(slides)

Cartan MASAs and Exact Sequences of Inverse
Semigroups
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 a
weak-∗ dense subset in M;
2
there is a normal, faithful conditional expectation E : M → 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 a
weak-∗ dense subset in M;
2
there is a normal, faithful conditional expectation E : M → D.
Alternatively
1
the partial isometries V ∈ M such that V DV ∗ , V ∗ DV ⊆ D
span a weak-∗ dense subset in M;
2
there is a normal, faithful conditional expectation E : M → 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 a
weak-∗ dense subset in M;
2
there is a normal, faithful conditional expectation E : M → D.
Alternatively
1
the partial isometries V ∈ M such that V DV ∗ , V ∗ DV ⊆ D
span 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 normalizing
partial isometries groupoid normalizers, written GM (D).
Examples of Cartan Pairs
Example
Let Mn be the n × n complex matrices, and let Dn be the diagonal
n × n matrices. Then (Mn , Dn ) is a Cartan pair:
1
the matrix units normalize Dn and generate Mn ;
2
The map
E : [aij ] 7→ diag[a11 , . . . , ann ]
gives a faithful normal conditional expectation.
Examples of Cartan Pairs
Example
Let Mn be the n × n complex matrices, and let Dn be the diagonal
n × n matrices. Then (Mn , Dn ) is a Cartan pair:
1
the matrix units normalize Dn and generate Mn ;
2
The map
E : [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 irrational
rotation. Then L∞ (T) is a Cartan MASA in L∞ (T) oα Z.
Examples of Cartan Pairs
Example
Let
G=
and let
a b
0 1
: a, b ∈ R, a 6= 0 ,
1 b
H=
:b∈R .
0 1
Then H is a normal subgroup of G and L(H) is Cartan MASA in
L(G ).
Feldman & Moore approach
Feldman and Moore (1977) explored Cartan pairs (M, D) where
M∗ is separable and D = L∞ (X , µ). They showed:
1
there is a measurable equivalence relation R on X with
countable 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., an
element of G could look like


0 λ 0
V = µ 0 0  ,
0 0 γ
with λ, µ, γ ∈ T.
Let P = G ∩ Dn . And let S = G/P.
form

0 1

S= 1 0
0 0
So elements of S are of the

0
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 the
continuous functions from the idempotents of S into T. From S
and P we can construct G , since every element of G is the product
of 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 algebraic
rather 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 a
unique “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 element
s ∈ S, ss † ∈ E(S).
Inverse Semigroups
A semigroup S is an inverse semigroup if for each s ∈ S there is a
unique “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 element
s ∈ 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 : VDn V ∗ ⊆ Dn , V ∗ Dn V ⊆ Dn }.
Then G is an inverse semigroup:
if V , W ∈ G then
(VW )Dn (VW )∗ = V (WDn W ∗ )V ∗ ⊆ Dn ,
so VW ∈ G ;
Matrix example
Example
Consider the Cartan pair (Mn , Dn ) again. Again, let
G = GMn (Dn )
= {partial isometries V ∈ Mn : VDn V ∗ ⊆ Dn , V ∗ Dn V ⊆ Dn }.
Then G is an inverse semigroup:
if V , W ∈ G then
(VW )Dn (VW )∗ = V (WDn W ∗ )V ∗ ⊆ Dn ,
so VW ∈ G ;
the “inverse” of V is V ∗ ;
Matrix example
Example
Consider the Cartan pair (Mn , Dn ) again. Again, let
G = GMn (Dn )
= {partial isometries V ∈ Mn : VDn V ∗ ⊆ Dn , V ∗ Dn V ⊆ Dn }.
Then G is an inverse semigroup:
if V , W ∈ G then
(VW )Dn (VW )∗ = V (WDn W ∗ )V ∗ ⊆ Dn ,
so VW ∈ G ;
the “inverse” of V is V ∗ ;
the idempotents are the projections in Dn ;
Matrix example
Example
Consider the Cartan pair (Mn , Dn ) again. Again, let
G = GMn (Dn )
= {partial isometries V ∈ Mn : VDn V ∗ ⊆ Dn , V ∗ Dn V ⊆ Dn }.
Then G is an inverse semigroup:
if V , W ∈ G then
(VW )Dn (VW )∗ = V (WDn W ∗ )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 normalizers
GM (D) form an inverse semigroup.
if V , W ∈ GM (D) then
(VW )D(VW )∗ = V (W DW ∗ )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 isomorphism
from E(P) to E(S).
An idempotent separating extension of S by P is an inverse
semigroup 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 (the
Munn congruence) ∼ on G by
s ∼ t if ses † = tet † for all e ∈ E(G ).
The Munn Congruence
Let G be an inverse semigroup. Define an equivalence relation (the
Munn congruence) ∼ on G by
s ∼ t if ses † = tet † for all e ∈ E(G ).
If s ∼ t and u ∼ v then
su ∼ tv .
Thus S = G / ∼ is an inverse semigroup.
The Munn Congruence
Let G be an inverse semigroup. Define an equivalence relation (the
Munn congruence) ∼ on G by
s ∼ t if ses † = tet † for all e ∈ E(G ).
If s ∼ t and u ∼ v then
su ∼ tv .
Thus S = G / ∼ is an inverse semigroup.
Let P = {v ∈ G : v ∼ e for some e ∈ E(G )}. Then P is an inverse
semigroup.
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 : v Dv ∗ ⊆ D and v ∗ Dv ⊆ 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 : v Dv ∗ ⊆ 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 extension
P ,→ 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.
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 idempotents
are the projection lattice of an abelian W ∗ -algebra;
3
4
S is a meet semilattice under the natural partial order on S;
W
for every pairwise orthogonal family F ⊆ S, F exists in S.
5
S contains 1 and 0.
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 idempotents
are the projection lattice of an abelian W ∗ -algebra;
3
4
S is a meet semilattice under the natural partial order on S;
W
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 a
Cartan inverse monoid.
Matrix example
Example
In the matrix example (Mn , Dn ), the semigroups P, G and S are
the semigroups discussed earlier:
1
G is the partial isometries V such that
VDn V ∗ , V ∗ Dn V ⊆ 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(S
e
i )). The isomorphism α induces an isomorphism α
from D1 to D2 .
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(S
e
i )). The isomorphism α induces an isomorphism α
from D1 to D2 .
Definition
Let S1 and S2 be isomorphic Cartan inverse monoids. Let Pi be
the partial isometries in Di . Extensions Gi of Si by Pi are
equivalent if there is an isomorphism α : G1 → G2 such that
ι
q1
ι
q2
P1 −−−1−→ G1 −−−−→




αy
α
ey
S1


αy
P2 −−−2−→ G2 −−−−→ S2 .
commutes.
More on Extensions of Inverse Monoids
It was shown by Laush (1975) that there is one-to-one
correspondence between extensions of S by P and the second
cohomology group H 2 (S, P).
It is also shown that every extension of S by P is determined by
cocycle function σ : S × S → P.
Uniqueness of Extension
Theorem
Let (M1 , D1 ) and (M2 , D2 ) be two Cartan pairs with associated
extensions
Pi ,→ Gi → Si
for i = 1, 2.
There is a normal isomorphism θ : M1 → M2 such θ(D1 ) = D2 if
and only if the two associated extensions are equivalent.
Going in the other direction
[ and let P be
Let S be a Cartan inverse monoid. Let D = C (E(S)),
the partial isometries in D. Given an extension
P ,→ G → S
we want to construct a Cartan pair (M, D) with associated
extension (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 an
isomorphism.
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 an
isomorphism.
Define a map
K: S ×S →D
by K (s, t) = j(s † t ∧ 1).
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 an
isomorphism.
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).
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 an
isomorphism.
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 and
s1 , . . . , sk ∈ S
X
ci cj K (si , sj ) ≥ 0.
i,j
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
X
X
X
h
ci ksi ,
dj ktj i =
ci dj K (si , tj )
i,j
defines a D-valued inner product on A0 . Let A be completion of
A0 .
Thus A is a reproducing kernel Hilbert D-module of functions from
S into D.
A left representation of G
For g ∈ G define an adjointable operator λ(g ) on A by
λ(g )ks = kq(g )s σ(g , s),
where σ : G × S → P is a “cocycle-like” function (related to the
cocycles 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 of
an element in j(S) by an element in P.
A left representation of G
For g ∈ G define an adjointable operator λ(g ) on A by
λ(g )ks = kq(g )s σ(g , s),
where σ : G × S → P is a “cocycle-like” function (related to the
cocycles 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 of
an 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. We
can form a Hilbert space A ⊗π H by completing A ⊗ H with
respect to the inner product
ha ⊗ h, b ⊗ ki := hh, π(ha, bi)ki.
A left representation of G on a Hilbert space
Let π be a faithful representation of D on a Hilbert space H. We
can form a Hilbert space A ⊗π H by completing A ⊗ H with
respect to the inner product
ha ⊗ h, b ⊗ ki := hh, π(ha, bi)ki.
Then π determines a faithful representation π̂ of L(A) on the
Hilbert space A ⊗π H by
π̂(T )(a ⊗ h) = (Ta) ⊗ h.
A left representation of G on a Hilbert space
Let π be a faithful representation of D on a Hilbert space H. We
can form a Hilbert space A ⊗π H by completing A ⊗ H with
respect to the inner product
ha ⊗ h, b ⊗ ki := hh, π(ha, bi)ki.
Then π determines a faithful representation π̂ of L(A) on the
Hilbert space A ⊗π H by
π̂(T )(a ⊗ h) = (Ta) ⊗ h.
Thus, we have a faithful representation of G on the hilbert space
A ⊗π H by
λπ : g 7→ π̂(λ(g )).
Creating Cartan pairs
Let Mq = λ(G )00 , and Dq = λ(E(S))00 . Then (Mq , Dq ) is a
Cartan pair such that
1
The pair (Mq , Dq ) is independent of choice of j and π;
Creating Cartan pairs
Let Mq = λ(G )00 , and Dq = λ(E(S))00 . Then (Mq , Dq ) is a
Cartan pair such that
1
2
The pair (Mq , Dq ) is independent of choice of j and π;
[
Dq is isomorphic to D = C (E(S));
Creating Cartan pairs
Let Mq = λ(G )00 , and Dq = λ(E(S))00 . Then (Mq , Dq ) is a
Cartan pair such that
1
2
3
The pair (Mq , Dq ) is independent of choice of j and π;
[
Dq is isomorphic to D = C (E(S));
The conditional expectation E : Mq → Dq is induced from
the map
S → E(S)
s 7→ s ∧ 1.
Creating Cartan pairs
Let Mq = λ(G )00 , and Dq = λ(E(S))00 . Then (Mq , Dq ) is a
Cartan pair such that
1
2
3
The pair (Mq , Dq ) is independent of choice of j and π;
[
Dq is isomorphic to D = C (E(S));
The conditional expectation E : Mq → Dq is induced from
the map
S → E(S)
s 7→ s ∧ 1.
4
The extension associated to (Mq , Dq ) is equivalent to
q
P ,→ G −
→S
(the extension we started with).
Main Theorem
Theorem (Feldman-Moore; Donsig-F-Pitts)
q
If S is a Cartan inverse monoid and P ,→ G −
→ S is an
∗
extension of S by P := p.i.(C (E(S)), then the extension
determines a Cartan pair (M, D) which is unique up to
isomorphism. Equivalent extensions determine isomorphic
Cartan pairs.
Every Cartan pair (M, D) determines uniquely an extension of
q
a Cartan inverse semigroup S by P, P ,→ G −
→ S.

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