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C-bialgebra defined by the direct sum of Cuntz algebras
We show that a tensor product among representation of certain
C-algebras induces a bialgebra. Let be the
smallest unitization of the direct sum of Cuntz algebras We
show that there exists a non-cocommutative comultiplication and a
counit of . From \Delta,\vep and the
standard algebraic structure, is a C-bialgebra.
Furthermore we show the following: (i) The antipode on
never exist. (ii) There exists a unique Haar state on .
(iii) For a certain one-parameter bialgebra automorphism group of , a KMS state on exists.Comment: 18 page
Domains of uniqueness for -semigroups on the dual of a Banach space
Let be a Banach space. In general, for a
-semigroup \semi on , its adjoint semigroup \semia
is no longer strongly continuous on the dual space . Consider on the topology of uniform
convergence on compact subsets of denoted by , for which the usual semigroups in literature
becomes -semigroups. The main purpose of this paper is to prove that only
a core can be the domain of uniqueness for a -semigroup on . As application, we show that the
generalized Schr\"odinger operator , , is -unique. Moreover, we
prove the -uniqueness of weak solution for the Fokker-Planck
equation associated with
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