A Liouville Theorem for Matrix‐Valued Harmonic Functions on Nilpotent Groups

Let σ be a non‐degenerate positive M n ‐valued measure on a locally compact group G with ‖σ‖ = 1. An M n ‐valued Borel function f on G is called σ‐harmonic if f ( x ) = ∫ G f ( x y − 1) d σ ( y )for all x ∈ G . Given such a function f which is bounded and left uniformly continuous on G , it is shown that every central element in G is a period of f . Further, it is shown that f is constant if G is nilpotent or central. 2000 Mathematics Subject Classification 31C05, 43A05, 45E10, 46G10.