Remarks on CCR and CAR flows over closed convex cones

For a closed convex cone [Formula: see text] in [Formula: see text] which is spanning and pointed, i.e. [Formula: see text] and [Formula: see text] we consider a family of [Formula: see text]-semigroups over [Formula: see text] consisting of a certain family of CCR flows and CAR flows over [Formula: see text] and classify them up to the cocycle conjugacy.

Convex Fuzzy Cones of Hypervector Spaces

The aim of this paper is to study some main properties of convex fuzzy cones of hypervector spaces. In this regard, a characterization of fuzzy subhyperspaces is presented based on ordinary subhyperspaces. Also the convex cone spanned by non-empty subsets of real hypervector spaces is obtained. Moreover, by introducing the notion of fuzzy cone, the smallest fuzzy subhyperspace of V containing µ and the largest fuzzy subhyperspace of V contained in µ is achieved, for a convex fuzzy cone µ of real hypervector space V .

SPECIAL VINBERG CONES

The paper is devoted to the generalization of the Vinberg theory of homogeneous convex cones. Such a cone is described as the set of “positive definite matrices” in the Vinberg commutative algebra ℋn of Hermitian T-matrices. These algebras are a generalization of Euclidean Jordan algebras and consist of n × n matrices A = (aij), where aii ∈ ℝ, the entry aij for i < j belongs to some Euclidean vector space (Vij ; 𝔤) and $$ {a}_{ji}={a}_{ij}^{\ast }=\mathfrak{g}\left({a}_{ij},\cdot \right)\in {V}_{ij}^{\ast } $$ a ji = a ij ∗ = g a ij ⋅ ∈ V ij ∗ belongs to the dual space $$ {V}_{ij}^{\ast }. $$ V ij ∗ . The multiplication of T-Hermitian matrices is defined by a system of “isometric” bilinear maps Vij × Vjk → Vij ; i < j < k, such that |aij ⋅ ajk| = |aij| ⋅ |aik|, alm ∈ Vlm. For n = 2, the Hermitian T-algebra ℋn= ℋ2 (V) is determined by a Euclidean vector space V and is isomorphic to a Euclidean Jordan algebra called the spin factor algebra and the associated homogeneous convex cone is the Lorentz cone of timelike future directed vectors in the Minkowski vector space ℝ1,1⊕ V . A special Vinberg Hermitian T-algebra is a rank 3 matrix algebra ℋ3(V; S) associated to a Clifford Cl(V )-module S together with an “admissible” Euclidean metric 𝔤S.We generalize the construction of rank 2 Vinberg algebras ℋ2(V ) and special Vinberg algebras ℋ3(V; S) to the pseudo-Euclidean case, when V is a pseudo-Euclidean vector space and S = S0 ⊕ S1 is a ℤ2-graded Clifford Cl(V )-module with an admissible pseudo-Euclidean metric. The associated cone 𝒱 is a homogeneous, but not convex cone in ℋm; m = 2; 3. We calculate the characteristic function of Koszul-Vinberg for this cone and write down the associated cubic polynomial. We extend Baez’ quantum-mechanical interpretation of the Vinberg cone 𝒱2 ⊂ ℋ2(V ) to the special rank 3 case.

Bad projections of the PSD cone

The image of the cone of positive semidefinite matrices under a linear map is a convex cone. Pataki characterized the set of linear maps for which that image is not closed. The Zariski closure of this set is a hypersurface in the Grassmannian. Its components are the coisotropic hypersurfaces of symmetric determinantal varieties. We develop the convex algebraic geometry of such bad projections, with focus on explicit computations.

THE METRIC PROJECTIONS ONTO CLOSED CONVEX CONES IN A HILBERT SPACE

We study the metric projection onto the closed convex cone in a real Hilbert space $\mathscr {H}$ generated by a sequence $\mathcal {V} = \{v_n\}_{n=0}^\infty $ . The first main result of this article provides a sufficient condition under which the closed convex cone generated by $\mathcal {V}$ coincides with the following set: $$ \begin{align*} \mathcal{C}[[\mathcal{V}]]: = \bigg\{\sum_{n=0}^\infty a_n v_n\Big|a_n\geq 0,\text{ the series }\sum_{n=0}^\infty a_n v_n\text{ converges in } \mathscr{H}\bigg\}. \end{align*} $$ Then, by adapting classical results on general convex cones, we give a useful description of the metric projection onto $\mathcal {C}[[\mathcal {V}]]$ . As an application, we obtain the best approximations of many concrete functions in $L^2([-1,1])$ by polynomials with nonnegative coefficients.

CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS

Let ($$ \mathfrak{g} $$ g , τ) be a real simple symmetric Lie algebra and let W ⊂ $$ \mathfrak{g} $$ g be an invariant closed convex cone which is pointed and generating with τ(W) = −W. For elements h ∈ $$ \mathfrak{g} $$ g with τ(h) = h, we classify the Lie algebras $$ \mathfrak{g} $$ g (W, τ, h) which are generated by the closed convex cones $$ {C}_{\pm}\left(W,\tau, h\right):= \left(\pm W\right)\cap {\mathfrak{g}}_{\pm 1}^{-\tau }(h) $$ C ± W τ h ≔ ± W ∩ g ± 1 − τ h , where $$ {\mathfrak{g}}_{\pm 1}^{-\tau }(h):= \left\{x\in \mathfrak{g}:\tau (x)=-x\left[h,x\right]=\pm x\right\} $$ g ± 1 − τ h ≔ x ∈ g : τ x = − x h x = ± x . These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if $$ \mathfrak{g} $$ g (W, τ, h) is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms τ of $$ \mathfrak{g} $$ g with τ(W) = −W a list of possible subalgebras $$ \mathfrak{g} $$ g (W, τ, h) up to isomorphy.