126 Noncommutative Mathematics for Quantum Systems
successfully implemented by Woronowicz ([Wo]) and led to the
theory of compact quantum groups, which is of fundamental
importance to the lectures of Uwe Franz in the first part of this
volume.
2.1.3 Cuntz algebras
Examples ofC∗-algebras are usually defined eitherabstractly, as
completions of certain∗-algebras with respect to a given norm
satisfying the C∗-condition, or concretely, as specific closed
∗-subalgebras of B(H). Below we present a very important
example of an abstractly definedC∗-algebra.
Definition 2.1.8 ([Cu 1 ]) Let N ∈ N,N ≥ 2. Consider a unital
∗-algebraANgenerated by NelementsS 1 ,... ,SN, which satisfy
the following relations:
S∗iSj=δi,j (^1) AN,
n
∑
i= 1
SiS∗i = (^1) AN.
The Cuntz algebraONis the completion ofANin the norm
‖x‖=sup{‖π(x)‖:π−unital∗-homomorphism fromANtoB(H)}.
In other words, the Cuntz algebra is the universalC∗-algebra
generated byNisometries with orthogonal ranges, whose range
projections sum up to identity. Note that the fact that the above
formula defines a norm onAN(and not only a seminorm) requires
a proof.
LetJk={1,... ,N}kdenote the set of all multi-indices of length
kwith values in{1,... ,N}. Forμ∈Jkput
Sμ:=Sμ 1 Sμ 2.. .Sμk,
|μ|=k.
Theorem 2.1.9 ([Cu 1 ]) The algebraONissimple, that is, does not
possess any non-trivial two-sided closed ideals. In particular any
C∗-algebra generated by N-isometries acting on a given Hilbert
spaceHand whose ranges are mutually orthogonal and together
fillHis isomorphic toON. The set
Lin{SμS∗ν:μ∈Jk,ν∈Jl,k,l∈N}