  The main purpose of this paper is to extend to a situation involving matrix
valued orthogonal polynomials and spherical functions, a result that traces its
origin and its importance to work of Claude Shannon in laying the mathematical
foundations of information theory and to a remarkable series of papers by D.
Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example
showing in a non-commutative setup that a bispectral property implies that the
corresponding global operator of "time and band limiting" admits a commuting
local operator. This is a noncommutative analog of the famous prolate
spheroidal wave operator.
