Marcoux, LaurentZhang, Yuanhang2024-01-312024-01-312021-01-01https://doi.org/10.1016/j.jfa.2020.108778http://hdl.handle.net/10012/20320The final publication is available at Elsevier via https://doi.org/10.1016/j.jfa.2020.108778. © 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/Specht’s Theorem states that two matrices Aand Bin Mn(C)are unitarily equivalent if and only if tr(w(A, A∗)) =tr(w(B, B∗))for all words w(x, y)in two non-commuting variables xand y. In this article we examine to what extent this trace condition characterises approximate unitary equivalence in uniformly hyperfinite (UHF) C∗-algebras. In particular, we show that given two elements a, bof the universal UHF-algebra Qwhich generate C∗-algebras satisfying the UCT, they are approximately unitarily equi-valent if and only if τ(w(a, a∗)) =τ(w(b, b∗))for all words w(x, y)in two non-commuting variables (where τdenotes the unique tracial state on Q), while there exist two elements a, bin the UHF-algebra M2∞which fail to be approximately unitarily equivalent despite the fact that they satisfy the trace condition. We also examine a consequence of these results for ampliations of matrices.enAttribution-NonCommercial-NoDerivatives 4.0 Internationalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Specht's Theoremapproximate unitary equivalenceUHF-algebrasapproximate absolute value conditionArticle