122 Noncommutative Mathematics for Quantum Systems
Notational conventions
For a subsetXof a Banach space the closed linear span ofXwill
be denoted byLinX(and the usual linear span by LinX). Notation
F ⊂⊂ Z will be used to signify that Fis a finite subset ofZ.
Sometimes we will write N 0 for N∪ { 0 }. All vector spaces
(algebras, Hilbert spaces, and so on) will be considered overC. As
in the first part of this volume, scalar products are linear on the
right(and not on the wrong!) side.
2.1.1 Noncommutative Mathematics – Gelfand–Naimark
Theorem
Definition 2.1.1 A Banach algebraAwith involution (that is, a
complex algebra with involutive antilinear antimultiplicative map
∗ :A→ A, equipped with a submultiplicative norm makingAa
Banach space and such that the involution is an isometry) is called
aC∗-algebra if
‖a∗a‖=‖a‖^2 , a∈A.For proofs of various general statements related toC∗-algebras
that will be used below we refer, for example, to the monograph
[Mu]. In these lecturesallC∗-algebras will be unital. A basic
example of aC∗-algebra isMn, the algebra ofnbynmatrices with
complex coefficients, equipped with the natural involution and
multiplication and with the operator norm induced by the
identification of each such matrix with a linear operator acting on
the Hilbert spaceCn. More generally, whenever His a Hilbert
space,B(H), that is, the algebra of all bounded linear operators on
H, is aC∗-algebra. Another example is given byC(X), the algebra
of all continuous complex-valued functions on a compact spaceX,
equipped with the natural algebraic operations and the supremum
norm (we will assume that all compact spaces are Hausdorff).
Theorem 2.1.2 [Gelfand-Naimark, 1943]: Every commutative
unitalC∗-algebra A is isometrically isomorphic to the algebra
C(XA)for some compact topological space XA. If Y is another
compact space then theC∗-algebrasAandC(Y)are isometrically
isomorphic if and only ifXAandYare homeomorphic.
The spaceXAis called thespectrumof the commutative algebraA.