Gq−→ S
(the extension we started with).
Creating Cartan pairs
Let Mq = λ(G )′′, and Dq = λ(E(S))′′. Then (Mq,Dq) is aCartan pair such that
1 The pair (Mq,Dq) is independent of choice of j and π;
2 Dq is isomorphic to D = C (E(S));
3 The conditional expectation E : Mq → Dq is induced fromthe map
S → E(S)
s 7→ s ∧ 1.
4 The extension associated to (Mq,Dq) is equivalent to
P ↪→ Gq−→ S
(the extension we started with).
Creating Cartan pairs
Let Mq = λ(G )′′, and Dq = λ(E(S))′′. Then (Mq,Dq) is aCartan pair such that
1 The pair (Mq,Dq) is independent of choice of j and π;
2 Dq is isomorphic to D = C (E(S));
3 The conditional expectation E : Mq → Dq is induced fromthe map
S → E(S)
s 7→ s ∧ 1.
4 The extension associated to (Mq,Dq) is equivalent to
P ↪→ Gq−→ S
(the extension we started with).
Creating Cartan pairs
Let Mq = λ(G )′′, and Dq = λ(E(S))′′. Then (Mq,Dq) is aCartan pair such that
1 The pair (Mq,Dq) is independent of choice of j and π;
2 Dq is isomorphic to D = C (E(S));
3 The conditional expectation E : Mq → Dq is induced fromthe map
S → E(S)
s 7→ s ∧ 1.
4 The extension associated to (Mq,Dq) is equivalent to
P ↪→ Gq−→ S
(the extension we started with).
Main Theorem
Theorem (Feldman-Moore; Donsig-F-Pitts)
If S is a Cartan inverse monoid and P ↪→ Gq−→ S is an
extension of S by P := p.i .(C ∗(E(S)), then the extensiondetermines a Cartan pair (M,D) which is unique up toisomorphism. Equivalent extensions determine isomorphicCartan pairs.
Every Cartan pair (M,D) determines uniquely an extension of
a Cartan inverse semigroup S by P, P ↪→ Gq−→ S.