Polynomial endomorphisms of the Cuntz algebras arising from permutations . I — General theory —

1.1. Main theorem. In usual, the irreducible decomposition of a representation of an operator algebra does not make sense because there is no uniqueness of such decomposition in general. This fact disturbs an intention to study an ordinary representation theory of operator algebras like that of semisimple Lie algebras and quantum groups. In spite of this, permutative representations of the Cuntz algebra ON ([3, 5, 6]) are completely reducible and their irreducible decompositions are unique up to unitary equivalences. Roughly speaking, there are two kinds of (cyclic)permutative representations, “cycle” and “chain”. This remarkable property assists to characterize endomorphisms of ON , too, in the following way: For N ≥ 2, let s1, . . . , sN be generators of ON and {1, . . . , N}k ≡ {(jl)l=1 : jl = 1, . . . , N, l = 1, . . . , k} for k ≥ 1. Theorem 1.1. For a permutation σ on {1, . . . , N}k, k ≥ 1, let ψσ be an endomorphism of ON defined by (1.1) ψσ(si) ≡ uσsi (i = 1, . . . , N) where uσ ≡ ∑ J∈{1,...,N}k sσ(J)(sJ) ∗ and sJ ≡ sj1 · · · sjk when J = (j1, . . . , jk). If (H, π) is a permutative representation, then (H, π ◦ψσ) is, too. Specially, if (H, π) has only cycles, then (H, π ◦ ψσ) does, too. Theorem 1.1 assures the completely reducibility of (H, π ◦ ψσ) for any permutative representation (H, π) and any permutation σ. The first aim of this article is a preparation of tools of analysis of endomorphisms of ON by representations.