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

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