Illinois Journal of Mathematics

Differential geometry of partial isometries and partial unitaries

Esteban Andruchow and Gustavo Corach

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Let $\a$ be a C$^*$-algebra. In this paper the sets $\ii$ of partial isometries and $\ii_\Delta\subset\ii$ of partial unitaries (i.e., partial isometries which commute with their adjoints) are studied from a differential geometric point of view. These sets are complemented submanifolds of $\a$. Special attention is paid to geodesic curves. The space $\ii$ is a homogeneous reductive space of the group $U_\a \times U_\a$, where $U_\a$ denotes the unitary group of $\a$, and geodesics are computed in a standard fashion. Here we study the problem of the existence and uniqueness of geodesics joining two given endpoints. The space $\ii_\Delta$ is \emph{not} homogeneous, and therefore a completely different treatment is given. A principal bundle with base space $\ii_\Delta$ is introduced, and a natural connection in it defined. Additional data, namely certain translating maps, enable one to produce a \emph{linear} connection in $\ii_\Delta$, whose geodesics are characterized.

Article information

Illinois J. Math., Volume 48, Number 1 (2004), 97-120.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L10: General theory of von Neumann algebras 46L30: States 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]


Andruchow, Esteban; Corach, Gustavo. Differential geometry of partial isometries and partial unitaries. Illinois J. Math. 48 (2004), no. 1, 97--120. doi:10.1215/ijm/1258136176.

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