Maps on matrix spaces

Abstract

It is well known that every automorphism of the full matrix algebra is inner. We give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras, characterizations of monotone and orthogonality preserving maps on idempotent matrices, some nonlinear preserver results, and some recent theorems concerning geometry of matrices. We show that all these topics are closely related and point out the connections with physics and geometry. Several open problems are posed.

Spectrally isometric elementary operators

“…Consequently, x ∈ P and hence P = SP . Since A/P ∼ = M n (C) for some n ∈ N, we conclude that S P is a Jordan automorphism by [ It is well known that every Jordan automorphism of M n (C), n > 1 is either of the form x → wxw −1 or x → wx t w −1 for some invertible w ∈ M n (C), where x t denotes the transpose of x; see, e.g., [20,Corollary 1.4]. Note that x → x t is the elementary operator T = n i,j=1 M e ji ,e ji , where e ij , 1 ≤ i, j ≤ n denotes the usual set of matrix units.…”

Schur Multiplicative Maps on Matrices

“…Linear maps, additive maps, and multiplicative maps on matrices mapping the set of rank-k matrices to itself have been studied by many researchers; for example, see [2][3][4]15] and the references therein. In this section, we characterize Schur multiplicative maps that map the set of rank-k matrices to itself.…”

“…The study of the Schur product is related to many pure and applied areas; see [9]. There has been considerable interest in studying linear maps, additive maps, and multiplicative maps f on matrices with some special properties such as f (S) ⊆ S for a certain subset of matrices, or ( f (A)) = (A) for a given function on matrices; for example, see [3,8,11,12,15] and the references therein. In this paper, we study Schur multiplicative maps on matrices with some of these special properties.…”

Maps on idempotent operators

“…An interested reader can find results on the finitedimensional case in [27]. Let us just mention that these results are closely connected to the geometry of matrices and to the geometry of Grassmann spaces (see [28]). …”

“…However, we expect that the finite-dimensional case can be substantially improved. To explain possible improvements 298 P.ŠEMRL we would need the notion of an injective degenerate order preserving map [27,28]. As the emphasis in this paper is on the infinite-dimensional case we omit the details here.…”