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 \cite{4} can be adapted to serve again in our construction. A multiplier Hopf group coalgebra is a family of algebras $A=\{A_{\alpha}\}_{\alpha \in \pi}$, ($\pi$ is a discrete group) equipped with a family of homomorphisms $\Delta=\{\Delta_{\alpha,\beta}:A_{\alpha\beta}\longrightarrow M(A_{\alpha}\otimes A_{\beta})\}_{\alpha,\beta \in \pi}$ which is called a comultiplication under some conditions, where $M(A_{\alpha}\otimes A_{\beta})$ is the multiplier algebra of $A_{\alpha}\otimes A_{\beta}$. 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 \cite{11}. 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 \\