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

Let O∗ denote the C∗-algebra defined by the direct sum of Cuntz algebras where we write O1 as C for convenience. We introduce a non-degenerate ∗-homomorphism Δφ from O∗ to O∗⊗O∗ which satisfies the coassociativity, and a ∗-homomorphism ε from O∗ to C such that (ε⊗id)○Δφ≅id≅(id⊗ε)○Δφ. Furthermore we show the following:(i)For the smallest unitization of O∗, there exists a unital extension of the pair (Δφ,ε) on such that is a unital bialgebra with the unital counit .(ii)The pair (O∗,Δφ) satisfies the cancellation law.(iii)There exists a unital ∗-homomorphism Γφ from O∞ to the multiplier algebra M(O∞⊗O∗) of O∞⊗O∗ 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 .