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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,… (More)

For a transformation F on a measure space (X,μ), we show that the Perron-Frobenius operator of F can be written by a representation (L2(X,μ), π) of the Cuntz-Krieger algebra OA associated with F when F satisfies some assumption. Especially, when OA is the Cuntz algebra ON and (L2(X,μ), π) in the above is some irreducible representation of ON , then there is… (More)

Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra O2 (or O2d more generally) are presented by using recursive constructions. As a typical example, an embedding of CAR onto the U(1)-invariant subalgebra of O2 is constructed explicitly. Generalizing this construction to the case of O2p , an embedding of CAR… (More)

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… (More)

For an infinitesimal symplectic action of a Lie algebra g on a symplectic manifold, we construct an infinitesimal crossed product of Hamiltonian vector fields and Lie algebra g. We obtain its second crossed product in case g = R and show an infinitesimal version for a theorem type of Takesaki duality.

We introduce a nonsymmetric, associative tensor product among representations of Cuntz algebras by using embeddings. We show the decomposition formulae of tensor products for permutative representations explicitly We apply decomposition formulae to determine properties of endomorphisms. Mathematics Subject Classifications (2000). 47L55, 81T05.

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… (More)

Katsunori Kawamura1 College of Science and Engineering Ritsumeikan University, 1-1-1 Noji Higashi, Kusatsu, Shiga 525-8577,Japan Abstract We show that there exists a non-cocommutative comultiplication ∆φ and a counit ε on the direct sum of Cuntz algebras O∗ ≡ C⊕O2 ⊕O3 ⊕O4 ⊕ · · · . From ∆φ, ε and the standard algebraic structure, the C ∗-algebra O∗ is a… (More)

Based on our previous work on the recursive fermion system in the Cuntz algebra, it is shown that a nonlinear transformation group of the CAR fermion algebra is induced from a U(2) action on the Cuntz algebra O2p with an arbitrary positive integer p. In general, these nonlinear transformations are expressed in terms of finite polynomials in generators. Some… (More)

Representation theory of the Cuntz algebras is studied by [4, 6, 7, 8, 9, 16, 17]. It is remarkable that representations of the Cuntz algebras in some class satisfy the uniqueness of irreducible decomposition. Furthermore these representations are related to quantum filed theory([1, 2, 3]), dynamical systems([11, 13, 14, 15]) and fractals([12]), and their… (More)