Noncommutative Mathematics for Quantum Systems

142 Noncommutative Mathematics for Quantum Systems

The proof will be divided into a series of lemmas, in whichN
andkwill be fixed (andρwill be as above). Fori,j∈Nput

Ai,j={SμS∗ν:μ∈Ji,ν∈Jj}, Fi,j=LinAi,j.

Lemma 2.3.2 For anyi,j,m∈N

ρm(Ai,j)⊂Fi+m(k− 1 ),j+m(k− 1 ).

Proof It suffices to consider the case ofm=1 (do you see why?).
Let thenμ∈Ji,ν∈Jjand note that

ρ(Sμ) =UσSμ 1 ···UσSμi∈Fk,k− 1 ···Fk,k− 1 ⊂Fk+i−1,k− 1.

Similarly

ρ(Sν)∗∈Fk−1,k+j− 1.

Hence

ρ(SμS∗ν)∈Fk+i−1,k+j− 1.

For each l ∈ N introduce the following (injective)
∗-homomorphismΨl:ON→M
Nl(ON):

Ψl(X) = ∑

μ,ν∈Jl

eμ,ν⊗S∗μXSν, X∈ON. (2.3.2)

We identify above the algebraMNl(ON)with the tensor product
MNl⊗ON, index the set{1,... ,Nl}by indices inJland denote by
eμ,νthe relevant matrix unit – the matrix whose only non-zero entry
is 1 in the ‘μ’-th row and ‘ν’-th column.

Exercise 2.3.1 Check that the formula (2.3.2) indeed defines a
unital∗-homomorphism.

Lemma 2.3.3 Letl,i,j∈N,l≥i>j. LetX∈Fi,j. Then for each
μ∈Ji−jthere existsTμ∈MNlsuch that‖Tμ‖≤‖X‖and

Ψl(X) = ∑

μ∈Ji−j

Tμ⊗Sμ. (2.3.3)