Minimal Representations of Cuntz Algebras

Abstract

Suppose that On (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 On are unitary equivalent, k(π, Ω) = k(π′, Ω′). Applying the number, one can define a minimal representation of On and give a sufficient condition of the minimality for a representation of On. 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 On are investigated, respectively.