3(2) : 144-145
Article Type: Original Article
Abstract: In 1955, H. Dye defined certain projections of a C∗-matrix algebra byPi,j(a)=(1+aa∗)−1⊗Ei,i+(1+aa∗)−1a⊗Ei,j+a∗(1+aa∗)−1⊗Ej,i+a∗(1+aa∗)−1a⊗Ej,j which was used to show that in the case of factors not of type I2n, the unitary group determines the algebraic type of that factor. We study these projections and 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 have A≃𝕄n(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's proof and we use the same construction to show that any unitary close to the unity can be written as a product of eleven ∗-symmetries, eight of such are of the form 1−2Pi,j(ω),\ω∈U(A). In simple, unital purely infinite C∗-algebras having trivial K1-group, we prove that all Pi,j(ω) have trivial K0-class. Consequently, we prove that every unitary of On can be written as a finite product of ∗-symmetries, of which a multiple of eight are conjugate as group elements.