Noncommutative Mathematics for Quantum Systems

Quantum Dynamical Systems from the Point of View of Noncommutative Mathematics 133

Exercise 2.2.3 Check that the formula (2.2.2) indeed defines an
endomorphism ofON.

Proposition 2.2.3 The shift endomorphism is a permutation
endomorphism given by the ‘transposition’σ:J 2 → J 2 ,σ(i,j) =
(j,i)fori,j=1,.. .N.

Proof It suffices to note that for eachi=1,... ,N

ρσ(Si) =

N

∑

j,k= 1

SjSk(SkSj)∗Si=

N

∑

j= 1

SjSiS∗j =Φ(Si).

Theorem 2.2.4 LetTdenote the left-sided shift onCNgiven by the
formula

T((wk)∞k= 1 )l=wl+ 1 , l∈N.

The restriction of the shift endomorphismΦtoCNis the map
induced byT; precisely speaking (see Theorem 2.1.10 and the
formula (2.1.1)):

γ◦Φ|CN=αT◦γ.

Proof This time it suffices to check that for eachμ∈

⋃
k∈NJkthe
following equality holds:

γ(Φ(SμS∗μ)) =αT(Zμ),

whereZμ⊂CNis defined as in Theorem 2.1.10. We leave the details
as an (enlightening!) exercise.

Exercise 2.2.4 Is the identity map on ON a permutation
endomorphism? Is the shift Φ an automorphism (that is, is it
surjective)? What other natural classes of unitaries inON one
could consider in this context (for example using Exercise 2.1.4)?

In recent years Conti, Hong, and Szymanski in a series of papers ́
have undertaken a deep analysis of various classes of
endomorphisms of ON – for the information on this line of
research we refer to the survey [CHS 1 ] and references therein.