Minimal Representations of Cuntz Algebras

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.