On certain projections of c*-matrix algebras

In 1955, H. Dye defined certain projections of a C*-matrix algebraby [formula] which was used to show that in the case of factors not of type I_2n, the unitarygroup determines the algebraic type of that factor. We study these projectionsand we show that in M₂(C), the set of such projections includes all the projections. For infinite C*-algebra A, having a system of matrix units, we haveA ≃ M_n(A). M. Leen proved that in a simple, purely infinite C*-algebra A, the *-symmetries generate u₀(A). Assuming K₁(A) is trivial, we revise Leen’sproof and we use the same construction to show that any unitary close to theunity can be written as a product of eleven *-symmetries, eight of such are ofthe form 1 − 2P_{i, j} (ω), ω ∈ u (A). In simple, unital purely infinite C*-algebrashaving trivial K₁-group, we prove that all P_{i, j} (ω) have trivial K₀-class. Consequently, we prove that every unitary of O_n can be written as a finite productof *-symmetries, of which a multiple of eight are conjugate as group elements.