Approximation of Generalized Left Derivations

Abstract

We need to take account of the superstability for generalized left derivations (resp., generalized derivations) associated with a Jensen-type functional equation, and we also deal with problems for the Jacobson radical ranges of left derivations (resp., derivations).

1. Introduction

Let 𝒜 be an algebra over the real or complex field 𝔽. An additive mapping 𝑑∶𝒜→𝒜 is said to be a left derivation (resp., derivation) if the functional equation 𝑑(𝑥𝑦)=𝑥𝑑(𝑦)+𝑦𝑑(𝑥) (resp., 𝑑(𝑥𝑦)=𝑥𝑑(𝑦)+𝑑(𝑥)𝑦) holds for all 𝑥,𝑦∈𝒜. Furthermore, if the functional equation 𝑑(𝜆𝑥)=𝜆𝑑(𝑥) is valid for all 𝜆∈𝔽 and all 𝑥∈𝒜, then 𝑑 is a linear left derivation (resp., linear derivation). An additive mapping 𝐺∶𝒜→𝒜 is called a generalized left derivation (resp., generalized derivation) if there exists a left derivation (resp., derivation) 𝛿∶𝒜→𝒜 such that the functional equation 𝐺(𝑥𝑦)=𝑥𝐺(𝑦)+𝑦𝛿(𝑥) (resp., 𝐺(𝑥𝑦)=𝑥𝐺(𝑦)+𝛿(𝑥)𝑦) is fulfilled for all 𝑥∈𝒜. In addition, if the functional equations 𝐺(𝜆𝑥)=𝜆𝐺(𝑥) and 𝛿(𝜆𝑥)=𝜆𝛿(𝑥) hold for all 𝜆∈𝔽 and all 𝑥∈𝒜, then 𝐺 is a linear generalized left derivation (resp., linear generalized derivation).

It is of interest to consider the concept of stability for a functional equation arising when we replace the functional equation by an inequality which acts as a perturbation of the equation. The study of stability problems had been formulated by Ulam [1] during a talk in 1940: “Under what condition does there exists a homomorphism near an approximate homomorphism?” In the following year 1941, Hyers [2] was answered affirmatively the question of Ulam for Banach spaces, which states that if 𝜀>0 and 𝑓∶𝒳→𝒴 is a map with 𝒳, a normed space, 𝒴, a Banach space, such that‖𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)‖≤𝜀, (1.1)for all 𝑥,𝑦∈𝒳, then there exists a unique additive map𝑇∶𝒳→𝒴such that‖𝑓(𝑥)−𝑇(𝑥)‖≤𝜀, (1.2)for all 𝑥∈𝒳. Moreover, if 𝑓(𝑡𝑥) is continuous in 𝑡∈ℝ for each fixed 𝑥 in 𝒳, where ℝ denotes the set of real numbers, then 𝑇 is linear. This stability phenomenon is called the Hyers-Ulam stability of the additive functional equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦). A generalized version of the theorem of Hyers for approximately additive mappings was given by Aoki [3] and for approximate linear mappings was presented by Rassias [4] in 1978 by considering the case when the inequality (1.1) is unbounded. Due to the fact that the additive functional equation 𝑓(𝑥+𝑦)=𝑓(𝑥)+𝑓(𝑦) is said to have the Hyers-Ulam-Rassias stability property. The stability result concerning derivations between operator algebras was first obtained by Šemrl [5]. Recently, Badora [6] gave a generalization of the Bourgin's result [7]. He also dealt with the Hyers-Ulam stability and the Bourgin-type superstability of derivations in [8].

In 1955, Singer and Wermer [9] obtained a fundamental result which started investigation into the ranges of linear derivations on Banach algebras. The result, which is called the Singer-Wermer theorem, states that any continuous linear derivation on a commutative Banach algebra maps into the Jacobson radical. They also made a very insightful conjecture, namely, that the assumption of continuity is unnecessary. This was known as the Singer-Wermer conjecture and was proved in 1988 by Thomas [10]. The Singer-Wermer conjecture implies that any linear derivation on a commutative semisimple Banach algebra is identically zero which is the result of Johnson [11]. After then, Hatori and Wada [12] showed that a zero operator is the only derivation on a commutative semisimple Banach algebra with the maximal ideal space without isolated points. Note that this differs from the above result of B.E. Johnson. Based on these facts and a private communication with Watanabe [13], Miura et al. proved the Hyers-Ulam-Rassias stability and Bourgin-type superstability of derivations on Banach algebras in [13]. Various stability results are given by Moslehian and Park, see, for example, [14–18].

The main purpose of the present paper is to consider the superstability of generalized left derivations (resp., generalized derivations) on Banach algebras associated to the following Jensen type functional equation:𝑓(𝑥+𝑦𝑘)=𝑓(𝑥)𝑘+𝑓(𝑦)𝑘, (1.3)where 𝑘>1 is an integer. This functional equation is introduced in [19]. Moreover, we will investigate the problems for the Jacobson radical ranges of left derivations (resp., derivations) on Banach algebras. We use the direct method and some ideas of Amyari et al. [19].

2. Main Results

Throughout this paper, the element 𝑒 of an algebra will denote a unit. We now establish the superstability of a generalized left derivation associated with the Jensen type functional equation as follows.

Theorem 2.1. Let 𝒜 be a Banach algebra with unit. Suppose that 𝑓∶𝒜→𝒜 is a mapping with 𝑓(0)=0 for which there exists a mapping 𝑔∶𝒜→𝒜 such that the functional inequality:‖𝑓(𝑥+𝑦𝑘+𝑧𝑤)−𝑓(𝑥)𝑘−𝑓(𝑦)𝑘−𝑧𝑓(𝑤)−𝑤𝑔(𝑧)‖≤𝜀, (2.1)for all 𝑥,𝑦,𝑧,𝑤∈𝒜. Then, 𝑓 is a generalized left derivation, and 𝑔 is a left derivation.

Proof. Substituting 𝑤=0 in (2.1), we get‖𝑓(𝑥+𝑦𝑘)−𝑓(𝑥)𝑘−𝑓(𝑦)𝑘‖≤𝜀, (2.2)for all 𝑥,𝑦∈𝒜. Let us take 𝑦=0 and replace 𝑥 by 𝑘𝑥 in the above relation. Then, it becomes‖𝑓(𝑥)−𝑓(𝑘𝑥)𝑘‖≤𝜀, (2.3)for all 𝑥∈𝒜. An induction implies that‖𝑓(𝑘^𝑛𝑥)𝑘^𝑛𝑘−𝑓(𝑥)‖≤1𝑘−1(1−𝑘^𝑛)𝜀, (2.4)for all 𝑥∈𝒜. By virtue of (2.4), one can easily check that for 𝑛>𝑚,‖𝑓(𝑘^𝑛𝑥)𝑘^𝑛−𝑓(𝑘^𝑚𝑥)𝑘^𝑚1‖=𝑘^𝑚‖𝑓(𝑘^𝑛−𝑚⋅𝑘^𝑚𝑥)𝑘^𝑛−𝑚−𝑓(𝑘^𝑚≤1𝑥)‖𝑘^𝑚−11(𝑘−1)(1−𝑘^𝑛−𝑚)𝜀, (2.5)for all 𝑥∈𝒜. So, the sequence {𝑓(𝑘^𝑛𝑥)/𝑘^𝑛} is Cauchy. Since 𝒜 is complete, {𝑓(𝑘^𝑛𝑥)/𝑘^𝑛} converges. Let 𝑑∶𝒜→𝒜 be the mapping defined by (𝑥∈𝒜)𝑑(𝑥)∶=lim_𝑛→∞𝑓(𝑘^𝑛𝑥)𝑘^𝑛. (2.6)By letting 𝑛→∞ in (2.4), we get𝑘‖𝑓(𝑥)−𝑑(𝑥)‖≤𝑘−1𝜀, (2.7)for all 𝑥∈𝒜.
Now, we assert that 𝑑 is additive. Replacing 𝑥 and 𝑦 by 𝑘^𝑛𝑥 and 𝑘^𝑛𝑦 in (2.2), respectively, we have‖1𝑘^𝑛𝑘𝑓(^𝑛𝑥+𝑘^𝑛𝑦𝑘1)−𝑘𝑓(𝑘^𝑛𝑥)𝑘^𝑛−1𝑘𝑓(𝑘^𝑛𝑦)𝑘^𝑛1‖≤𝑘^𝑛𝜀, (2.8)for all 𝑥,𝑦∈𝒜, taking the limit as 𝑛→∞, we obtain𝑑(𝑥+𝑦𝑘)=𝑑(𝑥)𝑘+𝑑(𝑦)𝑘, (2.9)for all 𝑥,𝑦∈𝒜. Letting 𝑦=0 in the previous identity yields 𝑑(𝑥/𝑘)=𝑑(𝑥)/𝑘 for all 𝑥∈𝒜. So, (2.9) becomes 𝑑(𝑥+𝑦)=𝑑(𝑥)+𝑑(𝑦), for all 𝑥,𝑦∈𝒜, namely, 𝑑 is additive.
To demonstrate the uniqueness of the additive mapping 𝑑 subject to (2.7), we assume that there exists another additive mapping 𝐷∶𝒜→𝒜 satisfying the inequality (2.7), for all 𝑥∈𝒜. Since 𝐷(𝑘^𝑛𝑥)=𝑘^𝑛𝐷(𝑥) and 𝑑(𝑘^𝑛𝑥)=𝑘^𝑛𝑑(𝑥), we see that1‖𝐷(𝑥)−𝑑(𝑥)‖=𝑘^𝑛‖𝐷(𝑘^𝑛𝑥)−𝑑(𝑘^𝑛≤1𝑥)‖𝑘^𝑛[‖𝐷(𝑘^𝑛𝑥)−𝑓(𝑘^𝑛𝑥)‖+‖𝑓(𝑘^𝑛𝑥)−𝑑(𝑘^𝑛≤2𝑥)‖]𝑘^𝑛−1(𝑘−1)𝜀, (2.10)for all 𝑥∈𝒜. By letting 𝑛→∞ in this inequality, we conclude that 𝐷=𝑑, that is, 𝑑 is unique.
Next, we are going to prove that 𝑓 is a generalized left derivation. If we take 𝑥=𝑦=0 in (2.1), we also have‖𝑓(𝑧𝑤)−𝑧𝑓(𝑤)−𝑤𝑔(𝑧)‖≤𝜀, (2.11)for all 𝑧,𝑤∈𝒜. Moreover, if we replace 𝑧 and 𝑤 with 𝑘^𝑛𝑧 and 𝑘^𝑛𝑤 in (2.11), respectively, and then divide both sides by 𝑘^2𝑛, we get‖𝑓(𝑘^2𝑛𝑧𝑤)𝑘^2𝑛−𝑧𝑓(𝑘^𝑛𝑤)𝑘^𝑛−𝑤𝑔(𝑘^𝑛𝑧)𝑘^𝑛1‖≤𝑘^2𝑛𝜀, (2.12)for all 𝑧,𝑤∈𝒜. Letting 𝑛→∞, we obtainlim_𝑛→∞𝑤𝑔(𝑘^𝑛𝑧)𝑘^𝑛=𝑑(𝑧𝑤)−𝑧𝑑(𝑤), (2.13)for all 𝑧,𝑤∈𝒜. Suppose that 𝑤=𝑒 in the above equation. Then, it followslim_𝑛→∞𝑔(𝑘^𝑛𝑧)𝑘^𝑛=𝑑(𝑧)−𝑧𝑑(𝑒), (2.14)for all 𝑧∈𝒜. Thus, if 𝛿(𝑧)∶=𝑑(𝑧)−𝑧𝑑(𝑒), then by the additivity of 𝑑, we get𝛿(𝑥+𝑦)=𝑑(𝑥)+𝑑(𝑦)−𝑥𝑑(𝑒)−𝑦𝑑(𝑒)=(𝑑(𝑥)−𝑥𝑑(𝑒))+(𝑑(𝑦)−𝑦𝑑(𝑒))=𝛿(𝑥)+𝛿(𝑦), (2.15)for all 𝑥∈𝒜. Hence, 𝛿 is additive.
Let Δ(𝑧,𝑤)=𝑓(𝑧𝑤)−𝑧𝑓(𝑤)−𝑤𝑔(𝑧), for all 𝑧,𝑤∈𝒜. Since, 𝑓 and 𝑔 satisfy the inequality given in (2.11), thenlim_𝑛→∞Δ(𝑘^𝑛𝑧,𝑤)𝑘^𝑛=0, (2.16)for all 𝑧,𝑤∈𝒜. We note that𝑑(𝑧𝑤)=lim_𝑛→∞𝑓(𝑘^𝑛𝑧𝑤)𝑘^𝑛=lim_𝑛→∞𝑓(𝑘^𝑛𝑧⋅𝑤)𝑘^𝑛=lim_𝑛→∞𝑘^𝑛𝑧𝑓(𝑤)+𝑤𝑔(𝑘^𝑛𝑧)+Δ(𝑘^𝑛𝑧,𝑤)𝑘^𝑛=lim_𝑛→∞{𝑧𝑓(𝑤)+𝑤𝑔(𝑘^𝑛𝑧)𝑘^𝑛+Δ(𝑘^𝑛𝑧,𝑤)𝑘^𝑛}=𝑧𝑓(𝑤)+𝑤𝛿(𝑧), (2.17)for all 𝑧,𝑤∈𝒜. Since 𝛿 is additive, we can rewrite (2.17) as𝑘^𝑛𝑧𝑓(𝑤)+𝑘^𝑛𝑤𝛿(𝑧)=𝑑(𝑘^𝑛𝑧⋅𝑤)=𝑑(𝑧⋅𝑘^𝑛𝑤)=𝑧𝑓(𝑘^𝑛𝑤)+𝑘^𝑛𝑤𝛿(𝑧), (2.18)for all 𝑧,𝑤∈𝒜. Based on the above relation, one has 𝑧𝑓(𝑤)=𝑧(𝑓(𝑘^𝑛𝑤)/𝑘^𝑛), for all 𝑧,𝑤∈𝒜. Moreover, we can obtain 𝑧𝑓(𝑤)=𝑧𝑑(𝑤), for all 𝑧,𝑤∈𝒜 as 𝑛→∞. If 𝑧=𝑒, we also have that 𝑓=𝑑. Therefore, we get𝑓(𝑧𝑤)=𝑧𝑓(𝑤)+𝑤𝛿(𝑧), (2.19)for all 𝑧,𝑤∈𝒜.
We now want to verify that 𝛿 is a left derivation using the equations developed in the previous part. Indeed, using the facts that 𝑓 satisfies (2.19), we have𝛿(𝑥𝑦)=𝑓(𝑥𝑦)−𝑥𝑦𝑓(𝑒)=𝑥𝑓(𝑦)+𝑦𝛿(𝑥)−𝑥𝑦𝑓(𝑒)=𝑥(𝑓(𝑦)−𝑦𝑓(𝑒))+𝑦𝛿(𝑥)=𝑥𝛿(𝑦)+𝑦𝛿(𝑥), (2.20)for all 𝑥,𝑦∈𝒜, which means that 𝑓 is a generalized left derivation.
We finally need to show that 𝑔 is a left derivation. Let us replace 𝑤 by 𝑘^𝑛𝑤 in (2.11). Then,‖𝑓(𝑘^𝑛𝑧𝑤)𝑘^𝑛−𝑧𝑓(𝑘^𝑛𝑤)𝑘^𝑛1−𝑤𝑔(𝑧)‖≤𝑘^𝑛𝜀, (2.21)for all 𝑧,𝑤∈𝒜. Passing the limit as 𝑛→∞, we get𝑑(𝑧𝑤)−𝑧𝑑(𝑤)−𝑤𝑔(𝑧)=0, (2.22)for all 𝑧,𝑤∈𝒜. This implies that 𝑑(𝑧𝑤)=𝑧𝑑(𝑤)+𝑤𝑔(𝑧), for all 𝑧,𝑤∈𝒜, and thus if 𝑤=𝑒, we deduce that 𝑔(𝑧)+𝑧𝑑(𝑒)=𝑑(𝑧), for all 𝑧∈𝒜. Hence, we get 𝑔(𝑧)=𝑑(𝑧)−𝑧𝑑(𝑒)=𝛿(𝑧), for all 𝑧∈𝒜. Since, 𝛿 is a left derivation, we can conclude that 𝑔 is a left derivation as well. This completes the proof of the theorem.

Employing the similar way as in the proof of Theorem 2.1, we get the following result for a generalized derivation.

Theorem 2.2. Let 𝒜 be a Banach algebra with unit. Suppose that 𝑓∶𝒜→𝒜 is a mapping with 𝑓(0)=0 for which there exists a mapping 𝑔∶𝒜→𝒜 such that‖𝑓(𝑥+𝑦𝑘+𝑧𝑤)−𝑓(𝑥)𝑘−𝑓(𝑦)𝑘−𝑧𝑓(𝑤)−𝑔(𝑧)𝑤‖≤𝜀, (2.23)for all 𝑥,𝑦,𝑧,𝑤∈𝒜. Then, 𝑓 is a generalized derivation, and 𝑔 is a derivation.

In view of the Thomas' result [10], derivations on Banach algebras now belong to the noncommutative setting. Among various noncommutative version of the Singer-Wermer theorem, Brešar and Vukman [20] proved the following. Any continuous linear left derivation on a Banach algebra maps into its Jacobson radical and also any left derivation on a semiprime ring is a derivation which maps into its center.

The following is the functional inequality with the problem as in the above Brešar and Vukman's result.

Theorem 2.3. Let 𝒜 be a semiprime Banach algebra with unit. Suppose that 𝑓∶𝒜→𝒜 is a mapping with 𝑓(0)=0 for which there exists a mapping 𝑔∶𝒜→𝒜 such that the functional inequality:‖𝑓(𝛼𝑥+𝛽𝑦𝑘+𝑧𝑤)−𝛼𝑓(𝑥)𝑘−𝛽𝑓(𝑦)𝑘−𝑧𝑓(𝑤)−𝑤𝑔(𝑧)‖≤𝜀, (2.24)for all 𝑥,𝑦,𝑧,𝑤∈𝒜 and all 𝛼,𝛽∈𝕌={𝑧∈𝐶∶|𝑧|=1}. Then, 𝑓 is a linear generalized left derivation. In this case, 𝑔 is a linear derivation which maps 𝒜 into the intersection of its center 𝑍(𝒜) and its Jacobson radical rad(𝒜).

Proof. We consider 𝛼=𝛽=1∈𝕌 in (2.24) and then 𝑓 satisfies the inequality (2.1). It follows from Theorem 2.1 that 𝑓 is a generalized left derivation, and 𝑔 is a left derivation, where𝑓(𝑥)∶=lim_𝑛→∞𝑓(𝑘^𝑛𝑥)𝑘^𝑛,𝑔(𝑥)∶=𝑓(𝑥)−𝑥𝑓(𝑒), (2.25)for all 𝑥∈𝒜. Letting 𝑤=0 in (2.24), we have‖𝑓(𝛼𝑥+𝛽𝑦𝑘)−𝛼𝑓(𝑥)𝑘−𝛽𝑓(𝑦)𝑘‖≤𝜀, (2.26)for all 𝑥,𝑦∈𝒜 and all 𝛼,𝛽∈𝕌. If we also replace 𝑥 and 𝑦 with 𝑘^𝑛𝑥 and 𝑘^𝑛𝑦 in (2.26), respectively, and then divide both sides by 𝑘^𝑛, we see that‖1𝑘^𝑛𝑓(𝛼𝑘^𝑛𝑥+𝛽𝑘^𝑛𝑦𝑘1)−𝛼𝑘𝑓(𝑘^𝑛𝑥)𝑘^𝑛1−𝛽𝑘𝑓(𝑘^𝑛𝑦)𝑘^𝑛1‖≤𝑘^𝑛𝜀⟶0, (2.27)for all 𝑥,𝑦∈𝒜 and all 𝛼,𝛽∈𝕌, as 𝑛→∞. So, we get𝑓(𝛼𝑥+𝛽𝑦𝑘)=𝛼𝑓(𝑥)𝑘+𝛽𝑓(𝑦)𝑘, (2.28)for all 𝑥,𝑦∈𝒜 and all 𝛼,𝛽∈𝕌. From the additivity of 𝑓, we find that𝑓(𝛼𝑥+𝛽𝑦)=𝛼𝑓(𝑥)+𝛽𝑓(𝑦), (2.29)for all 𝑥,𝑦∈𝒜 and all 𝛼,𝛽∈𝕌. Let us now assume that 𝜆 is a nonzero complex number and that 𝐿 a positive integer greater than |𝜆|. Then by applying a geometric argument, there exist 𝜆₁,𝜆₂∈𝕌 such that 2(𝜆/𝐿)=𝜆₁+𝜆₂. In particular, due to the additivity of 𝑓, we obtain 𝑓((1/2)𝑥)=(1/2)𝑓(𝑥) for all 𝑥∈𝒜. Thus, we have𝐿𝑓(𝜆𝑥)=𝑓(2𝜆⋅2⋅𝐿1𝑥)=𝐿𝑓(2𝜆⋅2⋅𝐿𝐿𝑥)=2𝑓((𝜆₁+𝜆₂=𝐿)𝑥)2(𝜆₁+𝜆₂𝐿)𝑓(𝑥)=2𝜆⋅2⋅𝐿𝑓(𝑥)=𝜆𝑓(𝑥), (2.30)for all 𝑥∈𝒜. Also, it is obvious that 𝑓(0𝑥)=0=0𝑓(𝑥), for all 𝑥∈𝒜, that is, 𝑓 is ℂ-linear. Therefore, 𝑓 is a linear generalized left derivation, and so 𝑔 is also a linear left derivation. According to the Brešar and Vukman's result which tells us that 𝑔 is a linear derivation which maps 𝒜 into its center 𝑍(𝒜). Since 𝑍(𝒜) is a commutative Banach algebra, the Singer-Wermer conjecture tells us that 𝑔_|𝑍(𝒜) maps 𝑍(𝒜) into rad(𝑍(𝒜))=𝑍(𝒜)∩rad(𝒜) and thus 𝑔²(𝒜)⊆rad(𝒜). Using the semiprimeness of rad(𝒜) as well as the identity, we have2𝑔(𝑥)𝑦𝑔(𝑥)=𝑔²(𝑥𝑦𝑥)−𝑥𝑔²(𝑦𝑥)−𝑔²(𝑥𝑦)𝑥+𝑥𝑔²(𝑦)𝑥, (2.31)for all 𝑥,𝑦∈𝒜, we have 𝑔(𝒜)⊆rad(𝒜), that is, 𝑔 is a linear derivation which maps 𝒜 into the intersection of its center 𝑍(𝒜) and its Jacobson radical rad(𝒜). The proof of the theorem is ended.

The next corollary is the Brešar and Vukman's result.

Corollary 2.4. Let 𝒜 be a Banach algebra with unit. Suppose that 𝑓∶𝒜→𝒜 is a continuous mapping with 𝑓(0)=0 for which there exists a mapping 𝑔∶𝒜→𝒜 such that the functional inequality (2.26). Then, 𝑓∶𝒜→𝒜 is a linear generalized left derivation. In this case, 𝑔 maps 𝒜 into its Jacobson radical rad(𝒜).

Proof. On account of Theorem 2.3, 𝑔 is a linear left derivation on 𝒜. Hence, 𝑔 maps 𝒜 into its Jacobson radical rad(𝒜) by the Brešar and Vukman's result, which completes the proof.

With the help of Theorem 2.2, the following property can be derived along the same argument in the proof of Theorem 2.3.

Theorem 2.5. Let 𝒜 be a commutative Banach algebra with unit. Suppose that 𝑓∶𝒜→𝒜 is a mapping with 𝑓(0)=0 for which there exists a mapping 𝑔∶𝒜→𝒜 such that the functional inequality:‖𝑓(𝛼𝑥+𝛽𝑦𝑘+𝑧𝑤)−𝛼𝑓(𝑥)𝑘−𝛽𝑓(𝑦)𝑘−𝑧𝑓(𝑤)−𝑔(𝑧)𝑤‖≤𝜀, (2.32)for all 𝑥,𝑦,𝑧,𝑤∈𝒜 and all 𝛼,𝛽∈𝕌={𝑧∈𝐶∶|𝑧|=1}. Then, 𝑓 is a linear generalized derivation. In this case, 𝑔 maps 𝒜 into its Jacobson radical rad(𝒜).

Remark 2.6. We can generalize our results by substituting another functions or another forms satisfying suitable conditions (see, e.g., [19, 21]) for the bound 𝜀 of the functional inequalities connected to the Jensen type functional equation.

Acknowledgments

The authors would like to thank referees for their valuable comments regarding a previous version of this paper. The corresponding author dedicates this paper to his late father.