Jordan-Banach and Jordan-Lie algebras

Jordan-Banach and Jordan-Lie algebras

0.0.1

Definitions of Jordan-Banach, Jordan-Lie, and Jordan-Banach-Lie algebras

Firstly, a specific

algebra

consists of a

vector space

E

over a

ground field

(typically

ℝ

or

ℂ

)
equipped with a

bilinear

and

distributive

multiplication

∘

. Note that

E

is not
necessarily

commutative

or

associative

.

A

Jordan algebra

(over

ℝ

), is an algebra over

ℝ

for which:

S

∘

T

=

T

∘

S

,

S

∘

(

T

∘

S

2

)

=

(

S

∘

T

)

∘

S

2

,

for all

elements

S

,

T

of the algebra.

It is worthwhile noting now that in the algebraic theory of Jordan algebras, an important
role is played by the

Jordan triple product

{

S

⁢

T

⁢

W

}

as defined by:

{

S

⁢

T

⁢

W

}

=

(

S

∘

T

)

∘

W

+

(

T

∘

W

)

∘

S

-

(

S

∘

W

)

∘

T

,

which is linear in each factor and for which

{

S

⁢

T

⁢

W

}

=

{

W

⁢

T

⁢

S

}

. Certain examples entail
setting

{

S

⁢

T

⁢

W

}

=

1

2

⁢

{

S

⁢

T

⁢

W

+

W

⁢

T

⁢

S

}

.

A

Jordan

Lie algebra

is a

real vector space

𝔄

ℝ

together with a

Jordan product

∘

and

Poisson bracket

{

,

}

, satisfying :

1.

for all

S

,

T

∈

𝔄

ℝ

,

S

∘

T

=

T

∘

S

{

S

,

T

}

=

-

{

T

,

S

}

2.

the

Leibniz rule

holds

{

S

,

T

∘

W

}

=

{

S

,

T

}

∘

W

+

T

∘

{

S

,

W

}

for all

S

,

T

,

W

∈

𝔄

ℝ

, along with

3.

the

Jacobi identity

:

{

S

,

{

T

,

W

}

}

=

{

{

S

,

T

}

,

W

}

+

{

T

,

{

S

,

W

}

}

4.

for some

ℏ

2

∈

ℝ

, there is the

associator

identity

:

(

S

∘

T

)

∘

W

-

S

∘

(

T

∘

W

)

=

1

4

⁢

ℏ

2

⁢

{

{

S

,

W

}

,

T

}

.

0.0.2

Poisson algebra

By a

Poisson algebra

we mean a Jordan algebra in which

∘

is associative. The
usual

algebraic

types of morphisms

automorphism

,

isomorphism

, etc.) apply to Jordan-Lie
(Poisson) algebras (see Landsman, 2003).

Consider the classical

configuration space

Q

=

ℝ

3

of a moving particle whose phase space
is the

cotangent bundle

T

*

⁢

ℝ

3

≅

ℝ

6

, and for which the space of (classical)
observables is taken to be the real vector space of

smooth functions

𝔄

ℝ

0

=

C

∞

⁢

(

T

*

⁢

R

3

,

ℝ

)

. The usual

pointwise

multiplication of

functions

f

⁢

g

defines a bilinear map on

𝔄

ℝ

0

, which is seen to be
commutative and associative. Further, the Poisson bracket on functions

{

f

,

g

}

:=

∂

⁡

f

∂

⁡

p

i

⁢

∂

⁡

g

∂

⁡

q

i

-

∂

⁡

f

∂

⁡

q

i

⁢

∂

⁡

g

∂

⁡

p

i

,

which can be easily seen to

satisfy

the Liebniz rule above. The axioms above then set the stage of passage to quantum mechanical systems which the parameter

k

2

suggests.

0.0.3

C*–algebras (C*–A), JLB and JBW Algebras

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

*

.

A

C*–algebra

is a simultaneously a *–algebra and a

Banach space

𝔄

, satisfying for all

S

,

T

∈

𝔄

:

∥

S

∘

T

∥

≤

∥

S

∥

⁢

∥

T

∥

,

∥

T

*

⁢

T

∥

2

=

∥

T

∥

2

.

We can easily see that

∥

A

*

∥

=

∥

A

∥

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

Banach algebra

where the latter requires the above 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, and

conversely

, any C*–algebra is isomorphic to a norm–closed

*

–algebra in

ℒ

⁢

(

H

)

for some Hilbert space

H

.

For a C*–algebra

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

.

A

Jordan–Banach algebra

(a JB–algebra for short) is both a real Jordan algebra and a Banach space, where for all

S

,

T

∈

𝔄

ℝ

, we have

∥

S

∘

T

∥

≤

∥

S

∥

⁢

∥

T

∥

,

∥

T

∥

2

≤

∥

S

2

+

T

2

∥

.

A

JLB–algebra

is a JB–algebra

𝔄

ℝ

together with a Poisson bracket for which it becomes a Jordan–Lie algebra for some

ℏ

2

≥

0

. Such JLB–algebras often constitute the real part of several widely studied complex associative algebras.

For the purpose of

quantization

, there are fundamental relations between

𝔄

s

⁢

a

, JLB and Poisson algebras.

For further details see Landsman (2003) (Thm. 1.1.9).

A JB–algebra which is

monotone

complete

and admits a separating set of normal sets is
called a

JBW-algebra

. These appeared in the work of von Neumann who developed a
(orthomodular)

lattice theory

of

projections

on

ℒ

⁢

(

H

)

on which to study

quantum
logic

(see later). BW-algebras have the following property: whereas

𝔄

s

⁢

a

is a
J(L)B–algebra, the self adjoint part of a

von Neumann algebra

is a JBW–algebra.

A

JC–algebra

is a norm closed real

linear subspace

of

ℒ

⁢

(

H

)

s

⁢

a

which is closed under the bilinear

product

S

∘

T

=

1

2

⁢

(

S

⁢

T

+

T

⁢

S

)

(non–commutative and nonassociative). Since any norm
closed Jordan subalgebra of

ℒ

⁢

(

H

)

s

⁢

a

is a JB–algebra, it is natural to specify
the exact relationship between JB and JC–algebras, at least in finite

dimensions

. In order to
do this, one introduces the ‘exceptional’ algebra

H

3

⁢

(

𝕆

)

, the algebra of

3

×

3

Hermitian matrices

with values in the octonians

𝕆

. Then a finite dimensional JB–algebra is a
JC–algebra if and only if it does not contain

H

3

⁢

(

𝕆

)

as a (direct) summand

[

1

]

.

The above definitions and constructions follow the approach of Alfsen and Schultz (2003) and Landsman (1998).

References

1

Alfsen, E.M. and F. W. Schultz: Geometry of

State Spaces

of Operator Algebras, Birkhäuser, Boston-Basel-Berlin.(2003).

Title

Jordan-Banach and Jordan-Lie algebras

Canonical name

JordanBanachAndJordanLieAlgebras

Date of creation

2013-03-22 18:14:05

Last modified on

2013-03-22 18:14:05

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

32

Author

bci1 (20947)

Entry type

Topic

Classification

msc 08A99

Classification

msc 08A05

Classification

msc 08A70

Synonym

quantum operator algebras

Related topic

Algebras2

Related topic

CAlgebra3

Related topic

AlgebraicCategoryOfLMnLogicAlgebras

Related topic

NonAbelianStructures

Related topic

AbelianCategory

Related topic

AxiomsForAnAbelianCategory

Related topic

GeneralizedVanKampenTheoremsHigherDimensional

Related topic

AxiomaticTheoryOfSupercategories

Related topic

AlgebraicCategoryOfLMnLogicAlgebras

Related topic

Categorical

Defines

Jordan algebra

Defines

Jordan-Banach algebra

Defines

Jordan-Lie algebra

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