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Geometry of the sphere of a Hilbert module. (English) Zbl 0945.46042

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.

MSC:

46L08 \(C^*\)-modules
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
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