Noncommutative Mathematics for Quantum Systems

134 Noncommutative Mathematics for Quantum Systems

2.2.2 The shift transformation on a graphC∗-algebra

LetΛ = (V,E) be a finite directed graph with no sources (see
Section 2.1.4). It is tempting and reasonable to study
endomorphisms of C∗(Λ) following the ideas of the previous
section (see for example [CHS 2 ]); in general however, one cannot
hope for a complete correspondence similar to that of Theorem
2.2.1. On the other hand, each C∗(Λ) is equipped with a
counterpart of the shift endomorphism ofONthat was introduced
in Definition 2.2.2. Observe first that the space of infinite paths on
Λadmits a natural continuous shift transformation, defined as
follows:

TΛ(f)(i) =f(i+ 1 ), f∈PΛ∞,i∈N 0.

Consider nowΨΛ:C∗(Λ)→C∗(Λ)defined by the formula:

ΨΛ(X) =∑

e∈E

SeXSe∗, X∈C∗(Λ).

Proposition 2.2.5 The mapΨΛintroduced above is unital and
completely positive; moreover, it leaves invariant the subalgebras
CΛandFΛ. When restricted toCΛ, the mapΨΛcan be identified
with the map induced by the shift onPΛ∞:

γΛ◦ΨΛ=αTΛ◦γΛ.

Proof Unitality follows from the simple calculation:

ΨΛ( 1 ) =∑

e∈E

SeS∗e= ∑

v∈V

∑

e∈E:r(e)=v

SeSe∗= ∑

v∈V

Pv=1.

Complete positivity follows from the fact that each of the maps
X 7→ SeXSe∗ is completely positive. As it was for the
endomorphism shift on Cuntz algebras, we leave the proof of the
last part as an exercise.

2.2.3 Classical topological entropy as defined by Rufus Bowen
and extensions to topological pressure

Entropy is a numerical invariant associated with a dynamical
system, very roughly speaking related to the system’s chaotic/
mixing properties. It was originally introduced by Kolmogorov