0.1 C*- and von Neumann algebras: Quantum operator algebra in quantum theories
0.1.1 Introduction
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 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].
0.2 Basic definitions
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 0.1. 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 , (ST)∗ = T∗S∗ , (λT)∗ = λT∗ .
A *-algebra is said to be a complex associative algebra together with an operation of involution
∗ .
0.3 C*-algebra
Definition 0.2. 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{∥Tu∥ : 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{(Tu,Tu) : u ∈ H , (u,u) = 1} , and ∥Tu∥2 = (Tu,Tu) = (u,T∗Tu) ≤∥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(ST) = 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 . One can thus also define the category 𝒞∗ of C*-algebras and morphisms between
C*-algebras.
For a C*-algebra 𝔄, we say that T ∈ 𝔄 is self–adjoint if T = T∗ . Accordingly, the
self–adjoint part 𝔄sa of 𝔄 is a real vector space since we can decompose T ∈ 𝔄sa as
:
T = T′ + T′′ := 
(T + T∗) + ι(
)(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.
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