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In this paper we introduce two new classes of operators that we call strongly order continuous and strongly σ-order continuous operators. An operator T : E → F between two Riesz spaces is said to be strongly order continuous (resp. strongly σ-orderWe give some conditions under which order continuity will be equivalent to strongly order continuity of operators on Riesz spaces. We show that the collection of all so-continuous linear functionals on a Riesz space E is a band of E ∼ .

A continuous operator T between two normed vector lattices E and F is called unbounded order-norm continuous whenever x? uo? 0 implies ||Tx?|| ? 0, for each norm bounded net (x?)? ? E. Let E and F be two Banach lattices. A continuous operator T : E ? F is called unbounded norm continuous, if for each norm bounded net (x?)? ? E, x? un? 0 implies Tx? un? 0. In this manuscript, we study some properties of these classes of operators and investigate their relationships with the other classes of operators.

Let A and B be Banach algebras and T : B → A be a continuous homomorphism. nweak amenability of the Banach algebra A × T B (defined in Bade, W. G., Curtis, P. C., Dales, H. G.: Amenability and weak amenability for Beurling and Lipschitz algebras. Proc. London Math. Soc., 55(2), 359-377 (1987)) is studied. The new version of a Banach algebra defined with a continuous homomorphism is introduced and Arens regularity and various notions of amenability of this algebra are studied.

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