Noncommutative Mathematics for Quantum Systems

138 Noncommutative Mathematics for Quantum Systems

htα= sup

e>0,Ω⊂⊂A

lim sup

n→∞

(

1

n

log rcp(Ω(n),e)

)

(here and below we use the notationΩ(n)=

⋃n− 1

j= 0 α

j(Ω)).

Basic properties of htα:

• ([Vo]) if(A,α) = (C(X),αT), andXis a compact metric space,
  then

htαT=htop(T).

• ([Br]) ht does not increase when we pass to invariant
  subalgebras: ifBis aC∗-subalgebra ofA, such thatα(B)⊂B,
  then htα|B≤htα.

None of the above facts is trivial; the second requires in fact a
certain reformulation/extension of the definition (a subalgebra of
a nuclearC∗-algebra need not be nuclear – can you see where the
problem lies?). We will only prove the inequality

htαT≤htop(T). (2.2.3)

Proof of the inequality(2.2.3) Assume that(X,d)is a compact metric
space,T : X → Xis a continuous map. Fixe > 0 and a finite
setΩ⊂⊂C(X). Compactness ofXand finiteness ofΩimply the
existence ofδ>0 such that

∀f∈Ω∀x,y∈X d(x,y)<δ =⇒ |f(x)−f(y)|<e.

LetF={x 1 ,... ,xk}be an(n,δ)-spanning set forTsuch thatk=
s(n,δ). For eachi=1,... ,klet

Ui={x∈X:∀j=0,...,n− 1 d(Tj(xi),Tj(x))<δ}.

The family (Ui)ki= 1 is an open cover of X. Let (φi)ki= 1 be a
corresponding partition of unity, that is, a family of functions in
C(X) such that each φi is non-negative, supported in Ui and

moreover∑ki= 1 φi=1.
Define linear mapsψ:C(X)→Mkandφ:Mk→C(X)by the
formulas

ψ(f) =diag[f(x 1 ),... ,f(xk)], f∈C(X),