On Specht's Theorem in UHF C*-algebras

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Date

2021-01-01

Authors

Marcoux, Laurent
Zhang, Yuanhang

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Publisher

Elsevier

Abstract

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.

Description

The 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/

Keywords

Specht's Theorem, approximate unitary equivalence, UHF-algebras, approximate absolute value condition

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