LIFTING POSITIVE ELEMENTS IN C-STAR-ALGEBRAS

It is elementary that every positive element in a quotient C*-algebra lifts to a positive element 'upstairs'. In this note, we consider whether a positive matrix with coefficients in the quotient can be lifted when some of the lifted entries are already specified. The arguments used here are elementary, but the results appear to be new. I would like to thank Vern Paulsen for asking the question that lead to this work. He raised it in connection with his paper with Rodman [8], in which they consider the problem of completing a partially positive matrix with coefficients in a C*-algebra to a positive one. Certain connections with this work will be explored. I also wish to thank my colleague Hong Sheng Yin for several conversations on this topic. Also, I acknowledge that I have borrowed shamelessly from some ideas in chapter 1 of Pedersen's book [9] on C*-algebras. Indeed, I thank Larry Brown for pointing out that I borrowed even more directly than I knew for one of my lemmas; and citing the original reference of Combes. Perhaps our most straightforward and useful result is: Theorem 0.1 Let A be a C*-algebra, let T be an ideal in A, and let pi be the quotient map of A onto A/T. Suppose that [GRAPHICS] is a positive element in M2(A/T), and that A and D are positive elements of A such that pi-(A) = a and pi-(D) = d. Then there is an element B of A such that [GRAPHICS] is positive.