On Specht's Theorem in UHF C*-algebras

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.