Commutativity conditions in pseudo-Michael algebras

We consider the commutativity conditions in unital pseudo-Michael algebras. These kinds of algebras have interesting properties regarding the commutativity criteria. We prove several results, which generalize known results in the case of unital Arens-Michael algebras to the pseudo-convex cases. In this paper, we first derive some specific results for the differentiable and entire functions in pseudo-Michael algebras. Then we show how such results can be applied to obtain commutativity conditions for these algebras. In Section 3, we give simple conditions implying commutativity in the unital pseudo-Michael algebras. These conditions are equivalent to similar cases in unital locally $m$-convex algebras, in particular, in Banach algebras. The most outstanding results in this direction are due to Toma, who generalized the commutativity criteria of Banach algebras to locally $m$-convex algebras. Our conditions ensure the commutativity of pseudo-Michael algebras. In the proofs of some theorems, we apply the exponential functions and Liouville theorem for bounded holomorphic functions. The use of them allows us to give a very striking short proof. Finally as a consequence, we show that some commutativity results hold for $k$-Banach algebras.