Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 127

is a dense∗-subalgebra ofON.

We will be further interested in the following twoC∗-subalgebras
ofON:

FN=Lin{SμS∗ν:μ∈Jk,ν∈Jk,k∈N},

CN=Lin{SμS∗μ:μ∈Jk,k∈N}.

Exercise 2.1.2 Prove thatCNis an abelian subalgebra ofON(in
fact it is even amasa– a maximal abelian subalgebra).

In connection withCNconsider now the Cantor set, presented in
the form given by a collection of infinite words built of letters from
the alphabet{1,... ,N}:

CN={w= (wk)∞k= 1 :∀k∈Nwk∈{1,... ,N}}.

The setCNis equipped with a natural metric

d(w,v) =

{

0 if w=v,

1

k if wk^6 =vk,wi=vifori<k.

It is easy to check that(CN,d)is a compact metric space (and the
resulting topology coincides with the Tikhonov topology of the
infinite Cartesian product of the discrete sets{1,... ,N}).

Theorem 2.1.10 The map sending eachSμS∗μ(μ∈

⋃

k∈NJk) to the

characteristic function of the setZμ:={w∈CN:(wn)|nμ=| 1 =μ}
extends to a∗-isomorphism betweenCNandC(CN), which will be
denoted in what follows byγ.

Proof Exercise.

We can thus think ofONas a certain noncommutativeC∗-algebra
containing the algebra of continuous functions on the Cantor set as
a subalgebra.

Exercise 2.1.3 LetX,Ybe compact spaces and assume that we
have a natural (unital, injective,∗-homomorphic) inclusion ofC(X)
intoC(Y)(in other words,C(X)is a unital subalgebra ofC(Y)).
What can be then said about the setsXandY? On the other hand,