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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).

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.

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




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




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).

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.

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.


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.


i ,j




K : S × S → D


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




K : S × S → D


se = te = s ∧ t.


i ,j



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)).

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.

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