DIAGONALIZING PROJECTIONS IN MULTIPLIER ALGEBRAS AND IN MATRICES OVER A C-STAR-ALGEBRA

Assume that A is a C*-algebra with the FS property ([3] and [16]). We prove that every projection in Mn(A) (n ≥ 1) or in L(KA) is homotopic to a projection whose diagonal entries are projections of A and off-diagonal entries are zeros. This yields partial answers for Questions 7 and 8 raised by M. A. Rieffel in [18]. If A is σ-unital but non-unital, then every projection in the multiplier algebra M(A) is unitarily equivalent to a diagonal projection, and homotopic to a block-diagonal projection with respect to an approximate identity of A consisting of an increasing sequence of projections. The unitary orbits of self-adjoint elements of A and M(A) are also considered. © 1990 by Pacific Journal of Mathematics.