Domination by ergodic elements in ordered Banach algebras

摘要

We recall the definition and properties of an algebra cone in an ordered Banach algebra (OBA) and continue to develop spectral theory for the positive elements. An element a of a Banach algebra is called ergodic if the sequence of sums ∑_(k=0)~(n-1) a~k converges. If a and b are positive elements in an OBA such that 0 ≤ a ≤ b and if b is ergodic, an interesting problem is that of finding conditions under which a is also ergodic. We will show that in a semisimple OBA that has certain natural properties, the condition we need is that the spectral radius of b is a Riesz point (relative to some inessential ideal). We will also show that the results obtained for OBAs can be extended to the more general setting of commutatively ordered Banach algebras (COBAs) when adjustments corresponding to the COBA structure are made.