2025/05/09 by Andruchow, Esteban, Corach, Gustavo, Recht, Lázaro +1
#14M15 #46L05 #46L08 #47A05 #58B20 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2505.06434
Given the unital C^*-algebra A, the unitary orbit of the projector p0=\beginpmatrix1 & 0 0 & 0 \endpmatrix in the C^*-algebra M2(A) of 2× 2 matrices with coefficients in A is called in this paper, the Riemann sphere R of A. We show that R is a homogeneous reductive C^∞ manifold of the unitary group U2(A)⊂ M2(A) and carries the differential geometry deduced from this structure (including an invariant Finsler metric). Special attention is paid to the properties of geodesics and the exponential map. If the algebra A is represented in a Hilbert space H, in terms of local charts of R, elements of the Riemann sphere may be identified with (graphs of) closed operators on H (bounded or unbounded). In the first part of the paper, we develop several geometric aspects of R including a relation between the exponential map of the reductive connection and the cross-ratio of subspaces of H× H. In the last section we show some applications of the geometry of R, to the geometry of operators on a Hilbert space. In particular, we define the notion of bounded deformation of an unbounded operator and give some relevant examples.