A Lie product formula for one parameter groups of isometries on Banach spaces

A Lie ideal of a division ring [Formula: see text] is an additive subgroup [Formula: see text] of [Formula: see text] such that the Lie product [Formula: see text] of any two elements [Formula: see text] is in [Formula: see text] or [Formula: see text]. The main concern of this paper is to present some properties of Lie ideals of [Formula: see text] which may be interpreted as being dual to known properties of normal subgroups of [Formula: see text]. In particular, we prove that if [Formula: see text] is a finite-dimensional division algebra with center [Formula: see text] and [Formula: see text], then any finitely generated [Formula: see text]-module Lie ideal of [Formula: see text] is central. We also show that the additive commutator subgroup [Formula: see text] of [Formula: see text] is not a finitely generated [Formula: see text]-module. Some other results about maximal additive subgroups of [Formula: see text] and [Formula: see text] are also presented.

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The centraliser of the injective tensor product

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700029841 ◽

1991 ◽

Vol 44 (3) ◽

pp. 357-365 ◽

Cited By ~ 1

Author(s):

Wend Werner

Keyword(s):

Tensor Product ◽

Banach Spaces ◽

Product Formula ◽

Injective Tensor Product ◽

Image Position

The aim of this note is to obtain an intrinsic product formula for the centraliser of the injective tensor product of a couple of Banach spaces, Z(). More precisely, we are going to prove thatHere, the spaces and depend only on X and Y, respectively, and Xk denotes the topological k-product.A Counterexaple used to demonstrate that the k-product cannot beavoided serves as an answer to a question posed by W. Rueß and D. Werner concerning the behaviour of M-ideals on Y.

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The Trotter–Lie product formula for Gibbs semigroups

Journal of Mathematical Physics ◽

10.1063/1.527985 ◽

1988 ◽

Vol 29 (4) ◽

pp. 888-891 ◽

Cited By ~ 8

Author(s):

V. A. Zagrebnov

Keyword(s):

Product Formula ◽

Lie Product Formula

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Operator-Norm Convergence of the Trotter Product Formula on Hilbert and Banach Spaces: A Short Survey

Current Research in Nonlinear Analysis - Springer Optimization and Its Applications ◽

10.1007/978-3-319-89800-1_9 ◽

2018 ◽

pp. 229-247 ◽

Cited By ~ 2

Author(s):

Hagen Neidhardt ◽

Artur Stephan ◽

Valentin A. Zagrebnov

Keyword(s):

Banach Spaces ◽

Product Formula ◽

Operator Norm ◽

Norm Convergence ◽

Trotter Product Formula ◽

Short Survey

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A note on uniquely maximal Banach spaces

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500028091 ◽

1983 ◽

Vol 26 (1) ◽

pp. 85-87 ◽

Cited By ~ 1

Author(s):

E. R. Cowie

Keyword(s):

Banach Space ◽

Maximal Subgroup ◽

Banach Spaces ◽

Hilbert Spaces ◽

Complex Banach Space ◽

Complex Spaces ◽

Finite Dimensional ◽

Bounded Group ◽

Group Of Isometries ◽

Groups Of Isometries

Let X be a real or complex Banach space with norm ∥·∥· Let G denote the set of all isometric automorphisms on X. Then G is a bounded subgroup of the group of all invertible operators GL(X) in B(X). We shall call G the group of isometries with respect to the norm ∥·∥· A bounded subgroup of GL(X) is said to be maximal if it is not contained in any larger bounded subgroup. The Banach space X has maximal norm if G is maximal. Hilbert spaces have maximal norm. For the (real or complex) spaces c0, lp (1≦p<∞), Lp[0,1] (1≦p<∞), Pelczynski and Rolewicz have shown that the standard norms are maximal ([3], pp. 252–265). In finite dimensional spaces the only maximal groups of isometries are the groups of orthogonal transformations. Given any bounded group H in B(X), X can be renormed equivalently so that each T∈H is an isometry, by ‖x‖1=sup{|Tx‖; T∈H}. Therefore corresponding to every maximal subgroup G there is at least one maximal norm for which G is the group of isometries. In this paper we shall investigate those maximal groups G for which there is only one maximal norm with G as its group of isometries.

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