有限Blaschke符号与标量里程表\(C^*\) - 代数的\(K\) - 理论
Finite Blaschke Symbols and the $K$-Theory of Scalar Odometer $C^*$-Algebras
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中文总结 AI 辅助
研究标量里程表\(C^*\) - 代数\(\mathcal A_\xi\)，证明其含紧算子，等距标量符号下\(W_\xi\)是Fredholm的条件，以及其在Calkin代数中像与里程表边界商同构等，得出不同次数有限Blaschke符号生成非同构\(C^*\) - 代数的结论。
中文摘要 AI 辅助
设\(O_n\)为里程表半群，\(\mathcal A_\xi = C^*(S_1,\ldots,S_n,W_\xi)\subseteq\mathcal B(\mathcal F_n^2)\)为由左生成算子和与符号\(\xi\in\mathcal F_n^2\)相关的标量里程表映射生成的\(C^*\) - 代数。证明了对每个标量符号\(\mathcal A_\xi\)包含紧算子。对于等距标量符号，证明\(W_\xi\)是Fredholm当且仅当相关内函数是有限Blaschke积。还表明\(\mathcal A_\xi\)在Calkin代数中的像与里程表边界商\(\mathcal Q(O_n)\)典范同构。若相关有限Blaschke积次数为\(d\)，则\(\operatorname{ind}(W_\xi)= -d\)。得出不同次数的有限Blaschke符号生成非同构\(C^*\) - 代数的结论。
英文摘要
Let $O_n$ be the odometer semigroup, and let $\mathcal A_ξ=C^*(S_1,\ldots,S_n,W_ξ)\subseteq\mathcal B(\mathcal F_n^2)$ be the $C^*$-algebra generated by the left creation operators and the scalar odometer map associated with a symbol $ξ\in\mathcal F_n^2$. We show that $\mathcal A_ξ$ contains the compact operators for every scalar symbol. For an isometric scalar symbol, we prove that $W_ξ$ is Fredholm if and only if the associated inner function is a finite Blaschke product. We further show that the image of $\mathcal A_ξ$ in the Calkin algebra is canonically isomorphic to the odometer boundary quotient $\mathcal Q(O_n)$. If the associated finite Blaschke product has degree $d$, then $\operatorname{ind}(W_ξ)=-d$. For $d\geq 1$, we obtain $K_0(\mathcal A_ξ)\cong\mathbb Z\oplus\mathbb Z_{d(n-1)}$ and $K_1(\mathcal A_ξ)=0$, whereas for $d=0$, $K_0(\mathcal A_ξ)\cong\mathbb Z^2$ and $K_1(\mathcal A_ξ)\cong\mathbb Z$. Consequently, for fixed $n\geq 2$, finite Blaschke symbols of distinct degrees generate non-isomorphic $C^*$-algebras.
