When are positive *-linear matrix maps also completely positive?

Professor Sanne ter Horst (North-West University) will present his research in Analysis.

"There are many more positive maps than completely positive maps," see [5], there are also various applications in which positive linear matrix maps happen to be completely positive. One such instance is Nevanlinna-Pick interpolation, where a necessary and sufficient solution criteria can be phrased as a positivity condition (Lyapunov or Stein order, see [2]) while the more common Pick matrix criteria stems from complete positivity (via Choi's theorem). In this talk we analyze *-linear matrix maps (linear maps that preserve adjoints) via what we call Hill representations (after [1], see also [3]) and determine various classes of *-linear matrix maps under which positivity and complete positivity coincide [4]. It is based on work with my PhD student Alma van der Merwe.

[1] R.D. Hill, Linear transformations which preserve hermitian matrices, Linear Algebra Appl. 6 (1973), 257--262.

[2] S. ter Horst and A. van der Merwe, A Hill-Pick matrix criteria for the Lyapunov order, submitted.

[3] S. ter Horst and A. van der Merwe, Hill representations for *-linear matrix maps, Indag. Math. (N.S.), to appear.

[4] S. ter Horst and A. van der Merwe, Linear matrix maps for which positivity and complete positivity coincide, Linear Algebra Appl. 628 (2021), 140--181.

[5] I. Klep, S. McCullough, K. Sivic, A. Zalar, There are many more positive maps than completely positive maps, Int. Math. Res. Not. 11 (2019) 3313--3375.

The talks are held using Zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at b.g.sharp@leeds.ac.uk to request zoom coordinates.