Irreducible Positive Linear Maps on Operator Algebras

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible positive linear maps on von Neumann algebras are explicitly constructed, and a criterion for the irreducibility of decomposable positive maps on full matrix algebras is given. Let A and A denote a C∗-algebra and its cone of positive elements. A linear map φ : A→ A is positive if it leaves the cone A invariant; that is, φ(x) ∈ A for every x ∈ A. If A is an n-dimensional commutative C∗-algebra, then a positive map φ on A has a representation as an n× n matrix with nonnegative real entries. Therefore, the classical matrix theoretic results (see [7, Ch.8]) of O. Perron and G. Frobenius on matrices with nonnegative entries can be viewed as an important case of a more general theory that deals with positive maps on operator algebras. This is the viewpoint taken by D. Evans and R. Høegh-Krohn [6], S. Albeverio and R. Høegh-Krohn [2], and U. Groh [8], [9] in their works on the spectra of positive maps on operator algebras. In both the classical theory and its various generalisations [12], and in the one put forward in [6], [2], [8], and [9], a positive map that is “strictly” positive, or that is “irreducible”, will possess certain interesting spectral properties. Although the spectral theory of such maps has been studied, the issue of how one is to determine whether a given positive linear map on an operator algebra is strictly positive or irreducible (or neither) has received much less attention. For matrices with nonnegative entries a simple and readily verifiable criterion exists, dating back to Frobenius: φ is strictly positive if and only if each entry of φ is positive, and φ is irreducible if and only if the directed graph of φ is strongly connected. However for positive maps on noncommutative operator algebras, it is somewhat more difficult to make this determination. Indeed this difficulty appears to be hampered even further by the fact that there is no tractable structure theory for positive maps. Received by the editors May 2, 1995. 1991 Mathematics Subject Classification. Primary 46L05.