2 00 5 Multiplier Hopf group coalgebras from algebraic and analytical point of views

The Multiplier Hopf Group Coalgebra was introduced by Hegazi in 2002 [7] as a generalization of Hope group caolgebra, introduced by Turaev in 2000 [5], in the non-unital case. We prove that the concepts introduced by A.Van Daele in constructing multiplier Hopf algebra [3] can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras A = {A α } α∈π , (π is a discrete group) equipped with a family of homomorphisms ∆ = {∆ α,β : A αβ −→ M (A α ⊗ A β)} α,β∈π which is called a comultiplication under some conditions, where M (A α ⊗ A β) is the multiplier algebra of A α ⊗ A β. In 2003 A. Van Daele suggest a new approach to study the same structure by consider the direct sum of the algebras A p 's which will be a multiplier Hopf algebra called later group cograded multiplier Hope algebra [8]. And hence there exist a one to one correspondence between multiplier Hopf Group Coalgebra and group cograded multiplier Hopf algebra. By using this one-one correspondence we studied multiplier Hopf Group Coalgebra