Irreducible representations of the Cuntz algebras arising from polynomial embeddings
نویسنده
چکیده
For an embedding φ of the Cuntz algebra OM into ON , OM is identified with a subalgebra φ(OM) of ON . We construct irreducible representations of ON with continuous parameters by extending irreducible representations of OM . They are not unitarily equivalent to any generalized permutative representation, especially not to any permutative representation by Bratteli-Jorgensen and Davidson-Pitts. We show their unitary equivalence by parameters and give another characterization for them by states or eigenequations of cyclic vectors without the information of the embedding.
منابع مشابه
Polynomial endomorphisms of the Cuntz algebras arising from permutations. II —Branching laws of endomorphisms—
For a subgroup H of a group G, an irreducible decomposition of π|H for an irreducible representation π of G is one of main study in representation theory([12]). When such decomposition holds, the decomposition formula is called a branching law. This is reformulated as the branching which is brought by the inclusion map ι from H to G, that is, ι∗(π) ≡ π ◦ ι gives a map from RepG to RepH. In gene...
متن کاملRepresentations of the Cuntz - Krieger algebras . II — Permutative representations —
We generalize permutative representations of the Cuntz algebras for the Cuntz-Krieger algebra OA for any A. We characterize cyclic permutative representations by notions of cycle and chain, and show their existence and uniqueness. We show necessary and sufficient conditions for their irreducibility and equivalence. In consequence, we have a complete classification of permutative representations...
متن کاملIrreducible representations of crossed products of the Cuntz algebras and isotropy subgroups of the unitary group
For an element x of the Cuntz algebra ON , define the isotropy subgroup Gx ≡ {g ∈ U(N) : αg(x) = x} of the unitary group U(N) with respect to the canonical action α of U(N) on ON . We have irreducible representations of the crossed product ONoGx by extending irreducible generalized permutative representations of ON and irreducible representation of Gx. ¿From this, the Peter-Weyl theorem for Gx ...
متن کاملPolynomial 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 Cunt...
متن کاملGeneralized permutative representations of the Cuntz algebras
We introduce representations of the Cuntz algebra ON which are parameterized by sequences in the set of unit vectors inC . These representations are natural generalization of permutative representations by Bratteli-Jorgensen and Davidson-Pitts. We show their existence, irreducibility, equivalence, uniqueness of irreducible decomposition and decomposition formulae by using parameters of represen...
متن کامل