C* 代数的行列式秩
Determinant Rank of C*-algebras
摘要：
设$A$是一个有单位元的$C^*$ -代数，$U_0(A)$是$A$中与单位元有路连接的酉群.设$i_A^{(1, n)}\colon U_0(A)/CU(A)\rightarrow U_0(\mathrm M_n(A))/CU(\mathrm M_n(A))$是通过将$u$发送到${\rm diag}(u,1_n).$而定义的\hm，我们研究了当映射$i_A^{(1,n)}$对所有$n.$是同构时的问题，证明了当$A$具有稳定秩1时，它总是满射的，并且是内射的.当$A$是实秩零的单位$C^*$ -代数或$A$没有迹态时，它也是内射的。本文证明了当$A$是稳定秩$k>1.$的Villadsen单AH-代数时，该映射是同构的。同时，还证明了对于所有Blackadar的无单位投影可分单$C^*$ -代数，该映射是同构的。设$A=\mathrm M_n(C(X)),$，其中$X$是任意紧度量空间.注意到映射$i_A^{(1, n)}$是所有$n.$的同构。因此，映射$i_A^{(1, n)}$总是任何单位$C^*$ -代数$A$的同构，该单位$C^*$-代数是如上所述形式为$\mathrm M_n(C(X))$的$C^*$ -代数的有限直和的归纳极限。然而，我们证明了存在单位$C^*$ -代数$A$使得$i_A^{(1,2)}$不是同构.
英文摘要：
Let $A$ be a unital $C^*$-algebra and let $U_0(A)$ be the group of unitaries of $A$ which are path connected to the identity. Denote by $CU(A)$ the closure of the commutator subgroup of $U_0(A).$ Let $i_A^{(1, n)}\colon U_0(A)/CU(A)\rightarrow U_0(\mathrm M_n(A))/CU(\mathrm M_n(A))$ be the \hm\, defined by sending $u$ to ${\rm diag}(u,1_n).$ We study the problem when the map $i_A^{(1,n)}$ is an isomorphism for all $n.$ We show that it is always surjective and is injective when $A$ has stable rank one. It is also injective when $A$ is a unital $C^*$-algebra of real rank zero, or $A$ has no tracial state. We prove that the map is an isomorphism when $A$ is the Villadsen's simple AH--algebra of stable rank $k>1.$ We also prove that the map is an isomorphism for all Blackadar's unital projectionless separable simple $C^*$-algebras. Let $A=\mathrm M_n(C(X)),$ where $X$ is any compact metric space. It is noted that the map $i_A^{(1, n)}$ is an isomorphism for all $n.$ As a consequence, the map $i_A^{(1, n)}$ is always an isomorphism for any unital $C^*$-algebra $A$ that is an inductive limit of finite direct sum of $C^*$-algebras of the form $\mathrm M_n(C(X))$ as above. Nevertheless we show that there are unital $C^*$-algebras $A$ such that $i_A^{(1,2)}$ is not an isomorphism.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
