Derivations on Banach algebras of connected multiplicative linear functionals
- PDF / 369,197 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 66 Downloads / 252 Views
Derivations on Banach algebras of connected multiplicative linear functionals M. Ghasemi1 · M. J. Mehdipour1 Received: 19 April 2019 / Revised: 6 September 2020 / Accepted: 21 September 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract φ,γ Let A and B be Banach algebras with σ (B) = ∅. Let θ, φ, γ ∈ σ (B) and Der(A ×θ B) be the set of all linear mappings d : A× B → A× B satisfying d((a, b)·θ (x, y)) = d(a, b) ·φ (x, y) + (a, b) ·γ d(x, y) for all a, x ∈ A and b, y ∈ B. In this paper, we φ,γ characterize elements of Der(A ×θ B) in the case where A has a right identity. φ,γ We then investigate the concept of centralizing for elements of Der(A ×θ B) and φ,γ determine dependent elements of Der(A ×θ B). We also apply some results to group algebras. Keywords Derivations · θ -Lau products · Centralizing mappings · Dependent elements · Locally compact groups Mathematics Subject Classification 47B47 · 43A15 · 43A20
1 Introduction Throughout the paper, A is a Banach algebra with Jacobson radical rad(A), A is a Banach algebra with a right identity u and right annihilator ran(A) = {z ∈ A : az = 0 forall a ∈ A} and B is a Banach algebra with nonempty spectrum σ (B). Let also θ, φ and γ be elements of σ (B). In this paper, we endowed A∗∗ , the second dual of A, with the first
Communicated by Pedro Tradacete.
B
M. J. Mehdipour [email protected] M. Ghasemi [email protected]
1
Department of Mathematics, Shiraz University of Technology, 71555-313 Shiraz, Iran
123
M. Ghasemi, M. J. Mehdipour
Arens product “” defined by mn, τ = m, nτ , where nτ, a = n, τ a and τ a, x = τ, ax for all m, n ∈ A∗∗ , τ ∈ A∗ and a, x ∈ A. Let us recall that the topological center of A∗∗ is denoted by Zt (A∗∗ ) and is defined by Zt (A∗∗ ) = {m ∈ A∗∗ : themapping n → mn isweak ∗ − weak ∗ continuouson A∗∗ }. Following [25], the θ -Lau product A and B is denoted by A ×θ B and it is the direct product A × B together with the component wise addition and the multiplication (a, b) ·θ (x, y) = (ax + θ (y)a + θ (b)x, by). We note that in the case where B = C and θ is the identity map on C, the unitization A will be obtained. We also note that if we permit θ = 0, the θ -Lau product A ×θ B is the usual direct product of Banach algebras. Hence, we disregard the possibility that θ = 0. A linear mapping D : A → A is called centralizing if for every a ∈ A [D(a), a] ∈ Z(A), where for each a, x ∈ A [a, x] = ax − xa and Z(A) denotes the center of A. Also, D is called a derivation if for every a, x ∈ A D(ax) = D(a)x + aD(x). The set of all derivations on A is denoted by Der(A). For any x ∈ A, the derivation a → [a, x] on A is called an inner derivation and it is denoted by ad x . Similarly, one can define derivations on rings. The θ -Lau products A ×θ B were first introduced by Lau [17], for Banach algebras that are pre-duals of von Neumann algebras, and for which the identity of the dual is a multiplicative linear functional. Sanjani Monfared [25] extended this product to arbitrary Banach
Data Loading...
