132 Noncommutative Mathematics for Quantum Systems
shift 2.2.1 Endomorphisms of Cuntz algebras and a quantum
We will be interested in unital endomorphisms of Cuntz algebras
(we will just call them endomorphisms of ON). Theorem 2.1.9
implies that they are automatically injective.
Theorem 2.2.1 ([Cu 2 ]) There is a bijective correspondence
between unitaries inON (that is, elementsU ∈ ON such that
U∗U=UU∗= (^1) ON) and endomorphisms ofON. It is given by the
formulas
Uρ=
N
∑
i= 1
ρ(Si)Si∗,
ρU(Si) =USi, i=1,... ,N
Proof Straightforward computation. Note that the fact that the
second formula determines a∗-homomorphism fromONtoON
follows from the defining universal property ofON.
A special class of endomorphisms ofONis the class of so-called
permutation endomorphisms, introduced in [Ka]. Letk∈Nand let
σ:Jk→ Jkbe a permutation. Define a unitaryUσ∈ ONby the
formula
Uσ= ∑
μ∈Jk
SμS∗σ(μ). (2.2.1)
The endomorphism ofONcorresponding toUσvia Theorem 2.2.1
will be denotedρσand called a permutation endomorphism.
Exercise 2.2.1 Check that the formula (2.2.1) indeed defines a
unitary. Prove that each permutation endomorphism mapsCNinto
CN, andFNintoFN.
Exercise 2.2.2 LetUbe a unitary inFN ⊂ ON. Prove that the
associated endomorphismρUmapsFNintoFN(note however that
there exist endomorphismsρofONthat mapFNintoFN, but their
associated unitariesUρdo not belong toFN– see [CRS]).
Definition 2.2.2 The shift endomorphism (sometimes called also
a quantum shift) is the endomorphism ofONgiven by the formula:
Φ(X) =
N
∑
i= 1
SiXS∗i, X∈ON. (2.2.2)