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 $\phi :P\to P_a$ sending p to the unique $q\in P_a$ with the same range as p and ${\Omega }_a:P_a\to P_a$ 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...

On the unit sphere $𝕊$ in a real Hilbert space $𝐇$, we derive a binary operation $\odot$ such that $(𝕊,\odot )$ is a power-associative Kikkawa left loop with two-sided identity ${𝐞}_0$, i.e., it has the left inverse, automorphic inverse, and $A_l$ properties. The operation $\odot$ is compatible with the symmetric space structure of $𝕊$. $(𝕊,\odot )$ is not a loop, and the right translations which fail to be injective are easily characterized. $(𝕊,\odot )$ satisfies the left power alternative and left Bol identities “almost everywhere” but not everywhere....