On Specht's Theorem in UHF C⁎-algebras

Abstract

Specht's Theorem states that two matrices A and B in 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 x and y. In this article we examine to what extent this trace condition characterises approximate unitary equivalence in uniformly hyper nite (UHF) C -algebras. In particular, we show that given two elements a; b of the universal UHF algebra Q which generate C -algebras satisfying the UCT, they are approximately unitarily equivalent if and only if (w(a; a )) = (w(b; b )) for all words w(x; y) in two non-commuting variables (where denotes the unique tracial state on Q), while there exist two elements a; b in the UHF-algebra M21 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.