Minimal Representations of Cuntz Algebras

Abstract

Suppose that O n (2 ≤ n ≤ ∞) is a Cuntz algebra. There exists a number 1 ≤ k(π, Ω) ≤ ∞ such that if the cyclic representations (H, π, Ω) and (H , π , Ω ) of the Cuntz algebra O n are unitary equivalent, k(π, Ω) = k(π , Ω ). Applying the number, one can define a minimal representation of O n and give a sufficient condition of the minimality for a representation of O n . Moreover, the relations between minimal states and minimal representations of Cuntz algebras and the relations between the minimality and the irreducibility of the representation of O n are investigated, respectively.