Recent zbMATH articles in MSC 46Ahttps://zbmath.org/atom/cc/46A2024-11-01T15:51:55.949586ZWerkzeugExtremal structure of cones of positive homogeneous polynomialshttps://zbmath.org/1544.460052024-11-01T15:51:55.949586Z"Kusraeva, Zalina A."https://zbmath.org/authors/?q=ai:kusraeva.zalina-anatolevnaThe author extends the main result of \textit{A. W. Wickstead} [Q. J. Math., Oxf. II. Ser. 32, 239--253 (1981; Zbl 0431.47024)] to orthogonally additive homogeneous polynomials between vector lattices.
The author concludes that the extremal structure of the cone of positive orthogonally additive homogeneous polynomials is similar to that of the cone of positive linear operators.
Reviewer: S. S. Kutateladze (Novosibirsk)Orthogonal additivity of monomials in positive homogeneous polynomialshttps://zbmath.org/1544.460062024-11-01T15:51:55.949586Z"Kusraeva, Zalina A."https://zbmath.org/authors/?q=ai:kusraeva.zalina-anatolevna"Tamaeva, Victoria A."https://zbmath.org/authors/?q=ai:tamaeva.victoria-aGiven a few homogeneous polynomials, a monomial is a pointwise product of their powers. The authors' previous result in [\textit{Z. A. Kusraeva} and \textit{V. A. Tamaeva}, Math. Notes 114, No. 6, 1297--1305 (2023; Zbl 07820500); translation from Mat. Zametki 114, No. 6, 863--872 (2023)] reads that a finite set of positive linear operators from an Archimedean vector lattice into a unital Archimedean \(f\)-algebra is disjointness preserving iff the corresponding monomial is orthogonally additive. The authors prove that a similar result is valid when the target space is a uniformly complete vector lattice.
Reviewer: S. S. Kutateladze (Novosibirsk)Barycentric decompositions in the space of weak expectationshttps://zbmath.org/1544.460072024-11-01T15:51:55.949586Z"Bhattacharya, Angshuman"https://zbmath.org/authors/?q=ai:bhattacharya.angshuman"Kulkarni, Chaitanya J."https://zbmath.org/authors/?q=ai:kulkarni.chaitanya-jIn this interesting application of the Choquet-Bishop-de Leeuw theorem the authors consider a unital \(C^*\)-algebra \(A\) and a fixed nondegenerate representation \(\pi\colon A\to B(H)\) of \(A\) on a Hilbert space \(H\). A unital completely positive map \(\theta\colon B(H)\to \pi(A)^{''}\) is called a weak expectation of \(\pi\) if \(\theta(\pi(a))=\pi(a)\) for all \(a\in A\). If \(\operatorname{WE}(\pi)\) denotes the set of all weak expectations of \(\pi\), then it is a closed convex subset of the set of all completely positive maps from \(B(H)\) to \(\pi(A)^{''}\) with norm \({\le 1}\). The latter set is compact in the bounded weak topology and hence also the set \(\operatorname{WE}(\pi)\) is a compact convex set in the locally convex space of all completely bounded maps from \(B(H)\) to \(\pi(A)^{''}\). It is not necessary in general, that \(\operatorname{WE}(\pi)\neq\emptyset\), but in many cases this set is nonempty.
The above mentioned Choquet-Bishop-de Leeuw theorem asserts that, given a compact convex set \(X\) in locally convex space and \(x_0\in X\), there exists a Baire measure \(\mu_{x_0}\) pseudo-supported on the extreme points of \(X\) such that for any continuous affine function \(f\colon X\to \mathbb{C}\) one has \(f(x_0)=\int f \,d\mu_{x_0}\). In order to apply this theorem to the set \(\operatorname{WE}(\pi)\), the paper carefully identifies, using operator theoretic techniques, the set of extreme points of \(\operatorname{WE}(\pi)\) in the main result.
Reviewer: Jiří Spurný (Praha)On extreme points and representer theorems for the Lipschitz unit ball on finite metric spaceshttps://zbmath.org/1544.460112024-11-01T15:51:55.949586Z"Bredies, Kristian"https://zbmath.org/authors/?q=ai:bredies.kristian"Rodriguez, Jonathan Chirinos"https://zbmath.org/authors/?q=ai:rodriguez.jonathan-chirinos"Naldi, Emanuele"https://zbmath.org/authors/?q=ai:naldi.emanueleLet \(X = \{x_0,\ldots,x_n\}\) be a finite metric space, \(Y\) a strictly convex Banach space and consider the convex set
\[
\operatorname{Lip}_0^1(X,Y):= \{f:X\to Y\colon f\text{ is \(1\)-Lipschitz and }f(x_0)=0\}.
\]
The authors identify extreme points of \(\operatorname{Lip}_0^1(X,Y)\) and prove that any function from \(\operatorname{Lip}_0^1(X,Y)\) is a convex combination of \(n+1\) extreme points.
Reviewer: Marek Cúth (Praha)A new notion of inner product in a subspace of \(n\)-normed spaceshttps://zbmath.org/1544.460222024-11-01T15:51:55.949586Z"Nur, Muh"https://zbmath.org/authors/?q=ai:nur.muh"Idris, Mochammad"https://zbmath.org/authors/?q=ai:idris.mochammadSummary: Given an \(n\)-normed space \(X\) for \(n \geq 2\), we investigate the completeness of \(Y\) (as a subspace of \(X\)) with respect to a new norm that correspond to this new inner product on \(Y\). Next, we introduce the angle on a subspace \(Y\) of \(n\)-normed space \(X\).Remarks on weak compactness criteria in variable exponent Lebesgue spaceshttps://zbmath.org/1544.460262024-11-01T15:51:55.949586Z"Hernández, Francisco L."https://zbmath.org/authors/?q=ai:hernandez.francisco-l"Ruiz, César"https://zbmath.org/authors/?q=ai:ruiz.cesar"Sanchiz, Mauro"https://zbmath.org/authors/?q=ai:sanchiz.mauroLet \((\Omega,\mu)\) be a \(\sigma\)-finite measurable space. Given a \(\mu\)-measurable function \(p(\cdot):\Omega\to[1,\infty)\), the variable Lebesgue space \(L^{p(\cdot)}(\Omega)\) is defined as the set of all measurable functions \(f:\Omega\to\mathbb{R}\) such that \(\varrho_{p(\cdot)}(f/\lambda)<\infty\) for some \(\lambda=\lambda(f)>0\), where \(\varrho_{p(\cdot)}(f):=\int_\Omega |f(t)|^{p(t)} \, d\mu(t)\). The norm in \(L^{p(\cdot)}(\Omega)\) is defined by \(\|f\|_{p(\cdot)}:=\inf\{\lambda>0:\varrho_{p(\cdot)}(f/\lambda)\le 1\}\). The main result of the paper says that if \(\operatornamewithlimits{ess\,inf}_{t\in\Omega}p(t)<\infty\) and \(\mu(p^{-1}(\{1\}))=0\), then a subset \(S\subset L^{p(\cdot)}(\Omega)\) is relatively weakly compact if and only if \[
\lim_{\lambda\to 0}\sup_{f\in S}\frac{1}{\lambda} \int_\Omega |\lambda f(t)|^{p(t)}\, d\mu(t)=0.
\]
Reviewer: Oleksiy Karlovych (Lisboa)Freedman's theorem for unitarily invariant states on the CCR algebrahttps://zbmath.org/1544.460492024-11-01T15:51:55.949586Z"Crismale, Vitonofrio"https://zbmath.org/authors/?q=ai:crismale.vitonofrio"Del Vecchio, Simone"https://zbmath.org/authors/?q=ai:del-vecchio.simone"Monni, Tommaso"https://zbmath.org/authors/?q=ai:monni.tommaso"Rossi, Stefano"https://zbmath.org/authors/?q=ai:rossi.stefanoSummary: The set of states on \(\text{CCR}(\mathcal{H})\), the CCR algebra of a separable Hilbert space \(\mathcal{H}\), is here looked at as a natural object to obtain a non-commutative version of Freedman's theorem for unitarily invariant stochastic processes. In this regard, we provide a complete description of the compact convex set of states of \(\text{CCR}(\mathcal{H})\) that are invariant under the action of all automorphisms induced in second quantization by unitaries of \(\mathcal{H}\). We prove that this set is a Bauer simplex, whose extreme states are either the canonical trace of the CCR algebra or Gaussian states with variance at least 1.\((C,B)\)-resolvents of closed linear operatorshttps://zbmath.org/1544.470082024-11-01T15:51:55.949586Z"Chaouchi, Belkacem"https://zbmath.org/authors/?q=ai:chaouchi.belkacem"Kostić, Marko"https://zbmath.org/authors/?q=ai:kostic.markoSummary: In this note, we analyze \((C,B)\)-resolvents of closed linear operators in sequentially complete locally convex spaces. We provide a simple application in the qualitative analysis of solutions of abstract degenerate Volterra integro-differential equations.Partially defined operators on locally Hilbert spaceshttps://zbmath.org/1544.470302024-11-01T15:51:55.949586Z"Cismas, Emanuel-Ciprian"https://zbmath.org/authors/?q=ai:cismas.emanuel-ciprianSummary: We investigate partially defined operators on inductive limits of Hilbert spaces, in order to introduce an operator theory for a class of linear operators, outside the Hilbert setting. Some spaces of functions with compact support can fall into the locally Hilbert class and a specific approach is needed for the linear operators compatible with the locally Hilbert structure.Channel divergences and complexity in algebraic QFThttps://zbmath.org/1544.810892024-11-01T15:51:55.949586Z"Hollands, Stefan"https://zbmath.org/authors/?q=ai:hollands.stefan"Ranallo, Alessio"https://zbmath.org/authors/?q=ai:ranallo.alessioSummary: We consider a notion of divergence between quantum channels in relativistic continuum quantum field theory (QFT) that is derived from the Belavkin-Staszewski relative entropy and the concept of bimodules for general von Neumann algebras. Key concepts of the divergence that we shall prove based on a new variational formulation of that relative entropy are the subadditivity under composition and additivity under the tensor product between channels. Based on these properties, we propose to use the channel divergence relative to the trivial (identity-) channel as a novel measure of complexity. Using the properties of our channel divergence, we prove in the prerequisite generality necessary for the algebras in QFT that the corresponding complexity has several reasonable properties: (i) the complexity of a composite channel is not larger than the sum of its parts, (ii) it is additive for channels localized in spacelike separated regions, (iii) it is convex, (iv) for an \(N\)-ary measurement channel it is \(\log N\), (v) for a conditional expectation associated with an inclusion of QFTs with finite Jones index it is given by \(\log (\text{Jones Index})\).