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Geometry of oblique projections

E. Andruchow·G. Corach·D. Stojanoff
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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 in A. The maps f_p: P \to P_a sending p to the unique q in P_a with the same range as p and Ω_a: P_a \to P 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 in A with |q-r|<1 such that there exists a positive element a in A verifying that q, r are in P_a. In this case q and r can be joined by an unique short geodesic along the space of idempotents Q of A.

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E. Andruchow, G. Corach, D. Stojanoff. 1999-11-18. Geometry of oblique projections. https://arxiv.org/abs/math/9911133

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