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.
DOI: http://dx.doi.org/10.7900/jot.2012apr01.1953
Keywords: real rank zero, tracially approximate interval algebra
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