Determinant rank of $C^*$-algebras

Guihua Gong, Huaxin Lin and Yifeng Xue

Vol. 274 (2015), No. 2, 405–436

DOI: 10.2140/pjm.2015.274.405

Abstract

Let $A$ be a unital $C^{\ast }$-algebra and let $U_0\left(A\right)$ be the group of unitaries of $A$ which are path-connected to the identity. Denote by $CU\left(A\right)$ the closure of the commutator subgroup of $U_0\left(A\right)$. Let $i_A^{\left(1,n\right)}:U_0\left(A\right)∕CU\left(A\right)\to U_0\left(M_n\left(A\right)\right)∕C‘U\left(M_n\left(A\right)\right)$ be the homomorphism defined by sending $u$ to $diag\left(u,1_{n-1}\right)$. We study the problem of when the map $i_A^{\left(1,n\right)}$ is an isomorphism for all $n$. We show that it is always surjective and that it is injective when $A$ has stable rank one. It is also injective when $A$ is a unital $C^{\ast }$-algebra of real rank zero, or $A$ has no tracial state. We prove that the map is an isomorphism when $A$ is 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 $C^{\ast }$-algebras. Let $A=M_n\left(C\left(X\right)\right)$, where $X$ is any compact metric space. We note that the map $i_A^{\left(1,n\right)}$ is an isomorphism for all $n$. As a consequence, the map $i_A^{\left(1,n\right)}$ is always an isomorphism for any unital $C^{\ast }$-algebra $A$ that is an inductive limit of the finite direct sum of $C^{\ast }$- algebras of the form $M_n\left(C\left(X\right)\right)$ as above. Nevertheless we show that there is a unital $C^{\ast }$-algebra $A$ such that $i_A^{\left(1,2\right)}$ is not an isomorphism.

Dedicated to George A. Elliott on his seventieth birthday

Keywords

determinant rank for $C^{`*}\!$-algebras

Mathematical Subject Classification 2010

Primary: 46L06, 46L35

Secondary: 46L80

Milestones

Received: 16 May 2014

Accepted: 18 September 2014

Published: 1 April 2015