On Specht's Theorem in UHF C⁎-algebras

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 hyperfinite (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 τ(w(a,a⁎))=τ(w(b,b⁎)) for all words w(x,y) in two non-commuting variables (where τ denotes the unique tracial state on Q), while there exist two elements a,b in 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.