Commuting maps on some subsets of matrices that are not closed under addition

SOME REMARKS ON GENERALIZATIONS OF MULTIPLICATIVELY CLOSED SUBSETS

Let R be a commutative ring with identity and Mbe a unitary R-module. In this paper we generalize the conceptmultiplicatively closed subset of R and we study some propertiesof these genaralized subsets of M. Among the many results in thispaper, we generalize some well-known theorems about multiplicativelyclosed subsets of R to these generalized subsets of M. Alsowe show that some other well-kno...

Commuting maps on rank-k matrices

Maximal subsets of pairwise non-commuting elements of some finite p-groups

Let G be a group. A subset X of G is a set of pairwise noncommuting elements if xy ̸= yx for any two distinct elements x and y in X. If |X| ≥ |Y | for any other set of pairwise non-commuting elements Y in G, then X is said to be a maximal subset of pairwise non-commuting elements. In this paper we determine the cardinality of a maximal subset of pairwise non-commuting elements in any non-abelian...

Almost-commuting matrices are almost jointly diagonalizable

We study the relation between approximate joint diagonalization of self-adjoint matrices and the norm of their commutator, and show that almost commuting self-adjoint matrices are almost jointly diagonalizable by a unitary matrix.

Quotients of countably based spaces are not closed under sobrification

In this note we show that quotients of countably based spaces (qcb spaces) and topological predomains as introduced by M. Schröder and A. Simpson are not closed under sobrification. As a consequence replete topological predomains need not be sober, i.e. in general repletion is not given by sobrification. Our counterexample also shows that a certain tentative “equalizer construction” of repletio...