The aim of this paper is to give new characterizations of some fundamental
issues about idempotents. In the general setting of adjointable operators on
Hilbert Cβ-modules, a new term of quasi-projection pair is introduced. For
each idempotent Q, a projection m(Q), called the matched projection of Q,
is constructed. It is shown that Q and m(Q) as idempotents are homotopic,
and (m(Q),Q) is a quasi-projection pair. Some formulas for m(Q) are
derived. Based on these formulas, representations and norm estimations
associated with m(Q) are dealt with.Comment: The original submission entitled "Quasi-projection pairs on Hilbert
Cβ-modules" will be divided into three parts, and this is the first par