The Logarithmic Function and Trace Zero Elements in Finite Von Neumann Factors

In this short note we present a common characterization of the logarithmic function and the subspace of all trace zero elements in finite von Neumann factors. Our aim is to prove the following statement. Theorem 1. Let A be a von Neumann factor and f :]0,∞[→ R be a nonconstant continuous function. Set S A f = span{ f (AB A)− (2 f (A)+ f (B)) : A,B ∈A −1 + }. Then either we have S A f = A or we have S A f ( A in which case A is finite, f = a log holds with some constant a 6= 0, and S A f equals the space of all trace zero elements of A . Here span stands for the closed linear span relative to the norm topology in A . The above statement can be viewed as a common characterization of the logarithmic function and the space of all trace zero elements (and hence the trace itself) in factor von Neumann algebras of finite type. For the proof we need some preliminary preparations which follow. We call a linear functional l on an algebra A tracial if it satisfies l (X Y ) = l (Y X ) for any X ,Y ∈ A . If A is a *-algebra, a linear functional h : A →C is called Hermitian if h(X ∗) = h(X ) holds for all X ∈A . Assume now that A is a C∗-algebra. For a tracial bounded linear functional l on A , defining l1(X ) = (1/2)(l (X )+ l (X ∗)), l2(X ) = (1/2i )(l (X )− l (X ∗)), X ∈A we have Hermitian tracial bounded linear functionals l1, l2 such that l = l1 + i l2. 2010 Mathematics Subject Classification. 46L10.