We introduce representations of the Cuntz algebra $\con$ which are parameterized by sequences in the set of unit vectors in ${\bf C}^{N}$. These representations are natural generalizations 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 representations.

Generalized permutative representations of Cuntz algebras

Let OH be the extended Cuntz algebra over the Hilbert space H. Since its zero grade part O 0 is the C ∗ -inductive limit of B(H ⊗r ), we look for some family of representations on an inductive limit of H ⊗r as r →∞ . When such construction is shaped according to the structure of O 0 , von Neumann's notion of a reference sequence of unit vectors for Hilbert infinite tensor products emerges; after a further Rieffel induction step, ac lass IPR(H) of representations of OH arises. For any two such representations, we describe explicitly their associated intertwiners. Any two representations in IPR(H) are either disjoint or unitarily equivalent. Actions of the group Z by translation on sequences of unit vectors are involved, as well as the ideals of Z.

https://link.springer.com/content/pdf/10.1007%2Fs10231-002-0065-x.pdf

Induced product representations of extended Cuntz algebras

We initiate a study of infinite tensor products of projective unitary representations of a discrete group G. Special attention is given to regular representations twisted by 2-cocycles and to projective representations associated with CCR-representations of bilinear maps. Detailed computations are presented in the case where G is a finitely generated free abelian group. We also discuss an extension problem about product type actions of G, where the projective representation theory of G plays a central role.

/pdf/on-infinite-tensor-products-of-projective-unitary-3b6gfaumja.pdf

On infinite tensor products of projective unitary representations

We study conditions for an anti-automorphism of a C*-algebra to extend to the cross-product by a compact group dual action. As a byproduct, we show that under a few natural assumptions any PCT transformation θ of , the observable net of a local QFT, extends to the canonical field net . Then we discuss the validity of the relation for every DHR morphism ρ of with finite statistics.

Extension of anti-automorphisms and pct-symmetry

Amenable Groups and Generalized Cuntz Algebras

Given a unitary representation u of the locally compact group G on the Hilbert space H, we investigate the notion of infinite tensor power of u. Then we apply the results to study covariance of the associated canonical action of G on the generalized Cuntz algebra .

Asymptotic invariance, amenability and generalized Cuntz algebras

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Induced product representations of extended Cuntz algebras

Amenable Groups and Generalized Cuntz Algebras

Asymptotic invariance, amenability and generalized Cuntz algebras