论文标题
关于Calkin代数的预测测量学的注释
A note on geodesics of projections in the Calkin algebra
论文作者
论文摘要
令$ {\ cal c}（{\ cal h}）= {\ cal b}（{\ cal h}） / {\ cal k}（{\ cal h}）$是calkin algebra（$ {$ {\ cal b}（\ cal b}（\ cal b}） $ {\ cal k}（{\ cal h}）$是紧凑型操作员的理想和$π：{\ cal b}（{\ cal h}）\ to {\ cal c}（{\ cal c}（{\ cal h}）$ the suberient map），$ {\ p} _} $ {\ cal c}（{\ cal H}）$中的自助杂点投影。 $ {\ cal c}（{\ cal h}）$的投影$ p $可以将其提升为{\ cal b}中的投影$ p \ in {\ cal b}（{\ cal H}）$：$π（p）= p $。 We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal geodesic of ${\cal P}_{{\cal C}({\cal H})}$ which joins $p$ and $q$ if and only there exist lifting projections $P$ and $Q$ such $ n（p-q \ pm 1）$都是有限维度，或者都是无限的维度。如果$ p+q- 1 $具有微不足道的Anhihilator，则最小的大地测量是独一无二的。这里的断言是最小的断言，意味着它比其他任何其他分段平滑曲线$γ（t）\ in {\ cal p} _ {{\ cal c}（{\ cal c}（{\ cal c}（{\ cal h}} $ \ | \dotγ（t）\ | D T $。
Let ${\cal C}({\cal H})={\cal B}({\cal H}) / {\cal K}({\cal H})$ be the Calkin algebra (${\cal B}({\cal H})$ the algebra of bounded operators on the Hilbert space ${\cal H}$, ${\cal K}({\cal H})$ the ideal of compact operators and $π:{\cal B}({\cal H})\to {\cal C}({\cal H})$ the quotient map), and ${\cal P}_{{\cal C}({\cal H})}$ the differentiable manifold of selfadjoint projections in ${\cal C}({\cal H})$. A projection $p$ in ${\cal C}({\cal H})$ can be lifted to a projection $P\in{\cal B}({\cal H})$: $π(P)=p$. We show that given $p,q \in {\cal P}_{{\cal C}({\cal H})}$, there exists a minimal geodesic of ${\cal P}_{{\cal C}({\cal H})}$ which joins $p$ and $q$ if and only there exist lifting projections $P$ and $Q$ such that either both $N(P-Q\pm 1)$ are finite dimensional, or both infinite dimensional. The minimal geodesic is unique if $p+q- 1$ has trivial anhihilator. Here the assertion that a geodesic is minimal means that it is shorter than any other piecewise smooth curve $γ(t) \in {\cal P}_{{\cal C}({\cal H})}$, $t \in I$, joining the same endpoints, where the length of $γ$ is measured by $\int_I \|\dotγ(t)\| d t$.
