A Banach–Lie Approach to Ternary Symmetries in Nest Algebras

Let N be a nest on a Banach space, and let Alg(N) denote the associated nest algebra equipped with the operator norm. In this paper, we develop a Banach–Lie framework for bounded triple derivations and triple automorphisms on Alg(N). We prove that the space of bounded triple derivations is closed under the commutator bracket and hence forms a Lie algebra, while the set of triple automorphisms forms a norm-closed subgroup of GL(Alg(N)). We further establish an exponential–differential correspondence between these two classes: the exponential of a bounded triple derivation yields a one-parameter group of triple automorphisms, and conversely, the tangent space at the identity of the triple automorphism group is identified with the Lie algebra of bounded triple derivations. We also relate these objects to derivations and automorphisms of the standard embedding Lie algebra associated with the Lie triple system naturally induced by Alg(N). To illustrate the general theory, we finally determine explicit forms of triple derivations and triple automorphisms for the eight non-perfect three-dimensional real Lie algebras.