Firstly, a specific algebra consists of a vector space
over a ground field (typically
or
) equipped with a bilinear and distributive multiplication
. Note that
is not necessarily commutative or associative.
A Jordan algebra (over
), is an algebra over
for which:
,
for all elements
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
as defined by:
which is linear in each factor and for which
. Certain examples entail
setting
.
A Jordan Lie algebra is a real vector space
together with a Jordan product
and Poisson bracket
, satisfying :
for all
the Jacobi identity :
for some
, there is the associator identity :
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
of a moving particle whose phase space is the cotangent bundle
, and for which the space of (classical) observables
is taken to be the real vector space of smooth functions
. The usual pointwise multiplication of functions
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
suggests.
An involution on a complex algebra
is a real-linear map
such that for all
and
, we have also
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
, satisfying for all
:
One can easily see that
. 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
the space
of (bounded) linear operators
from
to
forms a Banach space, where for
, the space
is a Banach algebra with respect to the norm
In quantum field theory
one may start with a Hilbert space
, and consider the Banach algebra of bounded linear operators
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
we have :
and
By a morphism between C*-algebras
we mean a linear map
, such that for all
, the following hold :
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
in
is a C*-algebra, and conversely, any C*-algebra is isomorphic to a norm-closed
-algebra in
for some Hilbert space
.
For a C*-algebra
, we say that
is self-adjoint if
. Accordingly, the self-adjoint part
of
is a real vector space since we can decompose
as :
A commutative C*-algebra is one for which the associative multiplication is commutative. Given a commutative C*-algebra
, we have
, the algebra of continuous functions on a compact Hausdorff space
.
A Jordan-Banach algebra (a JB-algebra for short) is both a real Jordan algebra and a Banach space, where for all
, we have
A JLB-algebra is a JB-algebra
together with a Poisson bracket for which it becomes a Jordan-Lie algebra for some
. 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
, JLB and Poisson algebras.
Conversely, given a JLB-algebra
with
, its
complexification
is a
-algebra under the operations :
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
on which to study quantum logic. BW-algebras have the following property: whereas
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
which is closed under the bilinear product
(non-commutative and nonassociative). Since any norm closed Jordan subalgebra of
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
, the algebra of
Hermitian matrices with values in the octonians
. Then a finite dimensional JB-algebra is a
JC-algebra if and only if it does not contain
as a (direct) summand [1].
The above definitions and constructions follow the approach of Alfsen and Schultz (2003), and also reported earlier by Landsman (1998).
As of this snapshot date, this entry was owned by bci1.