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The interplay between measurable dynamics and the theory of von Neumann algebras has a long and successful history (see [4,7,12]). A high point in this development (through

We prove that Choi's map in M3 cannot be written as the sum of a 2-positive map and a 2-copositive map. We also provide other examples of positive maps in Mn which cannot be written as the sum of an n-positive map and a 2-copositive map.

A complex number which is not a nonpositive real number has a unique square root in the right half-plane. In this paper, we obtain an extension of this observation to general (complex) Banach algebras. Since the elements we study are regular, and have logarithms, the existence of square roots is not at stake. Even the existence of roots having the desired spectral properties is evident. The uniqueness result we suppose to be new. It subsumes a somewhat weakened version of the classical result for positive, hermitian, bounded operators on a Hilbert space (that such an operator has a unique positive hermitian and bounded square root): our theorem would apply only to the positive definite (regular) case. Whether the results of this paper can be extended to the case of not necessarily regular elements of a Banach algebra, we do not at present know.

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