Andruchow, Esteban; Corach, Gustavo; Stojanoff, Demetrio Geometry of the sphere of a Hilbert module. (English) Zbl 0945.46042 Math. Proc. Camb. Philos. Soc. 127, No. 2, 295-315 (1999). Let \(X\) be a right Hilbert \(C^*\)-module over a unital \(C^*\)-algebra \(B\) and let \(S= \{x\in X\mid\langle x,x\rangle= 1\}\) be its unit sphere. The authors show that \(S\) becomes a homogeneous \(C^\infty\) space by a certain action of the unitary group of the algebra of adjointable \(B\)-module operators of \(X\). They introduce a reductive structure for \(S\) and compute the geodesics of the linear connection induced in \(S\) by this structure. Also, a natural Fisher metric is defined and local minimality of certain geodesics is obtained. The fundamental group of \(S\) is computed in the case when \(B\) is a von Neumann algebra and \(X\) is self-dual. Reviewer: Kh.N.Boyadzhiev (Ada) Cited in 7 Documents MSC: 46L08 \(C^*\)-modules 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) Keywords:right Hilbert \(C^*\)-module; unital \(C^*\)-algebra; homogeneous \(C^\infty\) space; adjointable \(B\)-module operators; reductive structure; Fisher metric; geodesics; von Neumann algebra PDFBibTeX XMLCite \textit{E. Andruchow} et al., Math. Proc. Camb. Philos. Soc. 127, No. 2, 295--315 (1999; Zbl 0945.46042) Full Text: DOI