论文标题
$ c^*$ - 与Hilbert $ c^*$ - 模块上的两个预测相关的同构
$C^*$-isomorphisms associated with two projections on a Hilbert $C^*$-module
论文作者
论文摘要
本文由用于表征弗里德里奇角度的两个规范方程式的动机，研究$ c^*$ - 同构，通过引入匹配的三重和半折断的投影，与两个预测相关的同构。如果$ h $是hilbert $ c^*$ - 模块，$ p $和$ q $的三重$（p，q，h）$是匹配的。 $ c^*$ - $ \ MATHCAL {l}（H）$的$ c^*$ - 由$ \ {p-p \ wedge Q，q-p \ wedge q，q-p \ wedge q，i \} $和$ \ {p，q，q，q，q，q，q，q \ wedge q，wedge q，i \} $表示，$ i（p，p，p，h）$（p，h）$（p，h）p，事实证明，$ o（p，q，h）$的每个忠实表示$（π，x）$都可以诱导忠实的表示$（\widetildeπ，x）$ $ i（p，q，q，h）$，这样\ begin {align*} align*}＆\wideTildeπ（p-p \ wedge q） ＆\widetildeπ（q-p \ wedge q）=π（q）-π（p）\wedgeπ（q）。 \ end {align*}当$（p，q）$是半手机的，也就是说，$ \ edline {\ Mathcal {\ Mathcal {r}（p+q）} $和$ \ overline {\ Mathcal {r}（r}（r}（2i-p-q）} $都在$ h $ h $ h $ h $ h $ h $ h $ h $ h $ h $ h中$ i（i-q，i-p，h）$通过$ \ MATHCAL {l}（H）$中的单一操作员在单位上等效。构建反例，这表明当$（p，q）$无法半手机时，情况可能并非如此。同样，构建反例，使得$（p，q）$是半火山的，而$（p，i-q）$不是半手机。还提供了一些其他示例，这些示例表明还提供了作用于希尔伯特$ c^*$ - 模块的新现象。
Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies $C^*$-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple $(P,Q,H)$ is said to be matched if $H$ is a Hilbert $C^*$-module, $P$ and $Q$ are projections on $H$ such that their infimum $P\wedge Q$ exists as an element of $\mathcal{L}(H)$, where $\mathcal{L}(H)$ denotes the set of all adjointable operators on $H$. The $C^*$-subalgebras of $\mathcal{L}(H)$ generated by elements in $\{P-P\wedge Q, Q-P\wedge Q, I\}$ and $\{P,Q,P\wedge Q,I\}$ are denoted by $i(P,Q,H)$ and $o(P,Q,H)$, respectively. It is proved that each faithful representation $(π, X)$ of $o(P,Q,H)$ can induce a faithful representation $(\widetildeπ, X)$ of $i(P,Q,H)$ such that \begin{align*}&\widetildeπ(P-P\wedge Q)=π(P)-π(P)\wedge π(Q),\\ &\widetildeπ(Q-P\wedge Q)=π(Q)-π(P)\wedge π(Q). \end{align*} When $(P,Q)$ is semi-harmonious, that is, $\overline{\mathcal{R}(P+Q)}$ and $\overline{\mathcal{R}(2I-P-Q)}$ are both orthogonally complemented in $H$, it is shown that $i(P,Q,H)$ and $i(I-Q,I-P,H)$ are unitarily equivalent via a unitary operator in $\mathcal{L}(H)$. A counterexample is constructed, which shows that the same may be not true when $(P,Q)$ fails to be semi-harmonious. Likewise, a counterexample is constructed such that $(P,Q)$ is semi-harmonious, whereas $(P,I-Q)$ is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert $C^*$-modules are also provided.
