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),