Positivity (2017) 21:509–511
Positive maps which map the set of rank k projections
Received: 23 May 2016 / Accepted: 16 June 2016 / Published online: 22 June 2016
© Springer International Publishing 2016
Abstract Extending Wigner’s theorem we give a characterization of positive maps
of B(H ) into itself which map the set of rank k projections onto itself.
Keywords Positive maps · Jordan algebras · Vector states
Mathematics Subject Classiﬁcation 46K50 · 46L30 · 46L99
One form of the celebrated Wigner’s theorem  is that if φ is a linear map of the
bounded operators B(H ) on a Hilbert space H into itself with the property that it maps
the set of rank 1 projections bijectively onto itself, then φ is of the form
φ(a) = UaU
or φ(a) = Ua
is the transpose of a with respect to a ﬁxed orthonormal basis for H, and U is
a unitary operator. In the paper  Sarbicki, Chruscinski and Mozrzymas generalized
this to the case when H is of ﬁnite dimension n with n a prime number, and the set
of rank 1 projections is replaced by rank k projections, where k is a natural number
strictly smaller than n. They gave a counter example to the conclusion (*) when n is
not a prime. In that case φ is no longer a positive map. In the present note we make
the extra assumption that φ is a positive unital map. Then for any Hilbert space we
obtain the conclusion (*). Closely related results have been obtained by Molnar .
Recall that an atomic masa in B(H ) is a maximal abelian subalgebra A generated
by the rank 1 projections corresponding to the vectors in an orthonormal basis for H.
Department of Mathematics, University of Oslo, 0316 Oslo, Norway