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Transactions of the American Mathematical Society

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The rectifiable metric on the set of closed subspaces of Hilbert space


Author: Lawrence G. Brown
Journal: Trans. Amer. Math. Soc. 337 (1993), 279-289
MSC: Primary 46C99; Secondary 46L99, 47A05, 47A99, 47D99, 58B20
DOI: https://doi.org/10.1090/S0002-9947-1993-1155349-5
MathSciNet review: 1155349
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Abstract: Consider the set of selfadjoint projections on a fixed Hilbert space. It is well known that the connected components, under the norm topology, are the sets $ \{ p:{\text{rank}}\;p = \alpha ,{\text{rank}}(1 - p) = \beta \} $, where $ \alpha $ and $ \beta $ are appropriate cardinal numbers. On a given component, instead of using the metric induced by the norm, we can use the rectifiable metric $ {d_r}$ which is defined in terms of the lengths of rectifiable paths or, equivalently in this case, the lengths of $ \varepsilon $-chains. If $ \left\Vert {p - q} \right\Vert < 1$, then $ {d_r}(p,q) = {\sin ^{ - 1}}(\left\Vert {p - q} \right\Vert)$, but if $ \left\Vert {p - q} \right\Vert = 1$, $ {d_r}(p,q)$ can have any value in $ \left[ {\frac{\pi } {2},\pi } \right]$ (assuming $ \alpha $ and $ \beta $ are infinite). If $ {d_r}(p,q) \ne \frac{\pi } {2}$, a minimizing path joining $ p$ and $ q$ exists; but if $ {d_r}(p,q) = \frac{\pi } {2}$, a minimizing path exists if and only if $ {\text{rank}}(p \wedge (1 - q)) = {\text{rank}}(q \wedge (1 - p))$.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1155349-5
Keywords: Hilbert space, projection, rectifiable
Article copyright: © Copyright 1993 American Mathematical Society

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