A Variant of D’alembert’s Matrix Functional Equation

The aim of this paper is to characterize the solutions Φ : G → M2(ℂ) of the following matrix functional equations {{\Phi \left( {xy} \right) + \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, and {{\Phi \left( {xy} \right) - \Phi \left( {\sigma \left( y \right)x} \right)} \over 2} = \Phi \left( x \right)\Phi \left( y \right),\,\,\,\,\,\,x,y, \in G, where G is a group that need not be abelian, and σ : G → G is an involutive automorphism of G. Our considerations are inspired by the papers [13, 14] in which the continuous solutions of the first equation on abelian topological groups were determined.

Analytic Extensions of Representations of *-Subsemigroups Without Polar Decomposition

Let $(G,\tau )$ be a finite-dimensional Lie group with an involutive automorphism $\tau $ of $G$ and let ${{\mathfrak{g}}} = {{\mathfrak{h}}} \oplus{{\mathfrak{q}}}$ be its corresponding Lie algebra decomposition. We show that every nondegenerate strongly continuous representation on a complex Hilbert space ${\mathcal{H}}$ of an open $^\ast $-subsemigroup $S \subset G$, where $s^{\ast } = \tau (s)^{-1}$, has an analytic extension to a strongly continuous unitary representation of the 1-connected Lie group $G_1^c$ with Lie algebra $[{{\mathfrak{q}}},{{\mathfrak{q}}}] \oplus i{{\mathfrak{q}}}$. We further examine the minimal conditions under which an analytic extension to the 1-connected Lie group $G^c$ with Lie algebra ${{\mathfrak{h}}} \oplus i{{\mathfrak{q}}}$ exists. This result generalizes the Lüscher–Mack theorem and the extensions of the Lüscher–Mack theorem for $^\ast $-subsemigroups satisfying $S = S(G^\tau )_0$ by Merigon, Neeb, and Ólafsson. Finally, we prove that nondegenerate strongly continuous representations of certain $^\ast $-subsemigroups $S$ can even be extended to representations of a generalized version of an Olshanski semigroup.

Solutions and Stability of Generalized Kannappan’s and Van Vleck’s Functional Equations

We study the solutions of the integral Kannappan’s and Van Vleck’s functional equations ∫Sf(xyt)dµ(t)+∫Sf(xσ(y)t)dµ(t)= 2f(x)f(y), x,y ∈ S; ∫Sf(xσ(y)t)dµ(t)-∫Sf(xyt)dµ(t)= 2f(x)f(y), x,y ∈ S; where S is a semigroup, σ is an involutive automorphism of S and µ is a linear combination of Dirac measures ( ᵟ zi)I ∈ I, such that for all i ∈ I, ziis in the center of S. We show that the solutions of these equations are closely related to the solutions of the d’Alembert’s classic functional equation with an involutive automorphism. Furthermore, we obtain the superstability theorems for these functional equations in the general case, where σ is an involutive morphism.

A Variant of Jensen’s Functional Equation on Semigroups

We determine the solutions f : S → H of the following functional equationf(xy) + f(σ(y)x) = 2f(x); x; y ∈ S;and the solutions ffwhere S is a semigroup, M is a monoid, H is an abelian group 2-torsion free, and σ is an involutive automorphism.

INVOLUTIVE GRADINGS AND KERNELS IN JBW*-TRIPLES

Involutive automorphism, or bijective triple homorphisms of order two, on a JBW *-triple are in a one-to-one correspondence with involutive gradings and bicontractive projections. Such mappings are always isometric and weak*-continuous. This paper analyses the algebraic kernels of the 1-eigenspace B+ and -1-eigenspace B- of involutive automorphisms. It is shown that B+ and B- give rise to several decompositions of A and, remarkably, Ker (B+) and Ker (B-) are Peirce inner ideals.