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Abstract

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections $P_a$ determined by the different involutions $#_a$ induced by positive invertible elements a ∈ A. The maps $φ:P → P_a$ sending p to the unique $q ∈ P_a$ with the same range as p and $Ω_a : P_a → P_a$ sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q,r ∈ A with ||q-r|| < 1 such that there exists a positive element a ∈ A satisfying $q,r ∈ P_a$. In this case q and r can be joined by a unique short geodesic along the space of idempotents Q of A.