C*-algebra has evolved as a key concept in Quantum Operator Algebra 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 (Alfsen and Schultz, 2003 [1]) cannot be overestimated. The theory of C*-algebras has numerous applications in the theory of representations of groups and symmetric algebras, the theory of dynamical systems, statistical physics and quantum field theory, and also in the theory of operators on a Hilbert space.
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 second section.
The fundamental connections that exist between categories
of
-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].
Let us consider first the definition of an involution on a complex algebra
.
and
, we have
A *-algebra is said to be a complex associative algebra
together with an operation
of involution
.
One can easily verify that
.
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
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
.
One can thus also define the category
of C*-algebras and morphisms between C*-algebras.
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
.
The classification of
-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.
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