stably gelfand quantales cartan sub-c*-algebraspmr/slides-ic-aug-2009.pdfstably gelfand quantales,...

STABLY GELFAND QUANTALES, GROUPOIDS AND

CARTAN SUB-C*-ALGEBRAS

Pedro Resende

Instituto Superior Tecnico, Lisboa, Portugal

Fifth workshop “Categories, Logic and Foundations of Physics”Imperial College, August 6, 2009

P. RESENDE (IST) STABLY GELFAND QUANTALES 5CLFP 1 / 36

MOTIVATIONS

Algebraic quantum mechanics / QFT:

C*-algebras of operators contain observables and unitaries;Commutative subalgebras ∼ consistent sets of observers (prominentrole in the topos approach pioneered by Doring and Isham).

Dynamical systems:

Discrete group Γ acting on a compact Hausdorff space X yields anaction of Γ on the commutative C*-algebra C(X);The crossed product C*-algebra Γ n C(X) contains a copy of C(X);Commutative subalgebras B⊆ A ∼ generalized dynamical systems.

MOTIVATIONS

C*-algebras of operators contain observables and unitaries;

Commutative subalgebras ∼ consistent sets of observers (prominentrole in the topos approach pioneered by Doring and Isham).

Dynamical systems:

MOTIVATIONS

Dynamical systems:

MOTIVATIONS

Dynamical systems:

MOTIVATIONS

Dynamical systems:

Discrete group Γ acting on a compact Hausdorff space X yields anaction of Γ on the commutative C*-algebra C(X);

The crossed product C*-algebra Γ n C(X) contains a copy of C(X);Commutative subalgebras B⊆ A ∼ generalized dynamical systems.

MOTIVATIONS

Dynamical systems:

Discrete group Γ acting on a compact Hausdorff space X yields anaction of Γ on the commutative C*-algebra C(X);The crossed product C*-algebra Γ n C(X) contains a copy of C(X);

Commutative subalgebras B⊆ A ∼ generalized dynamical systems.

MOTIVATIONS

Dynamical systems:

Quantales

Examples: C*-algebras, groupoids and inverse semigroupsStably Gelfand quantales

The many groupoids of a C*-algebra

Sheaf theory

Quantales

Examples: C*-algebras, groupoids and inverse semigroups

Stably Gelfand quantales

Sheaf theory

Quantales

Sheaf theory

Quantales

Sheaf theory

Quantales

Sheaf theory

QUANTALES

By a quantale is meant a complete lattice Q equipped with an associativemultiplication (a,b) 7→ ab satisfying:

abi(∨i

(This is a semigroup in the monoidal category of sup-lattices, just like aring is a semigroup in the monoidal category of abelian groups.)

A homomorphism of quantales h : Q→ R is a homomorphism ofsemigroups that preserves

∨(hence, the analogue of a homomorphism of

rings):

QUANTALES

abi(∨i

rings):

QUANTALES

abi(∨i

rings):

MAJOR EXAMPLE 1 — LOCALES

EXAMPLE

1 The topology Ω(X) of a topological space X is a quantale:

UV def= U∩V.

2 More generally, any locale (aka “pointfree space”) L is a quantale:

for a,b ∈ L we define ab def= a∧b.

Hence, the multiplication of a quantale generalizes the intersection of opensets, and suggests the following:

SLOGAN

Quantales are “noncommutative (and nonidempotent) pointfreetopologies”.

EXAMPLE

UV def= U∩V.

SLOGAN

EXAMPLE

UV def= U∩V.

SLOGAN

EXAMPLE

UV def= U∩V.

SLOGAN

EXAMPLE

UV def= U∩V.

SLOGAN

EXAMPLE

UV def= U∩V.

SLOGAN

MAJOR EXAMPLE 2 — C*-ALGEBRAS

EXAMPLE

Let A be a semigroup.

The powerset 2A is a quantale with XY being the pointwise product X ∗Y:

X ∗Y def= xy | x ∈ X, y ∈ Y for all X,Y ∈ 2A

(X ∗Y)∗Z = X ∗ (Y ∗Z)

(⋃α

X ∗Yα(⋃α

)∗Y =

Xα ∗Y

EXAMPLE

(X ∗Y)∗Z = X ∗ (Y ∗Z)

(⋃α

X ∗Yα(⋃α

)∗Y =

Xα ∗Y

EXAMPLE

(X ∗Y)∗Z = X ∗ (Y ∗Z)

(⋃α

X ∗Yα(⋃α

)∗Y =

Xα ∗Y

EXAMPLE

(X ∗Y)∗Z = X ∗ (Y ∗Z)

(⋃α

X ∗Yα(⋃α

)∗Y =

Xα ∗Y

EXAMPLE

(X ∗Y)∗Z = X ∗ (Y ∗Z)

(⋃α

X ∗Yα(⋃α

)∗Y =

Xα ∗Y

EXAMPLE

If A is a topological semigroup:

complete lattice of closed sets f(A);XY = X ∗Y;X 7→ X is a surjective homomorphism 2A→ f(A).

If A is a complex algebra:

complete lattice of linear subspaces SubC(A);XY = 〈X ∗Y〉;X 7→ 〈X〉 is a surjective homomorphism 2A→ SubC(A).

If A is a topological complex algebra:

complete lattice of closed linear subspaces CSubC(A);XY = 〈X ∗Y〉;X 7→ 〈X〉 is a surjective homomorphism 2A→ CSubC(A).

EXAMPLE

complete lattice of closed sets f(A);

XY = X ∗Y;X 7→ X is a surjective homomorphism 2A→ f(A).

EXAMPLE

complete lattice of closed sets f(A);XY = X ∗Y;

X 7→ X is a surjective homomorphism 2A→ f(A).

EXAMPLE

complete lattice of linear subspaces SubC(A);

XY = 〈X ∗Y〉;X 7→ 〈X〉 is a surjective homomorphism 2A→ SubC(A).

EXAMPLE

complete lattice of linear subspaces SubC(A);XY = 〈X ∗Y〉;

X 7→ 〈X〉 is a surjective homomorphism 2A→ SubC(A).

EXAMPLE

complete lattice of closed linear subspaces CSubC(A);

XY = 〈X ∗Y〉;X 7→ 〈X〉 is a surjective homomorphism 2A→ CSubC(A).

EXAMPLE

complete lattice of closed linear subspaces CSubC(A);XY = 〈X ∗Y〉;

X 7→ 〈X〉 is a surjective homomorphism 2A→ CSubC(A).

EXAMPLE

DEFINITION

Let A be a C*-algebra.

1 The quantale CSubC(A) is denoted by Max(A) and called thespectrum of A by Mulvey (1991).

2 Max(A) is an involutive quantale:

it is an involutive semigroup with the pointwise involution

V∗ def= a∗ | a ∈ V

and the involution preserves joins,(∨i

)∗=∨

3 If A has a unit 1 then Max(A) is unital with unit e = C1.

DEFINITION

V∗ def= a∗ | a ∈ V

)∗=∨

DEFINITION

V∗ def= a∗ | a ∈ V

)∗=∨

DEFINITION

V∗ def= a∗ | a ∈ V

)∗=∨

DEFINITION

V∗ def= a∗ | a ∈ V

)∗=∨

DEFINITION

V∗ def= a∗ | a ∈ V

)∗=∨

In a quantale Q the top is denoted by 1 and the elements z ∈ Q suchthat z1≤ z are right sided.

In Max(A) the right sided elements are the norm-closed right sidedideals.

Similarly for 1z≤ z (left sided) and 1z1≤ z (two sided).

The two-sided elements of Max(A) form a locale I(A) with I∧ J = IJfor any two I,J ∈ I(A).

This locale is isomorphic to the topology of the primitive spectrum ofA.

If A is commutative we obtain the usual spectrum.

If A is commutative I(A) suffices to recover A.

However, if A is noncommutative I(A) is certainly not enough.

There have been attempts to consider the quantale R(A) of closedright ideals instead: only works for post-liminal C*-algebras (Borceux,Rosicky and Van Den Bossche).

THEOREM (KRUML AND R 2004)For unital C*-algebras A and B we have

A∼= B ⇐⇒ Max(A)∼= Max(B)

Hence, the functor Max classifies unital C*-algebras!

REMARK

However, Aut(A) 6∼= Aut(Max(A)).

REMARK

The difficulties are:

“Too many” quantale homomorphisms.

Unknown algebraic characterization of Max(A).

DEFINITION

By a Gelfand quantale is meant an involutive quantale such thataa∗a = a for every right-sided a ∈ Q. (Mulvey)

DEFINITION

By a locally Gelfand quantale is meant an involutive quantale such thataa∗a = a for every a ∈ Q such that a≤ p and ap≤ p for some projection p(projection means p2 = p∗ = p). (Mulvey and Ramos)

(This has applications to quantale sheaves.)

DEFINITION

A stably Gelfand quantale is an involutive quantale satisfying thecondition

aa∗a≤ a ⇐⇒ aa∗a = a

THEOREM

1 Let A be a C*-algebra, and let V ∈Max(A) such that

V∗VV∗V ⊂ V∗V .

Then V ⊂ VV∗V.

2 Hence, Max(A) is stably Gelfand.

Indeed, this is what the standard approximate unit argument actuallyproves!

DEFINITION

THEOREM

Then V ⊂ VV∗V.

DEFINITION

THEOREM

Then V ⊂ VV∗V.

DEFINITION

THEOREM

Then V ⊂ VV∗V.

DEFINITION

THEOREM

Then V ⊂ VV∗V.

DEFINITION

THEOREM

Then V ⊂ VV∗V.

PROOF.V∗V is self-adjoint, and thus the condition V∗VV∗V ⊂ V∗V makes it asub-C*-algebra of A. Let (uλ ) be an approximate unit of the C*-algebraV∗V, consisting of positive elements in the unit ball of V∗V, and let a ∈ V.Then a∗a ∈ V∗V, and thus 0 = limλ a∗a(1−uλ ) (in the unitization ofV∗V). Hence,

||a−auλ ||2 = limλ

||(1−uλ )a∗a(1−uλ ) ||

≤ ||1−uλ || ||a∗a(1−uλ ) ||≤ lim

||a∗a(1−uλ ) ||= 0 ,

and thus auλ → a. Hence, denoting by U the closure of the linear span of(uλ ), we have V ⊂ VU ⊂ VV∗V.

MAJOR EXAMPLE 3 — GROUPOIDS

A groupoid G (in a category with “enough pullbacks”) is a structure asfollows, where G2 is the pullback of d and r, satisfying the usual axioms ofan internal category plus those asserting that i is an inversion operation:

G2m // G1

d// G0uoo

EXAMPLE

1 Topological groupoids: the underlying category is Top.

2 Lie groupoids: the underlying category is that of smooth manifolds;and d and r are required to be submersions (so that the neededpullbacks exist).

3 Localic groupoids: the underlying category is Loc.

G2m // G1

d// G0uoo

EXAMPLE

G2m // G1

d// G0uoo

EXAMPLE

G2m // G1

d// G0uoo

EXAMPLE

G2m // G1

d// G0uoo

EXAMPLE

A topological or localic groupoid is etale if d is a localhomeomorphism (⇒ all the structure maps are).

In both cases we have an associated unital involutive quantale O(G).

In the topological case the quantale consists of the open sets of G1with pointwise multiplication and involution:

UV = m((U×V)∩G2)U∗ = i(U)

e = u(G0)

(If d is open we have the same except that the quantale is not unital— in particular, any topological group has an associated quantale,which is unital if and only if the group is discrete.)

O(G) is stably Gelfand because U ⊂ UU∗U for all U.

UV = m((U×V)∩G2)U∗ = i(U)

e = u(G0)

UV = m((U×V)∩G2)U∗ = i(U)

e = u(G0)

UV = m((U×V)∩G2)U∗ = i(U)

e = u(G0)

UV = m((U×V)∩G2)U∗ = i(U)

e = u(G0)

The algebraic characterization of O(G) is simple (R 2007):

DEFINITION

By an inverse quantal frame is meant a locale Q equipped with theadditional structure of a unital involutive quantale such that, defining forall a ∈ Q

ς(a) = a1∧ e

I (Q) = s ∈ Q | ss∗ ≤ e, s∗s≤ e

the following three conditions are satisfied for all a ∈ Q:

ς(a) ≤ aa∗

a ≤ ς(a)a∨I (Q) = 1

DEFINITION

ς(a) = a1∧ e

I (Q) = s ∈ Q | ss∗ ≤ e, s∗s≤ e

ς(a) ≤ aa∗

a ≤ ς(a)a∨I (Q) = 1

DEFINITION

ς(a) = a1∧ e

I (Q) = s ∈ Q | ss∗ ≤ e, s∗s≤ e

ς(a) ≤ aa∗

a ≤ ς(a)a∨I (Q) = 1

From an inverse quantal frame Q one obtains a (localic) etalegroupoid G = G (Q) with ς being the direct image of ud : G1→ G1.

The elements of I (Q) are called the partial units of Q and there is abijection between them and the local bisections of G:

DEFINITION

Let G be a localic (resp. topological) etale groupoid. By a local bisectionof G is meant a local section s : U→ G1 of d, where U is an opensublocale (resp. subspace) of G0, such that r s : U→ G0 is a regularmonomorphism (an open embedding).

The local bisections of an etale groupoid G form an inverse semigroupof a special kind:

DEFINITION

In order to motivate the notion of inverse semigroup we recallpseudogroups:

EXAMPLE

If X is a topological space, a partial homeomorphism is ahomeomorphism h : U→ U′ between open subsets of X.

If h : U→ U′ and k : V→ V ′ are partial homeomorphisms their product hkis the composition k h with domain h−1(U′∩V) and codomain k(U′∩V).

This turns the set of partial homeomorphisms into an involutive semigroupwith involution h 7→ h−1.

The idempotents are the identities on open sets idU.

The involution does not define an inverse operation as in a group, buthh−1h = h holds for all h.

EXAMPLE

DEFINITION

An inverse semigroup S is an involutive semigroup S such that ss∗s = sfor all s ∈ S and all of whose idempotents commute; that is, for allf ,g ∈ E(S) we have fg = gf .

The natural order of S is the partial order defined by

s≤ t if s = ft

for some idempotent f ∈ E(S).

S is complete if every compatible subset X ⊂ S (i.e., such thatst∗ ∈ E(S) and s∗t ∈ E(S) for all s, t ∈ X) has a join

∨X in S.

If S is complete than it is an abstract complete pseudogroup if E(S) is alocale.

DEFINITION

s≤ t if s = ft

∨X in S.

DEFINITION

s≤ t if s = ft

∨X in S.

DEFINITION

s≤ t if s = ft

∨X in S.

DEFINITION

s≤ t if s = ft

∨X in S.

For a topological etale groupoid G, I (O(G)) is isomorphic to theinverse semigroup of local bisections of G.

Note that I (O(G)) is complete and the idempotents of I (O(G))form a locale isomorphic to G0.

From any abstract complete pseudogroup S we can form an inversequantal frame by completing with respect to joins of all subsets:

S 7→L ∨(S)

The locale spectrum of G = G (L ∨(S)) is the groupoid of germs ofS, which appears often in the literature: we can think of G (L ∨(S)) asthe “localic germ groupoid” (the actual germs are the points of G1).

(The relation between inverse semigroups and etale groupoids isanalogous to that between sheaves and local homeomorphisms.)

S 7→L ∨(S)

Inversequantalframes

Equiv. of cats.

Etale groupoidsbisections

Abstractcomplete

pseudogroups

“germs”pp

THE MANY GROUPOIDS OF A C*-ALGEBRA

THEOREM

Let Q be a stably Gelfand quantale and let p2 = p∗ = p ∈ Q (a projection).

The subset Ip(Q)⊂ Q of those s ∈ Q such that

ss∗ ≤ ps∗s ≤ p

sp ≤ sps ≤ s

is an abstract complete pseudogroup whose idempotents are the elementsb≤ p such that pb = bp = b.

EXAMPLE

1 If p = e we get the partial units of Q (even for Q more general thanan inverse quantal frame).

2 If p = 1 we get the locale I(Q) of two-sided elements of Q.

THEOREM

sp ≤ sps ≤ s

EXAMPLE

THEOREM

sp ≤ sps ≤ s

EXAMPLE

THEOREM

sp ≤ sps ≤ s

EXAMPLE

THEOREM

sp ≤ sps ≤ s

EXAMPLE

THEOREM

sp ≤ sps ≤ s

EXAMPLE

THEOREM

sp ≤ sps ≤ s

EXAMPLE

Proof.

Let us write S for Ip(Q).

ss∗s≤ sp≤ s⇒ ss∗s = s for all s ∈ S.

It is straightforward to show that S is a subsemigroup of Q with p asunit.

Write S(p) for s ∈ S | s≤ p.

Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

S(p) = E(S).

Proof.

S(p) = E(S).

Proof.

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).

Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.

And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.

Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.

Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).

Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).

Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.

Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof.

Write S(p) for s ∈ S | s≤ p.Let b ∈ S(p).Then bb∗ ≤ bp∗ = bp≤ b.And b = bb∗b≤ bb∗p = b(pb)∗ ≤ bb∗.Hence, bb∗ = b, and thus b = b∗, and also b = b2.Hence, S(p) ⊂ E(S).Conversely, let b ∈ E(S).Then b = bb, and thus b∗b = b∗bb≤ pb≤ b.Hence, b∗b≤ b∗, and thus b = bb∗b≤ bb∗ ≤ p, so b ∈ S(p).

S(p) = E(S).

Proof. (Cont.)

We have S(p) = E(S), and thus S(p), which is a unital subquantale ofQ with p as the unit, is also an idempotent quantale and thus it is alocale with ∧ as the multiplication (a well known characterization oflocales).

Hence, the idempotents of S commute, and thus S is an inversesemigroup.

Showing that S is complete is straightforward, and thus S is anabstract complete pseudogroup.

Proof. (Cont.)

COROLLARY

Let A be a C*-algebra and B⊂ A a sub-C*-algebra (commutative or not).Then there is an etale groupoid

G = G B(A)def= Germs(IB(Max(A)))

associated to B such that G0 = I(B).

The groupoid of germs of IB(Max(A)) has the primitive spectrum of B asits unit space.

If B is commutative then we obtain an etale groupoid whose unit space isthe spectrum of B.

COROLLARY

EXAMPLE

Let G be a locally compact etale groupoid (i.e., a topological etalegroupoid with Hausdorff G0 and second countable locally compact G1).

The reduced C*-algebra A = C∗red(G) of G is the completion in the norminherited from the (groupoid generalization of the) left regularrepresentation on L2(G1) of the convolution algebra of compactlysupported complex-valued functions Cc(G). The commutative subalgebraB = C0(G0) can be identified with a C*-subalgebra of A, and G isisomorphic to G B(A).

More generally, the same happens in the case of the convolution algebra ofcompactly supported sections of a Fell line bundle on G (∼ a “twistedgroupoid”). (Kumjian, Renault, Exel...)

The subalgebra B is a “diagonal” of A.

EXAMPLE

In particular, let X = 1, . . . ,n be a finite set and let G be the “pairgroupoid” of X (G0 = X, G1 = X×X, d = π1, r = π2, etc.):

d(x,y) = x r(x,y) = y m((x,y),(y,z)) = (x,z) (x,y)−1 = (y,x)

The algebra A is Mn(C), the algebra of complex n×n matrices.

The algebra B⊂ A is the subalgebra of diagonal matrices Dn(C).

Examples of elements of IB(A) with n = 3: 0 λ1 0

λ2 0 00 0 λ3

: λi ∈ C

0 λ1 0

λ2 0 00 0 0

: λi ∈ C

0 λ1 0

0 0 λ2λ3 0 0

: λi ∈ C

EXAMPLE

λ2 0 00 0 λ3

: λi ∈ C

0 λ1 0

λ2 0 00 0 0

: λi ∈ C

0 λ1 0

0 0 λ2λ3 0 0

: λi ∈ C

EXAMPLE

λ2 0 00 0 λ3

: λi ∈ C

0 λ1 0

λ2 0 00 0 0

: λi ∈ C

0 λ1 0

0 0 λ2λ3 0 0

: λi ∈ C

EXAMPLE

λ2 0 00 0 λ3

: λi ∈ C

0 λ1 0

λ2 0 00 0 0

: λi ∈ C

0 λ1 0

0 0 λ2λ3 0 0

: λi ∈ C

EXAMPLE

λ2 0 00 0 λ3

: λi ∈ C

0 λ1 0

λ2 0 00 0 0

: λi ∈ C

0 λ1 0

0 0 λ2λ3 0 0

: λi ∈ C

EXAMPLE

(Cont.)

The elements of IB(A) are in a bijective correspondence with the n×npartial permutation matrices.

For instance, the previous three examples with n = 3 correspond to thefollowing three matrices, 0 1 0

1 0 00 0 1

0 1 01 0 00 0 0

0 1 00 0 11 0 0

which correspond to partial bijections on the set 1,2,3.For general n: IB(A) is isomorphic to the pseudogroup of partial bijectionson X = 1, . . . ,n, whose germ groupoid is the pair groupoid on X.

EXAMPLE

(Cont.)

1 0 00 0 1

0 1 01 0 00 0 0

0 1 00 0 11 0 0

EXAMPLE

(Cont.)

1 0 00 0 1

0 1 01 0 00 0 0

0 1 00 0 11 0 0

which correspond to partial bijections on the set 1,2,3.

For general n: IB(A) is isomorphic to the pseudogroup of partial bijectionson X = 1, . . . ,n, whose germ groupoid is the pair groupoid on X.

EXAMPLE

(Cont.)

1 0 00 0 1

0 1 01 0 00 0 0

0 1 00 0 11 0 0

EXAMPLE

(Cont.)

If B = C.1 (for general n) the elements of IB(A) are the linear spans ofthe (total) n×n permutation matrices, such as the following for n = 3:

0 λ 0λ 0 00 0 λ

: λ ∈ C

In this case both IB(A) and G B(A) are isomorphic to the symmetric groupon n elements Sn.

EXAMPLE

(Cont.)

0 λ 0λ 0 00 0 λ

: λ ∈ C

EXAMPLE

(Cont.)

0 λ 0λ 0 00 0 λ

: λ ∈ C

EXAMPLE

(Cont.)

As an intermediate example, again with n = 3, take B to be the2-dimensional space containing the diagonal matrices of the following formfor all λ1,λ2 ∈ C: λ1 0 0

0 λ1 00 0 λ2

Now the associated groupoid G = G B(A) has G0 with two points, one ofthem with trivial isotropy and the other with isotropy group S2 ∼= Z2:

ZZ •

EXAMPLE

(Cont.)

0 λ1 00 0 λ2

ZZ •

EXAMPLE

(Cont.)

0 λ1 00 0 λ2

ZZ •

EXAMPLE

Let X be a compact Hausdorff space and let Γ be a discrete group thatacts on X.

Γ acts on the commutative C*-algebra B = C(X) of continuous functionson X, and we can define the cross-product (noncommutative) C*-algebraA = Γ n B, which contains B as a sub-C*-algebra.

In fact A is the groupoid algebra C∗red(G) where G = Γ n X is thecorresponding action groupoid.

EXAMPLE

In Renault’s terminology an etale groupoid is said to be topologicallyprincipal if the set of units whose isotropy is trivial is dense on G0. This isclosely related to G being the germ groupoid of a pseudogroup. (Moerdijkand Mrcun call the latter an effective groupoid.)

THEOREM (RENAULT)Let A be a C*-algebra and B⊂ A a subalgebra. The pair (A,B) arises froma Fell line bundle on a locally compact Hausdorff topological principal etalegroupoid if and only if B is a Cartan subalgebra in the following sense:

1 B contains an approximate unit of A;

2 B is maximal abelian (a “masa”);

3 B is regular (its normalizer N(B) generates A as a C*-algebra);

4 There exists a faithful conditional expectation P of A onto B.

SHEAVES

A sheaf on a space X (or locale) can be interpreted as a set Sequipped with an Ω(X)-valued equality (Higgs, Fourman–Scott):

the set S consists of all the local sections of the sheaf;if s and t are local sections over open sets U and V the value of theequality “s = t” is the largest open set Ast ⊂U∩V such that s|Ast = t|Ast .

The square matrix A : S×S→Ω(X) thus defined is idempotent andsymmetric.

Maps can be represented as “functional relations”, which again arerepresented by Ω-valued matrices, and we obtain a category ofΩ(X)-sets which is equivalent to the category of sheaves on X.

There is a distinction between complete Ω(X)-sets (those that areactually obtained from sheaves) and the sets that are just equippedwith projection matrices; however, the resulting categories areequivalent.

SHEAVES

the set S consists of all the local sections of the sheaf;

if s and t are local sections over open sets U and V the value of theequality “s = t” is the largest open set Ast ⊂U∩V such that s|Ast = t|Ast .

SHEAVES

Various theories of sheaves on quantales (and quantaloids) have beenput forward in the last thirty years, with this idea of sheaves asmatrices in the background.

Some are rather direct generalizations of the theory for locales, and allthe variations become equivalent if the quantales are stably Gelfand.

In particular, for a stably Gelfand quantale Q we obtain anequivalence between the category of complete Q-valued sets and thatof Q-sets (sets equipped with Q-valued projection matrices).

This and other facts have been noticed by Garraway (2005), who callsstably Gelfand quantales pseudo-rightsided.

Punchline: for stably Gelfand quantales the existing theories ofsheaves on involutive quantales converge to a single one!

SHEAVES

EXAMPLE

Let G be an etale groupoid.

The category of sheaves on the quantale O(G) is equivalent to the toposof equivariant sheaves on G (the classifying topos of G).

OPEN QUESTIONS

Quantale theory: Abelian sheaves? Cohomology of quantalesbeyond groupoid quantales? (Note that in general the quantalesheaves do not form toposes!)

Logic: Interpretation of higher order modal logic in sheaves overquantales (joint work with Palmigiano)

C*-algebras: We are provided with a “stable” notion of sheaf forquantales of the form Max(A):

Can an abelian version of these sheaves provide an appropriate notionof structure sheaf capable of doing away with the excessive abundanceof automorphisms of Max(A)?Alternative characterizations of groupoid C*-algebras?Relation with the K-theory sheaves of Dadarlat? (Classificationprogram for C*-algebras)

Physics: There is more to a commutative sub-C*-algebra than justits spectrum. Implications?

OPEN QUESTIONS

Can an abelian version of these sheaves provide an appropriate notionof structure sheaf capable of doing away with the excessive abundanceof automorphisms of Max(A)?

Alternative characterizations of groupoid C*-algebras?Relation with the K-theory sheaves of Dadarlat? (Classificationprogram for C*-algebras)

OPEN QUESTIONS

Can an abelian version of these sheaves provide an appropriate notionof structure sheaf capable of doing away with the excessive abundanceof automorphisms of Max(A)?Alternative characterizations of groupoid C*-algebras?

Relation with the K-theory sheaves of Dadarlat? (Classificationprogram for C*-algebras)

OPEN QUESTIONS

stably gelfand quantales cartan sub-c*-algebraspmr/slides-ic-aug-2009.pdfstably gelfand quantales,...

Documents