Generalized permutative representation of Cuntz algebra. II ...

{(V i , π i , Ω i )} q i=1 of GP representations with respect to a family {ξ iv} q i=1 of
parameters. Put
H ≡ V 1 ⊕ · · · ⊕ V q , Ω ≡ √ 1 q∑
Ω i ∈ H, π ≡ π 1 ⊕ · · · ⊕ π q .
q
i=1
Then π(s(v ⊗p ))Ω = π(s(v)) p Ω = Ω and dim W = q for W ≡ Lin <
{π(s(v)) l Ω} p−1
l=0 >. Note that vectors ((ξ i) k ) q k=1 and ((ξ j) k ) q k=1 in Cq are
linearly independent when i ≠ j. Hence Ω, π(s(v))Ω, . . ., π(s(v)) q−1 Ω ∈
Lin < {Ω i : i = 1, . . . , q} > are linearly independent in H. From this,
Ω 1 , . . . , Ω q ∈ Lin < {π(s(v)) i−1 Ω : i = 1, . . . , q} >. Therefore Ω i ∈ π(O N )Ω
for each i = 1, . . . , q. Hence
V i = π i (O N )Ω i ⊂ π(O N )Ω
for each i = 1, . . . , q. Therefore π(O N )Ω = H and (H, π, Ω) is cyclic. Since
(H, π, Ω) satisfies condition (2.1) with respect to v ⊗p and the proper period
of (H, π, Ω) is q, we obtain (H, π, Ω) in the statement.
By Lemma 4.4, GP (v ⊗p ) is not unique when p ≥ 2. Hence we classify
representations which satisfy the condition of GP (v ⊗p ) in the next section.
5 Irreducible decomposition of GP representation
For p ≥ 2 and v ∈ T S NP (C N ), we classify GP (v ⊗p ) and show decomposition
formulae. In order to classify periodic case, we introduce new symbols.
Definition 5.1 A representation (H, π, Ω) of O N is GP (v ⊗p ; q) if (H, π, Ω)
is GP (v ⊗p ) which has the proper period q.
Lemma 5.2 For each 1 ≤ q ≤ p, there exists GP (v ⊗p ; q).
Proof. By Lemma 4.4, it holds.
Theorem 5.3 Let p ≥ 2 and 1 ≤ q ≤ p. If (H, π, Ω) is GP (v ⊗p ; q) for
v ∈ T S NP (C N ), then there is a subset {ξ i } q i=1 ⊂ {e2π√ −1l/p : l = 0, . . . , p−1}
such that ξ i ≠ ξ j when i ≠ j and the following equivalence holds:
q⊕
(H, π) ∼ GP (ξ i v).
i=1
8