compact quantum groupoids related to C*-algebras

compact quantum groupoids related to C*-algebras

1

Compact quantum groupoids (CGQs) and C*-algebras

1.1

Introduction: von Neumann and C*-algebras. Quantum operator algebras in quantum theories

C*-

algebra

has evolved as a key

concept

in

quantum operator algebra

(QOA) after the

introduction

of the

von Neumann algebra

for the

mathematical foundation

of quantum mechanics. The von Neumann algebra classification is simpler and studied in greater depth than that of general C*-algebra classification theory. The importance of
C*-algebras for understanding the geometry of

quantum state spaces

(viz. Alfsen and Schultz, 2003

[

1

]

) cannot be overestimated. Moreover, the introduction of

non-commutative

C*-algebras in

noncommutative geometry

has already played important roles in expanding the

Hilbert space

perspective of quantum mechanics developed by von Neumann. Furthermore,

extended quantum symmetries

are currently being approached in terms of

groupoid

C*-

convolution algebra

and their

representations

; the latter also enter into the construction of

compact

quantum groupoids

as developed in the Bibliography cited, and also briefly outlined here in the third

section

. The fundamental

connections

that exist between

categories

of

C

*

-algebras and those of von Neumann and other quantum operator algebras, such as JB- or JBL- algebras are yet to be completed and are the subject
of in depth studies

[

1

]

.

1.2

Basic definitions

Let us recall first the basic definitions of C*-algebra and

involution

on a complex algebra.
Further details can be found in a separate entry focused on

C

*

-algebras (

http://planetmath.org/CAlgebra

).

A

C*-algebra

is simultaneously a *–algebra and a

Banach space

-with additional conditions- as defined next.

Let us consider first the definition of an

involution

on a complex algebra

𝔄

.

Definition 1.1

.

An

involution

on a complex algebra

𝔄

is a

real–

linear map

T

↦

T

*

such that for all

S

,

T

∈

𝔄

and

λ

∈

ℂ

, we have

T

*

*

=

T

,

(

S

⁢

T

)

*

=

T

*

⁢

S

*

,

(

λ

⁢

T

)

*

=

λ

¯

⁢

T

*

.

A

*-algebra

is said to be a complex

associative algebra

together with an involution

*

.

Definition 1.2

.

A

C*-algebra

is simultaneously a *-algebra and a Banach space

𝔄

,
satisfying for all

S

,

T

∈

𝔄

the following conditions:

∥

S

∘

T

∥

≤

∥

S

∥

⁢

∥

T

∥

,

∥

T

*

⁢

T

∥

2

=

∥

T

∥

2

.

One can easily verify that

∥

A

*

∥

=

∥

A

∥

.

By the above axioms a C*–algebra is a special case of a

Banach algebra

where the latter requires the above C*-norm property, but not the involution (

*

) property.

Given Banach spaces

E

,

F

the space

ℒ

⁢

(

E

,

F

)

of (

bounded

) linear operators from

E

to

F

forms a Banach space, where for

E

=

F

, the space

ℒ

⁢

(

E

)

=

ℒ

⁢

(

E

,

E

)

is a Banach algebra with respect to the norm

∥

T

∥

:=

sup

⁡

{

∥

T

⁢

u

∥

:

u

∈

E

,

∥

u

∥

=

1

}

.

In

quantum field theory

one may start with a Hilbert space

H

, and consider the Banach
algebra of

bounded linear operators

ℒ

⁢

(

H

)

which given to be

closed under

the usual
algebraic operations and taking

adjoints

, forms a

*

–algebra of

bounded operators

, where the
adjoint

operation

functions

as the involution, and for

T

∈

ℒ

⁢

(

H

)

we have :

∥

T

∥

:=

sup

⁡

{

(

T

⁢

u

,

T

⁢

u

)

:

u

∈

H

,

(

u

,

u

)

=

1

}

,

and

∥

T

⁢

u

∥

2

=

(

T

⁢

u

,

T

⁢

u

)

=

(

u

,

T

*

⁢

T

⁢

u

)

≤

∥

T

*

⁢

T

∥

⁢

∥

u

∥

2

.

By a

morphism

between C*-algebras

𝔄

,

𝔅

we mean a linear map

ϕ

:

𝔄

⟶

𝔅

, such that for all

S

,

T

∈

𝔄

, the following hold :

ϕ

⁢

(

S

⁢

T

)

=

ϕ

⁢

(

S

)

⁢

ϕ

⁢

(

T

)

,

ϕ

⁢

(

T

*

)

=

ϕ

⁢

(

T

)

*

,

where a

bijective

morphism is said to be an

isomorphism

(in which case it is then an

isometry

). A fundamental

relation

is that any norm-closed

*

-algebra

𝒜

in

ℒ

⁢

(

H

)

is a C*-algebra (

http://planetmath.org/CAlgebra3

), and

conversely

, any C*-algebra (

http://planetmath.org/CAlgebra3

) is isomorphic to a norm–closed

*

-algebra in

ℒ

⁢

(

H

)

for some Hilbert space

H

.
One can thus also define

the category

C

*

of C*-algebras and morphisms between C*-algebras

.

For a C*-algebra (

http://planetmath.org/CAlgebra3

)

𝔄

, we say that

T

∈

𝔄

is

self–adjoint

if

T

=

T

*

. Accordingly, the self–adjoint part

𝔄

s

⁢

a

of

𝔄

is a

real
vector space

since we can decompose

T

∈

𝔄

s

⁢

a

as :

T

=

T

′

+

T

′′

:=

1

2

⁢

(

T

+

T

*

)

+

ι

⁢

(

-

ι

2

)

⁢

(

T

-

T

*

)

.

A

commutative

C* -algebra is one for which the

associative

multiplication

is
commutative. Given a commutative C* -algebra

𝔄

, we have

𝔄

≅

C

⁢

(

Y

)

,
the algebra of

continuous functions

on a compact

Hausdorff space

Y

.

The classification of

C

*

-algebras

is far more complex than that of von Neumann algebras that provide
the fundamental

algebraic

content

of

quantum state

and operator spaces in

quantum theories

.

1.3

Quantum groupoids and the groupoid C*-algebra

Quantum groupoid (or their dual, weak Hopf coalgebras) and algebroid

symmetries

figure prominently both in the theory of dynamical

deformations

of

quantum groups

(or their dual

Hopf algebras

) and the quantum Yang–Baxter

equations

(Etingof et al., 1999, 2001;

[

12

,

E2k

]

). On the other hand, one can also consider the natural

extension

of

locally compact

(quantum) groups to locally compact
(proper)

groupoids

equipped with a

Haar measure

and a corresponding

groupoid representation

theory
(Buneci, 2003,

[

MB2k3

]

) as a major, potentially interesting source for locally compact (but
generally

non-Abelian

) quantum groupoids. The corresponding quantum groupoid representations on bundles of
Hilbert spaces extend quantum symmetries well beyond those of quantum groups and their dual Hopf algebras, and also beyond the simpler operator algebra representations, and are also

consistent

with the locally compact quantum group representations. The latter quantum groups are neither Hopf algebras, nor are they

equivalent

to Hopf algebras or their dual coalgebras. As pointed out in the previous section, quantum groupoid representations are, however, the next important step towards unifying quantum field theories with General Relativity in a locally covariant and quantized form. Such representations need not however be restricted to

weak Hopf algebra

representations, as the latter have no known connection to any type of GR theory and also appear to be inconsistent with GR.

Quantum groupoids were recently considered as weak C* -Hopf algebras, and were studied in relationship to the non- commutative symmetries of depth 2 von Neumann subfactors. If

A

⊂

B

⊂

B

1

⊂

B

2

⊂

…

(1.1)

is the Jones extension induced by a finite index depth

2

inclusion

A

⊂

B

of

I

⁢

I

1

factors, then

Q

=

A

′

∩

B

2

admits a quantum groupoid

structure

and acts on

B

1

, so that

B

=

B

1

Q

and

B

2

=

B

1

⋊

Q

. Similarly, ‘paragroups’ derived from weak C* -Hopf algebras comprise (quantum) groupoids of

equivalence classes

such as those associated with

6

⁢

j

-

symmetry groups

(relative to a fusion rules algebra). They correspond to type

I

⁢

I

von Neumann algebras in quantum mechanics, and arise as symmetries where the local subfactors (in the sense of containment of quantum observables within fields) have depth 2 in the
Jones extension. A related question is how a von Neumann algebra

W

*

, such as
of finite index depth 2, sits inside a weak Hopf algebra formed as the crossed product

W

*

⋊

A

.

1.4

Quantum compact groupoids

Compact quantum groupoids were introduced in Landsman (1998; ref.

[

L98

]

) as a
simultaneous

generalization

of a compact groupoid and a quantum group. Since this construction is relevant to the definition of locally compact quantum groupoids and their representations investigated here, its exposition is required before we can step up to the next level of generality. Firstly, let

𝔄

and

𝔅

denote C*–algebras equipped with a *–

homomorphism

η

s

:

𝔅

⟶

𝔄

, and a *–

antihomomorphism

η

t

:

𝔅

⟶

𝔄

whose images in

𝔄

commute. A non–commutative Haar measure is defined as a completely

positive map

P

:

𝔄

⟶

𝔅

which

satisfies

P

⁢

(

A

⁢

η

s

⁢

(

B

)

)

=

P

⁢

(

A

)

⁢

B

. Alternatively, the composition

ℰ

=

η

s

∘

P

:

𝔄

⟶

η

s

⁢

(

B

)

⊂

𝔄

is a

faithful

conditional expectation.

Next consider

𝖦

to be a (topological) groupoid, and let us denote by

C

c

⁢

(

𝖦

)

the space of smooth complex–valued functions with compact support on

𝖦

. In particular, for all

f

,

g

∈

C

c

⁢

(

𝖦

)

, the
function defined via convolution

(

f

*

g

)

⁢

(

γ

)

=

∫

γ

1

∘

γ

2

=

γ

f

⁢

(

γ

1

)

⁢

g

⁢

(

γ

2

)

,

(1.2)

is again an element of

C

c

⁢

(

𝖦

)

, where the

convolution product

defines the composition law on

C

c

⁢

(

𝖦

)

. We can turn

C

c

⁢

(

𝖦

)

into a

*

-algebra once we have defined the involution

*

, and this is done by specifying

f

*

⁢

(

γ

)

=

f

⁢

(

γ

-

1

)

¯

.

We recall that following Landsman (1998) a

representation

of a groupoid

𝔾

, consists of a
family (or field) of Hilbert spaces

{

ℋ

x

}

x

∈

X

indexed by

X

=

Ob

⁢

𝔾

, along with a

collection

of maps

{

U

⁢

(

γ

)

}

γ

∈

𝔾

, satisfying:

1.

U

⁢

(

γ

)

:

ℋ

s

⁢

(

γ

)

⟶

ℋ

r

⁢

(

γ

)

,
is

unitary

.

2.

U

⁢

(

γ

1

⁢

γ

2

)

=

U

⁢

(

γ

1

)

⁢

U

⁢

(

γ

2

)

, whenever

(

γ

1

,

γ

2

)

∈

𝔾

(

2

)

(the set of arrows).

3.

U

⁢

(

γ

-

1

)

=

U

⁢

(

γ

)

*

, for all

γ

∈

𝔾

.

Suppose now

𝖦

l

⁢

c

is a

Lie groupoid

. Then the

isotropy group

𝖦

x

is a

Lie group

, and for a (left or right) Haar
measure

μ

x

on

𝖦

x

, we can consider the Hilbert
spaces

ℋ

x

=

L

2

⁢

(

𝖦

x

,

μ

x

)

as exemplifying the
above sense of a representation. Putting aside some technical
details which can be found in Connes (1994) and Landsman (2006), the
overall idea is to define an

operator

of Hilbert spaces

π

x

⁢

(

f

)

:

L

2

⁢

(

𝖦

𝗑

,

μ

x

)

⟶

L

2

⁢

(

𝖦

x

,

μ

x

)

,

(1.3)

given by

(

π

x

⁢

(

f

)

⁢

ξ

)

⁢

(

γ

)

=

∫

f

⁢

(

γ

1

)

⁢

ξ

⁢

(

γ

1

-

1

⁢

γ

)

⁢

𝑑

μ

x

,

(1.4)

for all

γ

∈

𝖦

x

, and

ξ

∈

ℋ

x

. For each

x

∈

X

=

Ob

⁢

𝖦

,

π

x

defines an

involutive

representation

π

x

:

C

c

⁢

(

𝖦

)

⟶

ℋ

x

. We can define a norm on

C

c

⁢

(

𝖦

)

given by

∥

f

∥

=

sup

x

∈

X

⁡

∥

π

x

⁢

(

f

)

∥

,

(1.5)

whereby the

completion

of

C

c

⁢

(

𝖦

)

in this norm, defines

the

reduced

C*–algebra

C

r

*

⁢

(

G

)

of

G

l

⁢

c

. It is
perhaps the most commonly used C*–algebra for Lie groupoids
(groups) in noncommutative geometry.

The next step requires a little familiarity with the theory of

Hilbert modules

. We define a left

𝔅

–action

λ

and a right

𝔅

–action

ρ

on

𝔄

by

λ

⁢

(

B

)

⁢

A

=

A

⁢

η

t

⁢

(

B

)

and

ρ

⁢

(

B

)

⁢

A

=

A

⁢

η

s

⁢

(

B

)

. For the sake of

localization

of the
intended Hilbert module, we implant a

𝔅

–valued

inner
product

on

𝔄

given by

⟨

A

,

C

⟩

𝔅

=

P

⁢

(

A

*

⁢

C

)

. Let us recall that

P

is defined as a

completely positive map

.
Since

P

is faithful, we fit a new norm on

𝔄

given by

∥

A

∥

2

=

∥

P

⁢

(

A

*

⁢

A

)

∥

𝔅

. The completion of

𝔄

in this new
norm is denoted by

𝔄

-

leading then to a Hilbert
module over

𝔅

.

The

tensor product

𝔄

-

⊗

𝔅

𝔄

-

can be shown to be a Hilbert bimodule over

𝔅

,
which for

i

=

1

,

2

, leads to *–homorphisms

φ

i

:

𝔄

⟶

ℒ

𝔅

⁢

(

𝔄

-

⊗

𝔄

-

)

. Next is to define the (unital) C*–algebra

𝔄

⊗

𝔅

𝔄

as the C*–algebra

contained

in

ℒ

𝔅

⁢

(

𝔄

-

⊗

𝔄

-

)

that is

generated by

φ

1

⁢

(

𝔄

)

and

φ

2

⁢

(

𝔄

)

. The last stage of the recipe for defining a
compact quantum groupoid entails considering a certain

coproduct

operation

Δ

:

𝔄

⟶

𝔄

⊗

𝔅

𝔄

, together with a coinverse

Q

:

𝔄

⟶

𝔄

that it is both an algebra and
bimodule antihomomorphism. Finally, the following axiomatic
relationships are observed :

(

id

⊗

𝔅

Δ

)

∘

Δ

=

(

Δ

⊗

𝔅

id

)

∘

Δ

(1.6)

(

id

⊗

𝔅

P

)

∘

Δ

=

P

τ

∘

(

Δ

⊗

𝔅

Q

)

∘

Δ

=

Δ

∘

Q

where

τ

is a flip map :

τ

⁢

(

a

⊗

b

)

=

(

b

⊗

a

)

.

There is a natural extension of the above definition of quantum compact groupoids
to

locally compact

quantum groupoids by taking

𝖦

l

⁢

c

to be a

locally compact groupoid

(instead of a compact groupoid), and then following the steps in the above construction with the topological groupoid

𝖦

being replaced by

𝖦

l

⁢

c

. Additional integrability and Haar measure system conditions need however be also satisfied as in the general case of locally compact groupoid

representations

(for further details, see for example the monograph by Buneci (2003).

1.4.1

Reduced C*–algebra

Consider

𝖦

to be a topological groupoid. We denote by

C

c

⁢

(

𝖦

)

the space of smooth complex–valued functions with compact support on

𝖦

. In particular, for all

f

,

g

∈

C

c

⁢

(

𝖦

)

, the
function defined via convolution

(

f

*

g

)

⁢

(

γ

)

=

∫

γ

1

∘

γ

2

=

γ

f

⁢

(

γ

1

)

⁢

g

⁢

(

γ

2

)

,

(1.7)

is again an element of

C

c

⁢

(

𝖦

)

, where the convolution product
defines the composition law on

C

c

⁢

(

𝖦

)

. We can turn

C

c

⁢

(

𝖦

)

into a *–algebra once we have defined the involution

*

, and this is done by specifying

f

*

⁢

(

γ

)

=

f

⁢

(

γ

-

1

)

¯

.

We recall that following Landsman (1998) a

representation

of a groupoid

𝔾

, consists of a
family (or field) of Hilbert spaces

{

ℋ

x

}

x

∈

X

indexed by

X

=

Ob

⁢

𝔾

, along with a collection of maps

{

U

⁢

(

γ

)

}

γ

∈

𝔾

, satisfying:

1.

U

⁢

(

γ

)

:

ℋ

s

⁢

(

γ

)

⟶

ℋ

r

⁢

(

γ

)

,
is unitary.

2.

U

⁢

(

γ

1

⁢

γ

2

)

=

U

⁢

(

γ

1

)

⁢

U

⁢

(

γ

2

)

, whenever

(

γ

1

,

γ

2

)

∈

𝔾

(

2

)

(the set of arrows).

3.

U

⁢

(

γ

-

1

)

=

U

⁢

(

γ

)

*

, for all

γ

∈

𝔾

.

Suppose now

𝖦

l

⁢

c

is a Lie groupoid. Then the isotropy group

𝖦

x

is a Lie group, and for a (left or right) Haar
measure

μ

x

on

𝖦

x

, we can consider the Hilbert
spaces

ℋ

x

=

L

2

⁢

(

𝖦

x

,

μ

x

)

as exemplifying the
above sense of a representation. Putting aside some technical
details which can be found in Connes (1994) and Landsman (2006), the
overall idea is to define an operator of Hilbert spaces

π

x

⁢

(

f

)

:

L

2

⁢

(

𝖦

𝗑

,

μ

x

)

⟶

L

2

⁢

(

𝖦

x

,

μ

x

)

,

(1.8)

given by

(

π

x

⁢

(

f

)

⁢

ξ

)

⁢

(

γ

)

=

∫

f

⁢

(

γ

1

)

⁢

ξ

⁢

(

γ

1

-

1

⁢

γ

)

⁢

𝑑

μ

x

,

(1.9)

for all

γ

∈

𝖦

x

, and

ξ

∈

ℋ

x

. For each

x

∈

X

=

Ob

⁢

𝖦

,

π

x

defines an involutive representation

π

x

:

C

c

⁢

(

𝖦

)

⟶

ℋ

x

. We can define a norm on

C

c

⁢

(

𝖦

)

given by

∥

f

∥

=

sup

x

∈

X

⁡

∥

π

x

⁢

(

f

)

∥

,

(1.10)

whereby the completion of

C

c

⁢

(

𝖦

)

in this norm, defines

the reduced C*–algebra

C

r

*

⁢

(

G

)

of

G

l

⁢

c

.

It is perhaps the most commonly used C*–algebra for Lie groupoids
(groups) in noncommutative geometry.

References

1

E. M. Alfsen and F. W. Schultz:

Geometry of

State Spaces

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Title

compact quantum groupoids related to C*-algebras

Canonical name

CompactQuantumGroupoidsRelatedToCalgebras

Date of creation

2013-03-22 18:13:34

Last modified on

2013-03-22 18:13:34

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

125

Author

bci1 (20947)

Entry type

Topic

Classification

msc 81R40

Classification

msc 81R60

Classification

msc 81Q60

Classification

msc 81R50

Classification

msc 81R15

Classification

msc 46L05

Synonym

quantum compact groupoids

Synonym

weak Hopf algebras

Synonym

quantized locally compact groupoids with left Haar measure

Related topic

GroupoidCDynamicalSystem

Related topic

GroupoidAndGroupRepresentationsRelatedToQuantumSymmetries

Related topic

QuantumAlgebraicTopology

Related topic

GrassmanHopfAlgebrasAndTheirDualCoAlgebras

Related topic

NoncommutativeGeometry

Related topic

GroupoidCConvolutionAlgebra

Related topic

JordanBanachAndJordanLieAlgebras

Related topic

ClassesOfAlgebr

Defines

commutative C*-algebra

Defines

QOA

Defines

alternative definition of C*-algebra

Defines

C*-norm

Defines

morphism between C*-algebras

Defines

category of C*-algebras

Defines

quantum compact groupoid

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