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Let A A be a semisimple Banach algebra with ‖ ‖ \left \| \right \| , which is a dense subalgebra of a semisimple Banach algebra B B with | | \left | \right | such that ‖ ‖ \left \| \right \| majorizes | | \left | \right | on A A . The purpose of this paper is to investigate the dual property between the algebras A A and B B . Some well-known results follow from this paper.

We show that the algebra of double multipliers of a certain Banach algebra A A can be embedded in the second conjugate space A ∗ ∗ {A^{**}} of A A . This generalizes the previous work by the author for B ∗ {B^*} -algebras.

Let A be a dual B*-algebra and Ap the p-class in A. We show that the conjugate space of A1 is A**, the second conjugate space of A. We also obtain a three lines theorem for Ap (1 ≦p≦∞).

LetAbe a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebraBwith | · | such that ‖ · ‖ majorises | · | onA. The purpose of this paper is to investigate the annihilator property between the algebrasAandB.

We study quasi-complementors and complementors on annihilator Banach algebras. We give conditions for a quasi-complementor to be a complementor.

Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.

Let A A be a semisimple Banach algebra and M ( A ) M(A) the algebra of double multipliers of A A . We show that M ( A ) M(A) is isomorphic to ( A ∗ ∗ , ∘ ) ({A^{ * * }}, \circ ) if and only if A A has the following properties: (1) A A is Arens regular, (2) A A has a weak approximate identity, and (3) π ( A ) \pi (A) is an ideal of ( A ∗ ∗ , ∘ ) ({A^{ * * }}, \circ ) .

Let A be a dual B ∗ {B^\ast } -algebra. We give a minimax formula for the positive elements in A. By using this formula and some of its consequent results, we introduce and study the symmetric norms and symmetrically-normed ideals in A.

Let A be a modular complemented algebra. We give some necessary and sufficient conditions for some special classes of algebras A to be annihilator or dual algebras.

Let A A be a commutative B ∗ {B^ \ast } -algebra, Φ \Phi its carrier space and A ∗ {A^ \ast } the conjugate space of A A . Let A ′ A’ be the closed subspace of A ∗ {A^ \ast } spanned by Φ \Phi . We show that A A is a dual algebra if and only if A ′ = A ∗ A’ = {A^ \ast } and for each x ∈ A x \in A , the mapping T x : f → f ∗ x {T_x}:f \to f \ast x is a weakly completely continuous operator on A ∗ {A^ \ast } . This improves an early result by B. J. Tomiuk and the author. A similar result holds for general B ∗ {B^ \ast } -algebras.

In this paper, we study several problems in Banach algebras concerned with the Arens products.

The purpose of this paper is to generalize two results in a recent paper by B. J. Tomiuk and the author. Let A be a B ∗ {B^ \ast } -algebra and M ( A ) M(A) the algebra of double centralizers of A. We show that A is a dual algebra if and only if M ( A ) M(A) coincides with A ∗ ∗ {A^{ \ast \ast }} . We also obtain that if A is an annihilator A ∗ {A^\ast } -algebra, then π A ( A ) {\pi _A}(A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} .

LetA1,A2be commutative semi-simple Banach algebras andM(A1),M(A2) their multiplier algebras. Birtel in [2] has proved that every isomorphism ofA1ontoA2induces an isomorphism ofM(A1) ontoM(A2). In this note, we extend this result to the noncommutative case. We also show that ifAis a dualA*-algebra which is a dense two-sided ideal of aB*-algebraB, thenM(A) is isomorphic toM(B). Thus the converse of the previous result cannot hold. All algebras under consideration are over the complex field.

The purpose of this note is to generalize two recent results by the author for commutative Banach algebras. Let A be a commutative Banach algebra with carrier space X A {X_A} and π \pi the canonical embedding of A into its second conjugate space A ∗ ∗ {A^{ \ast \ast }} (with the Arens product). We show that if A is a semisimple annihilator algebra, then π ( A ) \pi (A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} . We also obtain that if A is a dense two-sided ideal of C 0 ( X A ) {C_0}({X_A}) , then π ( A ) \pi (A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} if and only if A is a modular annihilator algebra.