separability idempotents in c*-algebras (icm satellite conference … · 2014-08-13 · diﬀerent...

Separability idempotents in C ∗-algebras(ICM Satellite Conference at Cheongpung, KOREA)

Byung-Jay Kahng

Canisius College, Buffalo USA

August 2014

Byung-Jay Kahng (Canisius College, Buffalo USA)Separability idempotents in C∗-algebras August 2014 1 / 20

Intro.

In this talk ...

Purely algebraic case

Motivation from groupoids and weak multiplier Hopf algebras

Definition of a separability idempotent in the C ∗-algebra setting

Properties

An example

Outlook: Locally compact quantum groupoids

(*) This is based on a joint work with Alfons Van Daele (KU Leuven).


An example: Purely algebraic case

Let A be a (finite-dimensional) algebra (over a field k). Suppose we havean element E ∈ A⊗ Aop, given by E =

∑Ni=1 xi ⊗ yi , such that∑

i xiyi = 1A and∑

i axi ⊗ yi =∑

i xi ⊗ yia, for all a ∈ A, also∑i xi ⊗ ayi =

∑i xia⊗ yi , for all a ∈ A.

Properties:

1 E 2 = E

2 E ∈ B ⊗ C , where C ∼= Bop

3 There exist anti-isomorphisms γ : B → C and γ′ : C → B such that

E (b ⊗ 1) = E(1⊗ γ(b)

), and (1⊗ c)E =

(γ′(c)⊗ 1

)E ,

for b ∈ B, c ∈ C .


Purely algebraic case

Proof.

Consider two copies of E , written as E =∑

i xi ⊗ yi =∑

j x ′j ⊗ y ′j . Take

B = A, C = Aop. Then the product E 2 in B ⊗ C , expressed in A⊗ A is∑i ,j

x ′j xi ⊗ yiy′j =

∑i ,j

xi ⊗ yix′j y′j =

∑i

xi ⊗ yi

showing that E is an idempotent (E 2 = E ) in B ⊗ C .Moreover, with γ′ = idAop→A, we have:

(γ′(c)⊗ 1

)E = (1⊗ c)E .

Similar for γ = idA→Aop , such that E(1⊗ γ(b)

)= E (b ⊗ 1).

(*) Existence of such an element is a property of the algebra B.(*) Such elements naturally appear in the case of groupoids, or moregenerally, for weak multiplier Hopf algebras.


Groupoid: Review

A set G is called a groupoid over the base set G (0), written G ⇒ G (0),if it is equipped with a pair of maps s : G → G (0) and t : G → G (0),and the multiplication on G is partially defined:

i.e. (p, q) 7→ p · q is valid only when s(p) = t(q).We can naturally regard G (0) as embedded in G . There is also theinversion map p 7→ p−1. We have s(p−1) = t(p) and t(p−1) = s(p).

All groups are groupoids: G (0) = {e}All (equivalent) relations are groupoids: (x , y)(y , z) = (x , z)

Group actions give transformation groupoids

Many other examples, with more structure

we can also consider locally compact groupoids, together with asuitable (locally compact) topology on G and a left Haar system.


K (G ) as a weak (multiplier) Hopf algebra

For G a groupoid, consider A = K (G ), set of all complex-valued functionson G having finite support. So M(A) = C (G ), all complex functions.- We have a comultiplication ∆ : A → M(A⊗ A), a ∗-homomorphism:

(∆f )(p, q) :=

{f (pq) if s(p) = t(q)

0 otherwise

- ∆ will satisfy the coassociativity condition (loosely speaking“(id⊗∆)∆ = (∆⊗ id)∆”, but need some care).

- Here, consider the following element E :

E (p, q) =

{1 if s(p) = t(q)0 otherwise

- In particular, if G is finite, then 1 ∈ A, M(A) = A, E = ∆(1). And,(A,∆) becomes a weak Hopf algebra, with the antipode S : A → A givenby S(f )(p) = f (p−1). If G is not finite, (A,∆) is a weak multiplier Hopfalgebra. Then we have: E ∈ M(A⊗ A).- While E 2 = E and E (∆f ) = ∆f = (∆f )E , f ∈ A, but clearly, E 6= 1⊗ 1.




(∆f )(p, q) :=


0 otherwise

- ∆ will satisfy the coassociativity condition (loosely speaking“(id⊗∆)∆ = (∆⊗ id)∆”, but need some care).- Here, consider the following element E :

E (p, q) =


- In particular, if G is finite, then 1 ∈ A, M(A) = A, E = ∆(1). And,(A,∆) becomes a weak Hopf algebra, with the antipode S : A → A givenby S(f )(p) = f (p−1). If G is not finite, (A,∆) is a weak multiplier Hopfalgebra. Then we have: E ∈ M(A⊗ A).

- While E 2 = E and E (∆f ) = ∆f = (∆f )E , f ∈ A, but clearly, E 6= 1⊗ 1.




(∆f )(p, q) :=


0 otherwise

- ∆ will satisfy the coassociativity condition (loosely speaking“(id⊗∆)∆ = (∆⊗ id)∆”, but need some care).- Here, consider the following element E :

E (p, q) =


- In particular, if G is finite, then 1 ∈ A, M(A) = A, E = ∆(1). And,(A,∆) becomes a weak Hopf algebra, with the antipode S : A → A givenby S(f )(p) = f (p−1). If G is not finite, (A,∆) is a weak multiplier Hopfalgebra. Then we have: E ∈ M(A⊗ A).- While E 2 = E and E (∆f ) = ∆f = (∆f )E , f ∈ A, but clearly, E 6= 1⊗ 1.


Case of A = CG

- Consider instead A = CG = {∑

p f (p)λp : f ∈ K (G )}, equipped withthe convolution product, given by λpλq = λpq if pq makes sense and0 otherwise.- Define ∆ : A → A⊗ A by ∆(λp) = λp ⊗ λp. This also becomes aWHA/WMHA, together with S : λp 7→ λp−1 and E =

∑e∈G (0) λe ⊗ λe .

- Again, E 2 = E and E (∆f ) = ∆f = (∆f )E , f ∈ A, but E 6= 1⊗ 1.

- In the theory of weak multiplier Hopf algebras, the existence of acanonical idempotent element E plays a fundamental role.- There are of course other properties of such an E , but the following slidesuggest that E is a “separability idempotent”, in a certain sense:


E is a separability idempotent

Let E be the canonical idempotent of A = K (G ), A = CG , or anyWMHA. Let B ⊂ M(A) and C ⊂ M(A) be the “source algebra” andthe “target algebra”. For instance, in the case of A = K (G ), considerB = K (G (0)), viewed as a subalgebra of M(A) by pull-back of the sourcemap s : G → G (0), and similarly, C = K (G (0)) ⊂ M(A), but this time viathe target map. Among the properties of E , we have:

1 E 2 = E

2 E ∈ M(B ⊗ C ), where C ∼= Bop.

3 There exist anti-isomorphisms γ : B → C and γ′ : C → B such that

E (b ⊗ 1) = E(1⊗ γ(b)

), and (1⊗ c)E =

(γ′(c)⊗ 1

)E ,

for b ∈ B, c ∈ C . [Consider γ = S |B , γ′ = S |C , where S : A → A isthe antipode map for (A,∆).]


Separability idempotents in the C ∗-algebra setting

- We wish to define a notion in the C ∗-algebra framework that generalizesthe purely algebraic notion of a separability idempotent. That will beuseful in developing a general C ∗-algebraic theory of locally compactquantum groupoids.

- We will need to relax some conditions, but in some cases, the notioncould be more restrictive.- Among the biggest hurdles is the likelihood that the anti-homomorphismmaps γ, γ′ will be unbounded in general, and only densely-defined.- So it is not practical to directly generalize the purely algebraic definitionor even some of the characterizing properties. We will need a ratherdifferent approach: Use weights.(*) This is a similar philosophy as in the theory of general locally compactquantum groups, where we avoid introducing the antipode map in thedefinition but instead work with the Haar weights.



- We wish to define a notion in the C ∗-algebra framework that generalizesthe purely algebraic notion of a separability idempotent. That will beuseful in developing a general C ∗-algebraic theory of locally compactquantum groupoids.- We will need to relax some conditions, but in some cases, the notioncould be more restrictive.- Among the biggest hurdles is the likelihood that the anti-homomorphismmaps γ, γ′ will be unbounded in general, and only densely-defined.

- So it is not practical to directly generalize the purely algebraic definitionor even some of the characterizing properties. We will need a ratherdifferent approach: Use weights.(*) This is a similar philosophy as in the theory of general locally compactquantum groups, where we avoid introducing the antipode map in thedefinition but instead work with the Haar weights.



- We wish to define a notion in the C ∗-algebra framework that generalizesthe purely algebraic notion of a separability idempotent. That will beuseful in developing a general C ∗-algebraic theory of locally compactquantum groupoids.- We will need to relax some conditions, but in some cases, the notioncould be more restrictive.- Among the biggest hurdles is the likelihood that the anti-homomorphismmaps γ, γ′ will be unbounded in general, and only densely-defined.- So it is not practical to directly generalize the purely algebraic definitionor even some of the characterizing properties. We will need a ratherdifferent approach: Use weights.(*) This is a similar philosophy as in the theory of general locally compactquantum groups, where we avoid introducing the antipode map in thedefinition but instead work with the Haar weights.


Weights on C ∗-algebras: Preliminaries

- Let A be a C ∗-algebra. A weight ϕ on A is ϕ : A+ → [0,∞], such that ϕis additive and ϕ(λx) = λϕ(x) for λ ≥ 0 and x ∈ A+.- Given a weight ϕ, we can consider the following sets:

Nϕ ={x ∈ A : ϕ(x∗x) <∞

}, Mϕ = Nϕ

∗Nϕ.

Then Nϕ is a left ideal in A and Mϕ is a ∗-subalgebra of A. One cannaturally extend ϕ to ϕ : Mϕ → C.- Proper weights are densely-defined (semi-finite) and lowersemi-continuous. It is faithful if ϕ(x) = 0, x ∈ A+, means x = 0. Wemostly work with l.s.c., densely-defined, faithful weights on A.- One can perform the GNS construction (Hϕ,Λϕ, πϕ): Namely ...(i) The space Nϕ, together with (x , y) 7→ ϕ(y∗x), can be completed toobtain a Hilbert space, denoted Hϕ; (ii) Write Λϕ : Nϕ → Hϕ for the“GNS map”, which is the canonical injection; (iii) There also exists anon-degenerate ∗-representation πϕ of A on Hϕ, such thatπϕ(a)Λϕ(x) = Λϕ(ax), for a ∈ A, x ∈ Nϕ.


KMS weights

- A faithful proper weight ϕ on A is called a KMS weight, if there exists anorm-continuous one-parameter group of automorphisms (σt)t∈R of Asuch that ϕ ◦ σt = ϕ for all t ∈ R, and

ϕ(a∗a) = ϕ(σi/2(a)σi/2(a)

∗) for all a ∈ D(σi/2).

Here, σi/2 is the analytic generator of (σt), at z = i/2.

- It is known that a faithful KMS weight lifts to a normal, semi-finite,faithful (n.s.f.) weight ϕ̃ on the von Neumann algebra πψ(A)′′, for whichthere exists a well-developed “modular theory” (Tomita–Takesaki theory).- (A useful property): Let x ∈ Nϕ ∩N∗

ϕ and a ∈ N∗ϕ ∩ D(σ−i ) be such

that σ−i (a) ∈ Nϕ. Then ϕ(ax) = ϕ(xσ−i (a)

).

- In what follows, we will work with faithful KMS weights.


Separability idempotent E : Definition

Consider the following triple (E ,B, ν):

B is a C ∗-algebra;

ν is a faithful KMS weight on B, together with its continuousone-parameter group (σνt );

E ∈ M(B ⊗ C ) is a self-adjoint idempotent element, for someC ∗-algebra C such that there exists a ∗-anti-isomorphism R : B → C .(So C ∼= Bop as C ∗-algebras. Loosely: B: “source”, C : “target” ...)

For such a triple, assume that

(ν ⊗ id)(E ) = 1

(ν ⊗ id)(E (b ⊗ 1)

)= R

(σνi/2(b)

), for b ∈ Dom(σνi/2).

Then E is a separability idempotent.

We will write: γ(b) := (R ◦ σνi/2)(b), for b ∈ D(γ) = Dom(σνi/2).See next page for some consequences.


Results: E is a separability idempotent

Here are some results that follow from the definition in the previous page:

(B, ν) completely characterizes E .

γ is an injective, closed linear map, such that D(γ) is dense in B,Ran(γ) is dense in C .

γ is an anti-homomorphism: γ(bb′) = γ(b′)γ(b), for b, b′ ∈ D(γ).

There exists a faithful KMS weight µ on C , given by µ = ν ◦ R−1.

There exists a dense subset D(γ′) of C and a map γ′ : D(γ′) → B,which is again a densely-defined, injective, closed linear map having adense range. It is also an anti-homomorphism.

γ′(γ(b)∗

)∗= b, b ∈ D(γ) and γ

(γ′(c)∗

)∗= c , c ∈ D(γ′).

E (b ⊗ 1) = E(1⊗ γ(b)

), for b ∈ D(γ),

(1⊗ c)E =(γ′(c)⊗ 1

)E , for c ∈ D(γ′).

(*) We can see that E satisfies properties that are analogous to the purelyalgebraic notion of E being a “separability idempotent”.


Results: E is a separability idempotent

We also have ...

(γ ◦ γ′)(c) = σµ−i (c) and (γ−1 ◦ γ′−1)(b) = σν−i (b).

(σνt ⊗ σµ−t)(E ) = E , for any t.

(γ′ ⊗ γ)(σE ) = E and (γ ⊗ γ′)(E ) = σE , where σ denotes the flipmap on M(B ⊗ C ).

And, E is “full”:

span{(id⊗ω)(E (1⊗ c)) | c ∈ C , ω ∈ C ∗} is dense in B.

span{(id⊗ω)((1⊗ c)E ) | c ∈ C , ω ∈ C ∗} is dense in B.

span{(ω ⊗ id)(E (b ⊗ 1)) | b ∈ B, ω ∈ B∗} is dense in C .

span{(ω ⊗ id)((b ⊗ 1)E ) | b ∈ B, ω ∈ B∗} is dense in C .

If E (1⊗ c) = 0, c ∈ C , then necessarily c = 0. Other similar resultsalso hold.


Example (3)

(3). Let B = C = B0(H), the C ∗-algebra of compact operators on aHilbert space H. We wish to define a separability idempotentE ∈ B(H⊗H) = M(B ⊗ C ).

- Fix an orthonormal basis (ξj)j∈J for H. Define E0 :=∑

i ,j∈J ei ,j ⊗ ei ,j ,where ei ,j ∈ B(H) is defined by ei ,j(v) := 〈v, ξj〉 = vjξi , for v ∈ H. Bybasic linear algebra, we have: E0(v ⊗w) =

∑i ,j∈J vjwjξi ⊗ ξi .

- Then, for a ∈ B(H), we have: E0(a⊗ 1) = E0(1⊗ aT ).

- Next, consider r ∈ B(H), which is invertible and Tr(r∗r) = 1. So, r is acompact (Hilbert–Schmidt) operator, and this automatically means thatr−1 is unbounded in general, unless H is finite-dimensional. We thendefine E ∈ B(H⊗H) by

E := (r ⊗ 1)E0(r∗ ⊗ 1).


Example (3) continued ...

- Let E := (r ⊗ 1)E0(r∗ ⊗ 1),

- Clearly E ∗ = E , and one can show that E 2 = E using Tr(r∗r) = 1.

Proof.

E0(r∗r ⊗ 1)E0

=∑

i ,j ,k,l∈J

eij ⊗ eij(r∗r ⊗ 1)ekl ⊗ ekl =

∑i ,j ,k,l∈J

eij r∗rekl ⊗ δjkeil

=∑

i ,j ,l∈J

eij r∗rejl ⊗ eil =

∑i ,l∈J

Tr(r∗r)eil ⊗ eil

= Tr(r∗r)E0 = E0.

So E 2 = (r ⊗ 1)E0(r∗r ⊗ 1)E0(r

∗ ⊗ 1) = (r ⊗ 1)E0(r∗ ⊗ 1) = E .


Example (3) continued ...

-On B = B0(H), consider ν := Tr(r−1 · r∗−1) = Tr(p ·), wherep = r∗−1r−1 = (rr∗)−1. It is a faithful KMS weight on B, whoseassociated one parameter group of automorphisms is (σνt )t∈R, defined byσνt (b) = pitbp−it .

-Write r∗ = u|r∗|, where |r∗| = (rr∗)12 = p−

12 . Then, with the

∗-anti-isomorphism R : B → C defined by R(b) := (ubu∗)T ,we can indeed show that (E ,B, ν) forms a separability triple.

-In addition, we have, for b ∈ D(σνi/2):

E (b ⊗ 1) = E(1⊗ γ(b)

)= E

(1⊗ (r∗br∗−1)T

).

Also, for c ∈ D(σµ−i/2), with µ = ν ◦ R−1, we have:

(1⊗ c)E =(γ′(c)⊗ 1

)E =

(rcT r−1 ⊗ 1

)E .

Observe: γ′(γ(b)∗

)∗= b, γ

(γ′(c)∗

)∗= c , and γ, γ′ are

anti-homomorphisms.


Final comments: On locally compact quantum groupoids

- Clarifying the notion of the separability idempotent element in theC ∗-algebra setting helps us to generalize the weak multiplier Hopf algebrasto define a class of locally compact quantum groupoids. Such a quantumgroupoid is given by the data (A,∆,E ,B, ν, ϕ, ψ), from which otherresults/properties follow.- In particular, the left regular representation of such a quantum groupoidis described by a certain partial isometry W with W ∗W = E , such that

A ={(id⊗ω)(W ) : ω ∈ B(Hϕ)∗

}‖ ‖

and∆x = W ∗(1⊗ x)W ,

- The antipode map is constructed/defined in terms of a polardecomposition.


Final comments: On locally compact quantum groupoids

- The existence of a separability idempotent is more of a condition onthe base algebra B and the weight ν on it. It ensures the quasi-invarianceof ν, which is necessary even in the ordinary locally compact groupoidcase. On the other hand, it restricts the possible choice of B in that thebase C ∗-algebra B must be postliminal.

- The notion for a more general class of C ∗-algebraic (locally compact)quantum groupoids is rather technical and seems some time away fromfruition. But there have been recent developments in the purelyalgebraic setting (“multiplier Hopf algebroids”), and a full theory existsin the measurable setting (“measured quantum groupoids”).

(*) Even so, the class of locally compact quantum groupoids as definedabove contains all quantum groups as a subclass, and forms a self-dualcategory.


Thank you

Thank you for your attention!


separability idempotents in c*-algebras (icm satellite conference … · 2014-08-13 · diﬀerent...

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