On state space of measurable function in symmetric Δ-Banach algebra with new results.

The aim of this paper is to present new results obtained in the study of spaces with a Δ-symmetric Banach Algebra, we defined the δ-characters functional and discuss the space of measurable function(L^o(µ)) is Δ- symmetric and is Banach algebra. Also, we define s_ℂ(L^o(µ)) by depended on L^o(µ) and proved the spaces is satisfies new properties in symmetric Banach algebra and prove any function in s_ℂ(L^o(µ)) satisfies all conditions of state functional and found corresponding between the space s_ℂ(L^o(µ)) and the space of all o- characters functional δch(L^o(µ)) on L^o(µ) as well as well defined. Also, we proved the functional ψ from Æ into s_ℂ(L^o(µ)) is continuous and state functional if every non-zero element has an inverse. We proved the quotient s_ℂ(L^o(µ))/Ι is also symmetric Δ –Banach algebra. [ABSTRACT FROM AUTHOR]