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Analytic bundle structure on the idempotent manifold
Monatshefte für Mathematik ( IF 0.8 ) Pub Date : 2021-05-02 , DOI: 10.1007/s00605-021-01562-4
Chi-Wai Leung , Chi-Keung Ng

Let X be a (real or complex) Banach space (not necessarily a Hilbert space), and \(\mathcal {I}(X)\) be the set of all non-trivial idempotents; i.e., bounded linear operators on X whose squares equal themselves. We show that, when equipped with the Banach submanifold structure induced from \(\mathcal {L}(X)\), the subset \(\mathcal {I}(X)\) is a locally trivial analytic affine-Banach bundle over the Grassmann manifold \(\mathscr {G}(X)\), via the map \(\kappa \) that sends \(Q\in \mathcal {I}(X)\) to Q(X), such that the affine-Banach space structure on each fiber is the one induced from \(\mathcal {L}(X)\). Using this, we show that if H is a real or complex Hilbert space, then the assignment

$$\begin{aligned} (E,T)\mapsto T^*\circ P_{E^\bot } + P_{E}, \quad \text {where}\quad E\in \mathscr {G}(H)\quad \text {and}\quad T\in \mathcal {L}(E,E^\bot ), \end{aligned}$$

induces a real bi-analytic bijection from the total space of the tangent bundle, \(\mathbf {T}(\mathscr {G}(H))\), of \(\mathscr {G}(H)\) onto \(\mathcal {I}(H)\) (here, \(E^\bot \) is the orthogonal complement of E, \(P_E\in \mathcal {L}(H)\) is the orthogonal projection onto E, and \(T^*\) is the adjoint of T). Notice that this real bi-analytic bijection is an affine map on each tangent plane. Furthermore, if for every \(E\in \mathscr {G}(H)\), we identify \(\mathcal {L}(E,E^\bot )\) with a subspace of \(\mathcal {L}(H)\) via the embedding \(S\mapsto S\circ P_E\), then the inclusion map from \(\mathbf {T}(\mathscr {G}(H))\) to the trivial Banach bundle \(\mathscr {G}(H)\times \mathcal {L}(H)\) is a real analytic immersion. Through this, we give a concrete idempotent in \(M_{n^2}\big (C(\mathscr {G}(\mathbb {K}^n))\big )\) that represents the K-theory class of the tangent bundle \(\mathbf {T}(\mathscr {G}(\mathbb {K}^n))\), when \(\mathbb {K}\) is either the real field or the complex field.



中文翻译:

幂等流形上的解析束结构

X为(实数或复数)Banach空间(不一定是希尔伯特空间),而\(\ mathcal {I}(X)\)为所有非平凡等幂的集合;也就是说,X上的平方等于自己的有界线性算子。我们表明,当配备有从诱导的Banach子流形结构\(\ mathcal {L}(X)\) ,子集\(\ mathcal {I}(X)\)是一个局部琐碎解析仿射的Banach捆绑过格拉斯曼歧管\(\ mathscr {G}(X)\) ,经由地图\(\卡帕\)发送\(Q \在\ mathcal {I}(X)\)QX),这样每根光纤上的仿射-巴纳赫空间结构就是由\(\ mathcal {L}(X)\)引起的。使用该函数,我们表明如果H是实或复杂的希尔伯特空间,则赋值

$$ \ begin {aligned}(E,T)\ mapsto T ^ * \ circ P_ {E ^ \ bot} + P_ {E},\ quad \ text {where} \ quad E \ in \ mathscr {G}( H)\ quad \ text {and} \ quad T \ in \ mathcal {L}(E,E ^ \ bot),\ end {aligned} $$

诱导从切丛,总空间中的真实双解析双射\(\ mathbf横置(\ mathscr {G}(H))\) ,的\(\ mathscr {G}(H)\)\(\ mathcal {I}(H)\)(这里,\(E ^ \ bot \)E的正交补码,\(P_E \ in \ mathcal {L}(H)\)是到E\(T ^ * \)T的伴随。请注意,此实际双解析双射是每个切平面上的仿射图。此外,如果对于每个\(E \ in \ mathscr {G}(H)\),我们将\(\ mathcal {L}(E,E ^ \ bot)\)标识为\(\ mathcal {L }(H)\)通过嵌入\(S \ mapsto S \ circ P_E \),然后从\(\ mathbf {T}(\ mathscr {G}(H))\)到平凡的Banach束\(\ mathscr {G }(H)\ times \ mathcal {L}(H)\)是一个真正的解析沉浸。通过这一点,我们得到在混凝土等幂\({M_ N ^ 2} \大(C(\ mathscr {G}(\ mathbb {K} ^ N))\大)\)表示该ķ -理论类的当\(\ mathbb {K} \)是实数场或复数场时,切线束\(\ mathbf {T}(\ mathscr {G}(\ mathbb {K} ^ n))\)

更新日期:2021-05-03
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