Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

Abstract

Abstract. We establish the algebraic reflexivity of three isometry groups of operator structures: The group of all surjective isometries on the unitary group, the group of all surjective isometries on the set of all positive invertible operators equipped with the Thompson metric, and the group of all surjective isometries on the general linear group of B(H), the operator algebra over a complex infinite dimensional separable Hilbert space H. We show that those isometry groups coincide with certain groups of automorphisms of corresponding structures and hence we also obtain the reflexivity of some automorphism groups.

A Spherical Version of the Kowalski–słodkowski Theorem and Its Applications

“…3 gives an affirmative answer to the problem mentioned by Molnár. Mori proved the same statement in[29, Theorem 4.6] by a different argument.Next we consider the disk algebra.…”

“…Since any map in WC C is complex-linear, we infer by a simple calculation that T(0) = 0 and T is homogeneous with respect to a complex scalar. We see by (3)(4)(5)(6) that…”

“…Molnár [22] proved that each 2-local complex-linear isometry on certain C * -algebras is a surjective complex-linear isometry. Initiated by his result, there are a lot of studies on 2-local complex-linear isometries on operator algebras and function spaces assuring that each 2-local complex-linear isometry is in fact a surjective complex-linear isometry [1,3,7,9,12,13,17,22,24].…”

“…Inspired by his problem, Hatori and the author proved that a 2-local map in Iso(B, B) is an element of Iso(B, B), where B is the Banach space of all continuously differentiable functions or the Banach space of Lipschitz functions on the closed unit interval equipped with a certain norm [10]. S. Oi [3] The aim of this paper is to establish a generalization of the spherical variant of the Kowalski-Słodkowski theorem exhibited in [17]. Applying it, we prove that 2-local isometries on several function spaces are surjective isometries.…”

A generalization of the Kowalski -S\{l} odkowski theorem and its application to 2-local maps on function spaces

“…Molnár [24] proved that a 2-local complex-linear isometry on a certain C * -algebra is a surjective complex-linear isometry. Initiated by his result, there are a lot of studies on 2-local complexlinear isometries on operator algebras and function spaces assuring that a 2-local complex-linear isometry is in fact a surjective complex-linear isometry [1,3,7,9,12,13,17,23,24].…”