Fourier–Mukai transforms commuting with Frobenius over algebraic spaces

The paper contains a survey of known results on the structure of J-symmetric operator algebras in Pontryagin and Krein spaces, as well as on representations of groups and *-algebras in these spaces.

For a complete discrete valuation field K , we show that one may always glue a separated formal algebraic space $\mathfrak {X}$ over $\mathcal {O}_K$ to a separated algebraic space U over K along an open immersion of rigid spaces $j\colon \mathfrak {X}^{\mathrm {rig}}\to U^{\mathrm {an}}$ , producing a separated algebraic space X over $\mathcal {O}_K$ . This process gives rise to an equivalence between such ‘gluing triples’ $(U,\mathfrak {X},j)$ and separated algebraic spaces X over $\mathcal {O}_K$ , which one might interpret as a version of the Beauville–Laszlo theorem for algebraic spaces rather than coherent sheaves. Moreover, an analogous equivalence exists over any excellent base. Examples due to Matsumoto imply that the result of such a gluing might be a genuine algebraic space (not a scheme) even if U and the special fibre of $\mathfrak {X}$ are projective. The proof is a combination of the Nagata compactification theorem for algebraic spaces and of Artin’s contraction theorem. We give multiple examples and applications of this idea.

The aim of this research paper is to introduce the concept of bi-(Gamma)-algebra space (bi-gamma algebra space). The concept of bi-(mu)-measurable set in a bi-(Gamma)-algebra space is defined. With this concept, some properties of bi-(Gamma)-algebra space are proved. We then define various separation axioms for bi-(Gamma)-algebra space such as (M_{0},M_{1}, M_{2}, M_{3},) and (M_{4}); then the relationships between them are studied. In addition, the concept of measurable function between two bi-measurable spaces is introduced and some results are discussed.

In [We, Ch. 1], Weil defines a process of “adelization” of algebraic varieties over global fields. There is an alternative procedure, due to Grothendieck, using adelic points. One aim of this (largely) expository note is to prove that for schemes of finite type over global fields (i.e., without affineness hypotheses), and also for separated algebraic spaces of finite type over such fields, Weil’s adelization process naturally coincides (as a set) with the set of adelic points in the sense of Grothendieck (and that in the affine case the topologies defined by these two viewpoints coincide; Grothendieck’s approach doesn’t provide a topology beyond the affine case). The other aim is to prove in general that topologies obtained by Weil’s method satisfy good functorial properties, including expected behavior with respect to finite flat Weil restriction of scalars. The affine case suffices for most applications, but the non-affine case is useful (e.g., adelic points of G/P for connected reductive groups G and parabolic subgroups P ). We also discuss topologizing X(k) for possibly non-separated algebraic spaces X over locally compact fields k; motivation for this is given in Example 5.5. Although everything we prove (except perhaps for the case of algebraic spaces) is “well known” folklore, and [Oes, I, §3] provides an excellent summary in the affine case, some aspects are not so easy to extract from the available literature. Moreover, (i) some references that discuss the matter in the non-affine case have errors in the description of the topology on adelic points, and (ii) much of what we prove is needed in my paper [Con], or in arithmetic arguments in [CGP]. In effect, these notes can be viewed as an expanded version of [Oes, I, §3], and I hope they will provide a useful general reference on the topic of adelic points of algebro-geometric objects (varieties, schemes, algebraic spaces) over global fields. In §2 we carry out Grothendieck’s method in the affine case over any topological ring R, characterizing the topology on sets of R-points by means of several axioms. The generalization to arbitrary schemes of finite type via a method of Weil is developed in §3. We explore properties of these topologies in §4, especially for adelic points and behavior with respect to Weil restriction of scalars. Finally, in §5 everything is generalized to the case of algebraic spaces. Notation. We write AF to denote the adele ring of a global field F , and likewise A n F denotes Euclidean n-space over AF . There is no risk of confusion with the common use of such notation to denote affine n-space over SpecF since we avoid ever using this latter meaning for the notation.

In this paper, the author first introduces the concept of generalized algebraic cone metric spaces and some elementary results concerning generalized algebraic cone metric spaces. Next, by using these results, some new fixed point theorems on generalized (complete) algebraic cone metric spaces are proved and an example is given. As a consequence, the main results generalize the corresponding results in complete algebraic cone metric spaces and generalized complete metric spaces.

Granular computing (GrC) is an emerging computing paradigm, and it is an umbrella term exploring multilevel granularity. we present a generic abstract mathematical model of the granular system. Supposing the inter-granule structure as an algebra, we propose the algebraic quotient space model. In this model, the granulation is based on a congruence relation and all the congruence relations on a granular system form a complete semi-order lattice, which is the theoretical basis for transformation, composition and decomposition among different granularities. The different granulation rules between the topological quotient space model and the algebraic quotient space model lead to the dissimilarity while composing granularities. A real-world case study is presented that demonstrates how the algebraic quotient space model works in the network transmission by error-correcting code. These work shows that the granular system model and the algebraic quotient space model are powerful conceptual modeling and functional specification methodologies for GrC.

The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.

In this paper, we study the moduli space of 4-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. Because the space of 4-dimensional algebras is large, we only classify the non-nilpotent algebras in this paper.

Normal state spaces of Jordan and von Neumann algebras

This paper gives an introduction to the C ∗-algebra of a one-sided shift space. Focus will be given to the fundamental structure of the C ∗-algebra of a one-sided shift space, but some of the most important results about C ∗-algebras associated to shift spaces will also be presented.

The representation theory of operator algebras in Hilbert spaces lies at the intersection of functional analysis, abstract algebra, and quantum physics. Originating from the development of C*-algebras and von Neumann algebras in the early twentieth century, this field provides a rigorous framework for studying bounded and unbounded operators on Hilbert spaces. Representations of operator algebras are not only central to pure mathematics but also essential for modeling symmetries and observables in quantum mechanics, statistical physics, and noncommutative geometry. This study presents a conceptual study of the representation theory of operator algebras in Hilbert spaces, emphasizing structural insights rather than heavy technical formalism. We review the historical development of C*-algebra and von Neumann algebra representations, highlighting the Gelfand–Naimark–Segal (GNS) construction, cyclic and factor representations, and the role of commutants in von Neumann’s bicommutant theorem. The interplay between representation theory and physical models is discussed, with applications to quantum field theory, spectral theory, and ergodic analysis. Further, we examine classification schemes such as Type I, II, and III von Neumann algebras and their impact on understanding quantum states. Recent advances in noncommutative geometry and K-theory are also briefly surveyed, showing how representation theory continues to provide bridges between mathematics and physics. The study concludes by outlining potential directions, including applications to quantum information theory and topological phases of matter. By focusing on conceptual frameworks and examples, this study aims to provide both clarity and accessibility while maintaining mathematical rigor. The result is a balanced view of representation theory as a central pillar in the analysis of operator algebras in Hilbert spaces.

In this chapter, we study the vector algebra of 3-dimensional space. The term “algebra” is meant here in its mathematical sense, so that in addition to the usual vector-space manipulations, an associative multiplication of vectors is required. Relatively simple considerations lead us to what is called the geometric algebra(or Clifford algebra)of 3dimensional space, also known as the Pauli algebra. The standard matrix representation of this algebra replaces basis vectors by Pauli spin matrices (and hence the name “Pauli algebra”), but specific representations encumber the mathematics with unnecessary baggage; it is usually simpler to work directly in the algebra in component-free notation without reference to any matrices.

Recent developments in Banach space theory provided unexpected examples of unital Banach algebras that are isomorphic to Calkin algebras of Banach spaces, however no example of a unital Banach algebra that cannot be realised as a Calkin algebra has been found so far. This naturally led to the question of possible limitations of such assignments. In the present note we provide examples of unital Banach algebras meeting the necessary density condition for being the Calkin algebra of a separable Banach space that are not isomorphic to Calkin algebras of such spaces, nonetheless. The examples may be found of the form C ( X ) C(X) for a compact space X X , ℓ 1 ( G ) \ell _1(G) for some torsion-free Abelian group, and a simple, unital AF C ∗ C^* -algebra. Extensions to higher densities are also presented.

A realisation of coordinates and momenta in the algebraic space to describe vibrational excitations of ν-equivalent oscillators is obtained. The connection between algebraic and configuration spaces is carried out using the approach recently proposed [Mol. Phys. (2017), doi:10.1080/00268976.2017.1358829]. The realisation consists in an expansion in terms of the dynamical algebra generators with coefficients determined through a minimisation procedure and given in terms of matrix elements defined in configuration space. Two realisations are presented: one through an isomorphism with a harmonic oscillators basis and the second one using a mapping to Morse oscillators. In the case of the harmonic oscillator mapping, two chains associated with coordinate and momentum representation have been identified. Our approach allows us to establish the algebraic representation of any interaction and, consequently, of the Hamiltonian associated with ν-interacting oscillators, providing a formal approach to estimate the potential energy surface.

The aim of this note is to discuss the Weil restriction of schemes and algebraic spaces, highlighting pathological phenomena that appear in the theory and are not widely known. It is shown that the Weil restriction of a locally finite algebraic space along a finite flat morphism is an algebraic space.