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(ℋ𝒜) 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.
