Positive completions
supervised by
Dr Ying-Fen Lin
A partially defined n × n matrix is a matrix in which only some of the entries are specified
while the others are left undefined. A pattern is a subset J of {1, . . . , n} × {1, . . . , n}. A
partially defined n × n matrix T = (tij ) will be said to have pattern J if tij is specified if and
only if (i, j) ∈ J. A pattern J is called symmetric if (i, i) ∈ J for all i and (i, j) ∈ J implies
that (j, i) ∈ J. A partially defined matrix T is called symmetric provided that its pattern J
is symmetric, tii is real for all i, and whenever tij is specified then tji = tij . Then a positive
completion of a partially defined matrix is simply a specification of all unspecified entries such
that the resulting n × n matrix is positive. A partially defined matrix T is partially positive
if it is symmetric and if every principal specified submatrix of T is positive. It is clear that a
necessary condition for a partially defined symmetric matrix to have a positive completion is
that it is partially positive; however, not every partially positive matrix can be completed to
a positive matrix.
While positive completions of partially defined matrices were studied in [2], one can consider
the same problem in a much more general setting. One such setting is given as follows. Let
n > 2, and consider a sequence of matrix spaces {Mnk }k≥1 . For each k, there is a natural
unital *-homomorphism ϕk : Mnk → Mnk+1 given by ϕk (A) = A ⊗ In . The direct limit of
the sequence {Mnk }k≥1 with *-homomorphisms ϕk is an example of an AF C*-algebra. This
class of algebras has played a fundamental role in Operator Algebra Theory and has found
important applications in Theoretical Physics.
The aim of this project is to develop a theory of positive completions of partially defined
elements in AF C*-algebras. As a starting point, we will look for the right notions of partially
defined and partially positive elements of an AF C*-algebra, as well as the notion of a positive
completion of a partially positive element. We aim to obtain characterisations of positive
completions in this setting, using the powerful modern theory of completely positive maps [1].
References
[1] V.I. Paulsen, Completely bounded maps and operator algebras, Cambridge University Press, 2002.
[2] V.I. Paulsen, S. Power and R.R. Smith, Schur products and matrix completions, J. Funcr. Anal. 85
(1989), 151-178.