The universal C*-algebra generated by n projections has been described. As an immediate corollary one obtains structure theorem for a pair of projections and the solution to an associated index problem. This puts the study of a pair of projections in the proper perspective of noncommutative geometry. keyword: Index problem, crossed product, Fredholm module, pair of projections. MSC 46L87.

A subalgebra $B$ of a Lie algebra $L$ is called weak c-ideal if there subideal $C$ such that $L=B+C$ and $B\cap C\leq B_{L} $ where $B_{L}$ the largest ideal contained in $B.$ This analogous to concept weakly c-normal subgroups, which has been studied by number authors. We obtain some properties c-ideals use them give characterisations solvable supersolvable algebras. also note one-dimensional ...

We prove that if A is a C-algebra, then for each a ∈ A, Aa = {x ∈ A/x ≤ a} is itself a C-algebra and is isomorphic to the quotient algebra A/θa of A where θa = {(x, y) ∈ A×A/a∧ x = a∧ y}. If A is C-algebra with T , we prove that for every a ∈ B(A), the centre of A, A is isomorphic to Aa ×Aa′ and that if A is isomorphic A1 ×A2, then there exists a∈ B(A) such that A1 is isomorphic Aa and A2 is is...