On the multiplicative behavior of regular matrices

Abstract

Let T T be a bounded linear operator on C ( X ) , X C(X),X compact T 2 {T_2} , with T 1 = 1 T1 = 1 . We define M T {M_T} to be the subalgebra of C ( X ) C(X) consisting of g g such that T ( f g ) = T f T g T(fg) = TfTg for all f f , and give a characterization of M T {M_T} . We apply the characterization to the multiplicative behavior of regular matrices, considering these as linear operators on C ( β N ∖ N ) C(\beta N\backslash N) . We also relate invariance properties of a matrix under suitable mappings of the integers to topological properties of its support set in β N ∖ N \beta N\backslash N , and give an example of a nonnegative multiplicative matrix whose support set is nowhere dense in β N ∖ N \beta N\backslash N .