Narrow Results

Let $n>1$. Let ${\{U_{ij}\}}_{1\le i<j\le n}$ be $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ commuting unitaries on some Hilbert space ${\mathscr{H}}$, and suppose $U_{ji}:=U_{ij}^{\ast }$, $1\le i<j\le n$. An $n$-tuple of isometries $V=(V_1,\dots ,V_n)$ on ${\mathscr{H}}$ is called ${{𝒰}}_n$-twisted isometry with respect to ${\{U_{ij}\}}_{i<j}$ (or simply ${{𝒰}}_n$-twisted isometry if ${\{U_{ij}\}}_{i<j}$ is clear from the context) if $V_i$’s are in the commutator ${\{U_{st}:s\ne t\}}^{\prime }$, and $V_i^{\ast }{V}_j=U_{ij}^{\ast }{V}_j{V}_i^{\ast }$, $i\ne j$

We prove that each ${{𝒰}}_n$-twisted isometry admits a von Neumann–Wold type orthogonal decomposition, and prove that the universal $C^{\ast }$-algebra generated by ${{𝒰}}_n$-twisted isometries is nuclear. We exhibit concrete analytic models of ${{𝒰}}_n$-twisted isometries, and establish connections between unitary equivalence classes of the irreducible representations of the $C^{\ast }$-algebras generated by ${{𝒰}}_n$-twisted isometries and the unitary equivalence classes of the nonzero irreducible representations of twisted noncommutative tori. Our motivation of ${{𝒰}}_n$-twisted isometries stems from the classical rotation $C^{\ast }$-algebras and Heisenberg group $C^{\ast }$-algebras.

The purpose of this paper is to study semigroups of *-endomorphisms of type I factors, with parameters in a countable dense subgroup of the real line. Most of our results are motivated by the theory of semigroups of isometries. It is shown that every semigroup of *-endomorphisms can be extended to a group of *-automorphisms in a minimal way. In certain situations, it is shown that a semigroup of *-endomorphisms can be decomposed into three simpler semigroups. Shifts are defined and classified up to outer conjugacy.

We study the invariance properties of five-dimensional metrics and their corresponding geodesic equations of motion. In this context a number of five-dimensional models of the Einstein–Gauss–Bonnet (EGB) theory leading to black holes, wormholes and spacetime horns arising in a variety of situations are discussed in the context of variational symmetries of which each vector field, via Noether’s theorem (NT), provides a nontrivial conservation law. In particular, it is shown that algebraic structure of isometries and the variational conservation laws of the five-dimensional Einstein–Bonnet metric extend consistently from the well-known Minkowski, de-Sitter and Schwarzschild four-dimensional spacetimes to the considered five-dimensional ones. In the equivalent five-dimensional case, the maximal algebra of kvs is fifteen with eight additional Noether symmetries. Also, whereas the constant curvature five-dimensional case leads to fifteen kvs and one additional Noether symmetry and seven plus one in the minimal case, a number of metrics of the EGB theory in five dimensions give rise to algebras isomorphic a seven-dimensional algebra of kvs and a single additional Noether symmetry.

In this paper we study some properties of the polytope of belief functions on a finite referential. These properties can be used in the problem of identification of a belief function from sample data. More concretely, we study the set of isometries, the set of invariant measures and the adjacency structure. From these results, we prove that the polytope of belief functions is not an order polytope if the referential has more than two elements. Similar results are obtained for plausibility functions.

We provide geometrical conditions on the manifold for the existence of the Liao's factorization of stochastic flows [10]. If M is simply connected and has constant curvature, then this decomposition holds for any stochastic flow, conversely, if every flow on M has this decomposition, then M has constant curvature. Under certain conditions, it is possible to go further on the factorization: φ_t = ξ_t°Ψ_t° Θ_t, where ξ_t and Ψ_t have the same properties of Liao's decomposition and (ξ_t°Ψ_t) are affine transformations on M. We study the asymptotic behaviour of the isometric component ξ_t via rotation matrix, providing a Furstenberg–Khasminskii formula for this skew-symmetric matrix.

We study the class of higher-dimensional Kundt metrics admitting a covariantly constant null vector, known as CCNV spacetimes. We pay particular attention to those CCNV spacetimes with constant (polynomial) curvature invariants (CSI). We investigate the existence of an additional isometry in CCNV spacetimes, by studying the Killing equations for the general form of the CCNV metric. In particular, we list all CCNV spacetimes allowing an additional non-spacelike isometry for all values of the light-cone coordinate v, which are of interest due to the invariance of the metric under a translation in v. As an application we use our results to find all CSICCNV spacetimes with an additional isometry as well as the subset of these spacetimes in which the isometry is non-spacelike for all values v.

In this paper we extend two classical results concerning the isometries of strictly convex Hilbert geometries, and the characterisation of the isometry groups of Hilbert geometries on finite dimensional simplices, to infinite dimensions. The proofs rely on a mix of geometric and functional analytic methods.

Lie point symmetries of the geodesic equations of the Gödel’s metric are found. These form a ten dimensional Lie algebra. The Lie algebra contains a maximal seven-dimensional solvable sub-algebra. It also contains five dimensional subalgebra of isometries of the metric. The isometries are used to reduce the order of the geodesic system by one. The time-like trajectories of the Gödel’s metric are then derived and their graphs in the (r, ϕ) plane are displayed showing some interesting features of the dynamics in this universe.