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Journal of Operator Theory

Volume 71, Issue 2,  Spring  2014  pp. 341-379.

The kernel of the determinant map on certain simple $C^*$-algebras

Authors:  P.W. Ng
Author institution: Mathematics Department, Univ. of Louisiana at Lafayette, 217 Maxim D. Doucet Hall, P. O. Box 41010, Lafayette, LA, 70504-1010, U.S.A.

Summary: Let $\mathcal{A}$ be a unital separable simple $C^*$-algebra such that, either (1) $\mathcal{A}$ has real rank zero, strict comparison and cancellation of projections; or (2) $\mathcal{A}$ is TAI (tracially approximate interval). Let $\Delta_T : GL^0(\mathcal{A}) \rightarrow E_u/ T(K_0(\mathcal{A}))$ be the universal determinant of de la Harpe and Skandalis. Then for all $x \in GL^0(\mathcal{A})$, $\Delta_T(x) = 0$ if and only if $x$ is the product of $8$ multiplicative commutators in $GL^0(\mathcal{A})$. We also have results for the unitary case and other cases.

Keywords:  real rank zero, tracially approximate interval algebra

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