C ∗-bialgebra defined by the direct sum of Cuntz algebras

Abstract

Let O ∗ denote the C ∗-algebra defined by the direct sum of Cuntz algebras { O n : 1 ⩽ n < ∞ } where we write O 1 as C for convenience. We introduce a non-degenerate ∗-homomorphism Δ φ from O ∗ to O ∗ ⊗ O ∗ which satisfies the coassociativity, and a ∗-homomorphism ε from O ∗ to C such that ( ε ⊗ id ) ○ Δ φ ≅ id ≅ ( id ⊗ ε ) ○ Δ φ . Furthermore we show the following: (i) For the smallest unitization O ˜ ∗ of O ∗ , there exists a unital extension ( Δ ˆ φ , ε ˜ ) of the pair ( Δ φ , ε ) on O ˜ ∗ such that ( O ˜ ∗ , Δ ˆ φ ) is a unital bialgebra with the unital counit ε ˜ . (ii) The pair ( O ∗ , Δ φ ) satisfies the cancellation law. (iii) There exists a unital ∗-homomorphism Γ φ from O ∞ to the multiplier algebra M ( O ∞ ⊗ O ∗ ) of O ∞ ⊗ O ∗ such that ( Γ φ ⊗ id ) ○ Γ φ = ( id ⊗ Δ φ ) ○ Γ φ . (iv) There is no antipode for O ˜ ∗ . (v) There exists a unique Haar state on O ˜ ∗ . (vi) For a certain one-parameter bialgebra automorphism group of O ˜ ∗ , there exists a KMS state on O ˜ ∗ .