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

Let O-* denote the C*-algebra defined by the direct sum of Cuntz algebras {O-n: 1 <= n < infinity} where we write O-1 as C for convenience. We introduce a non-degenerate *-homomorphism Delta(phi) from O-* to O-* circle times O-* which satisfies the coassociativity, and a *-homomorphism epsilon from O-* to C such that (epsilon circle times id) o Delta(phi) congruent to id congruent to (id circle times epsilon) o Delta(phi). Furthermore we show the following: (i) For the smallest unitization (O) over tilde (*) of O-*, there exists a unital extension ((Delta) over cap (phi) , (epsilon) over tilde) of the pair (Delta(phi), epsilon) on (O) over tilde (*) such that ((O) over tilde (*), ($) over cap (phi)) is a unital bialgebra with the unital counit (epsilon) over tilde (ii) The pair (O-*, Delta(phi)) satisfies the cancellation law. (iii) There exists a unital *-homomorphism Gamma(phi) from to the multiplier algebra M(O-infinity circle times O-*) of O-infinity circle times O-* such that (Gamma(phi) circle times id) o Gamma(phi) = (id circle times Delta(phi)) o Gamma(phi). (iv) There is no antipode for (O) over tilde (*). (v) There exists a unique Haar state on (O) over tilde (*). (vi) For a certain one-parameter bialgebra automorphism group of (O) over tilde (*), there exists a KMS state on (O) over tilde (*). (c) 2008 Elsevier Inc. All rights reserved