A note on geodesics of projections in the Calkin algebra
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Archiv der Mathematik
A note on geodesics of projections in the Calkin algebra Esteban Andruchow Abstract. Let C(H) = B(H)/K(H) be the Calkin algebra (B(H) the algebra of bounded operators on the Hilbert space H, K(H) the ideal of compact operators, and π : B(H) → C(H) the quotient map), and PC(H) the differentiable manifold of selfadjoint projections in C(H). A projection p in C(H) can be lifted to a projection P ∈ B(H): π(P ) = p. We show that, given p, q ∈ PC(H) , there exists a minimal geodesic of PC(H) which joins p and q if and only if there exist lifting projections P and Q such that either both N (P − Q ± 1) are finite dimensional, or both are infinite dimensional. The minimal geodesic is unique if p + q − 1 has trivial anhihilator. Here the assertion that a geodesic is minimal means that it is shorter than any other piecewise smooth curve γ(t) ∈ PC(H) , t ∈ I, joining the same endpoints, where the length of γ is measured by γ(t)dt. ˙ I Mathematics Subject Classification. Primary 58B20; Secondary 46L05, 53C22. Keywords. Projections, Calkin algebra, Geodesics of projections.
1. Introduction. If A is a C∗ -algebra, let PA denote the set of (selfadjoint) projections in A. PA has a rich geometric structure, see for instance the papers [6] by H. Porta and L. Recht and [2] by G. Corach, H. Porta, and L. Recht. In these works, it was shown that PA is a differentiable (C∞ ) complemented submanifold of As , the set of selfadjoint elements of A, and has a natural linear connection, whose geodesics can be explicitly computed. A metric is introduced, called in this context a Finsler metric: since the tangent spaces of PA are closed (complemented) linear subspaces of As , they can be endowed with the norm metric. With this Finsler metric, Porta and Recht [6] showed that two projections p, q ∈ PA which satisfy p − q < 1 can be joined by a unique geodesic, which is minimal for the metric (i.e., it is shorter than any other smooth curve in PA joining the same endpoints).
E. Andruchow
Arch. Math.
In general, two projections p, q in A satisfy p − q ≤ 1, so that what remains to consider is what happens in the extremal case p − q = 1: under what conditions does there exist a geodesic, or a minimal geodesic, joining p and q. For general C∗ -algebras, this is too vast of a question. In this note, we shall consider it for the case of the Calkin algebra A = C(H) = B(H)/K(H), where B(H) is the algebra of bounded linear operators in a Hilbert space H and K(H) is the ideal of compact operators. Denote by π : B(H) → C(H) the quotient ∗-homomorphism. Let c ∈ C(H) be a selfadjoint element. We say that c has trivial anhihilator if cx = 0 for x ∈ C(H) implies x = 0. Note that since c∗ = c, this is equivalent to xc = 0 implies x = 0. Clearly, if p−q±1 have trivial anhihilators, then for any lifting projections P and Q (of p and q, respectively), dim N (P −Q±1) < ∞: if dim N (P −Q±1) = +∞, then (P − Q − 1)PN (P −Q−1) = 0 so that (p − q − 1)π(PN (P −Q−1) ) = 0 with π(PN (P −Q−1) ) = 0 (and similarly for P − Q + 1). Also, these co
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