124 Noncommutative Mathematics for Quantum Systems
Definition 2.1.4 A quantum (or noncommutative) dynamical
system is a pair (A,α), where A is a unital C∗-algebra and
α:A→Aa unital∗-homomorphism.
In these lectures we will study mainly discrete dynamics,
described by a single transformation. As in the classical setting
one can also investigate dynamical systems described by discrete
or continuous group actions (for example, families of
automorphisms of a givenC∗-algebra indexed byR+orR). In
quantum mechanics traditionally the most important role was
played by automorphisms ofB(H), which are automaticallyinner,
that is, given by the formulaa 7→ U∗aU, whereU ∈ B(H)is a
unitary(UU∗=U∗U=1) – see also Section 2.4.1 below.
An element of aC∗-algebraAis calledpositiveif it is equal to
b∗bfor someb ∈A. The collection of all positive elements inA,
denotedA+, is a closed cone inA; moreover, ifa,b∈A+then‖a+
b‖ ≥ ‖a‖. A linear map betweenC∗-algebras is calledpositiveif it
maps positive elements to positive elements. In particular, we can
talk about positive functionals; it is easy to show that a positive
functional is automatically bounded.
Definition 2.1.5 A state on aC∗-algebraA is a positive linear
functionalω:A→Csuch thatω( 1 ) =1.
The collection of all states on a given C∗-algebra A will be
denoted byS(A); a stateω∈S(A)is said to befaithfulif for a fixed
a ∈Athe equalityω(a∗a) = 0 implies thata = 0 andtracialif
ω(ab) = ω(ba)for alla,b ∈A. States have the role of quantum
probability measures – Riesz theorem says that ifXis a compact
space then there exists a bijective correspondence between states
onC(X)and Borel regular probability measures onX(the value of
the state given by a measureμon a functionf ∈C(X)is given by
the integration of f with respect toμ). Thus, tracial states (also
known simply as traces) are those which preserve vestiges of
commutativity: informally speaking, when we work with traces
on noncommutative algebras we think of ‘functions’ that
‘commute under the integral sign’.
A unital∗-homomorphism from aC∗-algebraAtoB(H)is called
arepresentationofA. We say that a representationπ:A→B(H)is
faithfulif it is injective (in that case it is automatically isometric;
more generally each ∗-homomorphism between C∗-algebras is