On certain projections of C∗-matrix algebras

Annals of Functional Analysis

Volume 3 - Number 2

Article Type: Original Article
Abstract: ‎In 1955‎, ‎H‎. ‎Dye defined certain projections of a‎ ‎C-matrix algebra by‎Pi,j(a)=(1+aa)1Ei,i+(1+aa)1aEi,j+a(1+aa)1Ej,i+a(1+aa)1aEj,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(), ‎the set of such projections includes all‎ ‎the projections‎. ‎For infinite C-algebra A‎, ‎having a system of‎ ‎matrix units‎, ‎we have A𝕄n(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.
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