Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 123
The assumption of unitality of theC∗-algebras that we study is
in a sense equivalent to assuming the compactness of the
underlying (quantum) space; for example each commutative
C∗-algebra is isomorphic to C 0 (X), the algebra of continuous
functions vanishing at infinity on a locally compact spaceX, and
of courseC 0 (X)is unital if and only ifXis compact. In general, the
procedure of adding a unit (unitisation) can be viewed as a
noncommutative counterpart of passing to a compactification of a
given locally compact space.
Exercise 2.1.1 LetHbe a Hilbert space and letKHdenote the
space of all compact operators onH; recall it is the norm closure of
the space of all finite-rank operators onH. Show thatKH is a
C∗-subalgebra ofB(H), unital if and only ifHis finite-dimensional.
Furthermore, show thatKH+C1 :={k+λ (^1) B(H):k∈ KH,λ∈C}
is a unitalC∗-subalgebra ofB(H).
2.1.2 Quantum topological dynamical systems and some
properties of transformations ofC∗-algebras
Definition 2.1.3 A (classical) topological dynamical system is a
pair(X,T), whereX is a compact space and T : X → X a
continuous map.
IfX,Yare compact spaces andT:X→Yis continuous, then we
obtain in a natural way a mapαT:C(Y)→C(X)defined by the
formula:
αT(f) =f◦T, f∈C(Y). (2.1.1)
Note that ‘the arrows are inverted’:
T: X → Y
C(X) ← C(Y) :αT.
It is easy to check thatαTis a unital∗-homomorphism. It turns out
that every unital∗-homomorphismα:C(Y)→C(X)arises in this
way from exactly one continuous transformation from XtoY.
Thus, the ‘classical’ theory of compact spaces and continuous
maps is in a natural way equivalent to the theory of commutative
unitalC∗-algebras and unital∗-homomorphisms.