Defines complex manifolds on commutative Banach algebras and studies continuous families of compact complex manifolds.
problem Defines complex manifolds on commutative Banach algebras.
method Introduces a new manifold structure on continuous cross sections of complex vector bundles.
result Finite-dimensional C(X)-manifolds can be constructed from continuous families of compact complex manifolds.
Defines holomorphic differential forms on complex manifolds over commutative Banach algebras.
problem Defines holomorphic differential forms on complex manifolds over commutative Banach algebras.
method Defines A-holomorphic vector bundles and differential forms on A-manifolds. result Defines cohomology groups as A-modules, providing a new perspective on cohomology. Study the Lax equation in infinite-dimensional Lie algebras and Lie groups.
problem Investigate the Lax equation in infinite-dimensional Lie algebras and Lie groups.
method Derived integral expansions and generalized Baker-Campbell-Hausdorff formula for Lie groups.
result Explicit representation of product integral in terms of exponential map.
We prove that the quasi-homogenous symbols on the projective space Pn(C) yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball Bn. These algebras are Banach but not C∗. We prove the existen…
On the predual of a von Neumann algebra, we define a differentiable manifold structure and affine connections by embeddings into non-commutative L_p-spaces. Using the geometry of uniformly convex Banach spaces and duality of the L_p and L_q spaces for 1/p+1/q=1, we show that we can introduce the α-divergence, for αin (…
Representations of C∗-algebras are realized on section spaces of holomorphic homogeneous vector bundles. The corresponding section spaces are investigated by means of a new notion of reproducing kernel, suitable for dealing with involutive diffeomorphisms defined on the base spaces of the bundles. Applications of th…
Researchers develop neural networks for approximating functions in Banach spaces.
problem Approximating Banach space valued continuous functions.
method Quasi-interpolation Banach space valued neural network operators using algebraic sigmoid functions.
result Jackson type inequalities for function approximation.
Explicit BCH series radii found for special Banach-Malcev shift algebras.
problem Finding convergence radii for BCH series in specific algebraic structures.
method Established explicit convergence radii using continuity estimates and algebraic properties.
result Explicit formula for convergence radii derived and validated for various shift algebras.
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
Study infinitesimal properties of idempotent sets in Banach algebras.
problem Investigate infinitesimal aspects of idempotent sets in Banach algebras.
method Introduce and study Stiefel bundles on flag manifolds, using connections on infinite-dimensional bundles.
result Developed new connections on infinite-dimensional bundles.
We investigate the Banach Lie groupoids and inverse semigroups naturally associated to W*-algebras. We also present statements describing relationship between these groupoids and the Banach Poisson geometry which follows in the canonical way from the W*-algebra structure.
The study integrates Banach manifolds into H-manifolds, integrating Lie algebras into H-groups.
problem Integrating Banach manifolds and Lie algebras into H-manifolds and H-groups.
method Investigating quotients of Banach manifolds with free actions of pseudogroups of local diffeomorphisms.
result Every real Banach-Lie algebra can be integrated to an H-group.
Generalizes Gelfand's spectrum to monogenic spinor fields on compact Riemannian manifolds.
problem Extend Gelfand's spectrum concept to non-commutative spaces of spinor fields.
method Use Clifford algebras and monogenic spinor fields, proving essential lemmas and a Stone-Weierstrass theorem.
result Spectrum of monogenic spinor fields on compact Riemannian manifolds is homeomorphic to the manifold itself.
Study Banach-Lie groupoids, revealing new insights.
problem Understanding infinite-dimensional manifolds.
method Adapt classical Lie groupoid results to Banach-Lie groupoids.
result Locally transitive Banach-Lie groupoids offer new perspectives.
We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, …
Study extends JB-algebra structure group results to infinite dimensions.
problem Extend results for real Jordan algebras to infinite dimensional JB-algebras.
method Prove structure groups, cone preserving groups, and automorphism groups are embedded Banach-Lie groups; describe components via cones, isotopes, and central projections; apply to special JB-algebra of self-adjoint operators.
result Full description of components of structure group and automorphism group, including their Banach-Lie algebras and connected components.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing ρ-commutative manifolds, Q-manifolds, and modular classes. result Generalized modular classes for non-commutative spaces.
New spectral theory for non-associative algebras with applications to Moufang dynamics.
problem Spectral theory of non-associative algebras and their applications.
method Introducing almost periodic Banach--Malcev algebras and analyzing their spectral properties.
result Spectral characterization and continuous functional calculus for almost periodic derivations.
In this note we give sufficient conditions to ensure that the weak Finsler structure of a complete Ck Finsler manifold M is determined by the normed algebra Cbk(M) of all real-valued, bounded and Ck smooth functions with bounded derivative defined on M. As a consequence, we obtain: (i) the Finsler structu…
Study Nijenhuis operators on homogeneous spaces related to C*-algebras.
problem Characterize Nijenhuis operators on homogeneous spaces of C*-algebras.
method Analyze vector bundle maps induced by admissible operators on C*-algebras.
result Identify conditions for vector bundle maps to be Nijenhuis operators.
Study examines linear Poisson structures tied to W*-algebras.
problem Understanding fiber-wise linear Poisson structures related to W*-algebras.
method Investigates structures defined by W*-algebra structure and shows their arrangement in a short exact sequence of VB-groupoids.
result Fiber-wise linear Poisson structures are arranged in a short exact sequence of VB-groupoids.
Homogeneous pseudo-length functions on groups imply zero commutator length.
problem Characterizing pseudo-length functions on groups that are homogeneous.
method Analyzing properties of pseudo-length functions and their implications on group commutators.
result Homogeneous pseudo-length functions imply zero commutator length.
A {1}-structure on a Banach manifold M (with model space E) is an E-valued 1-form on M that induces on each tangent space an isomorphism onto E. Given a Banach principal bundle P with connected base space and a {1}-structure on P, we show that its automorphism group can be turned into a Banach-Lie group acting smoothly…
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
New bialgebra structures for relative Poisson algebras are introduced.
problem Extending bialgebra structures from commutative differential algebras to relative Poisson algebras.
method Introducing new bialgebra structures (relative PCA bialgebras) and using commutative 2-cocycles.
result New bialgebra structures (relative PCA bialgebras) are equivalent to certain Manin triples.
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on GA(1,5). We study n-ary commutative superalgebras and L∞-algebras that possess a skew-symmetric invariant form, using the derived bracket formalism. This class of superalgebras includes for instance Lie algebras and their n-ary generalizations, commutative associative and Jordan algebras with an invariant form. We…
Proves that emergent algebras right-distributivity implies left-distributivity.
problem Proving the implication between emergent algebra distributivity conditions.
method Analyzing families of quasigroup operations indexed by commutative groups.
result Emergent algebras right-distributive imply left-distributive.
Study non-commutative function algebras using contact geometry.
problem Quantize non-commutative function algebras in several variables.
method Contact geometry and rational differential operators.
result Generalizes known constructions of classical equivariants.
Unified framework for generalized sparsity and RIP analysis.
problem Analyzing inverse problems with sparsity models.
method Proposed generalized notions of sparsity and a unified RIP framework.
result Extends RIP analysis to broader contexts including tensor products.
Study first-order locally convex Lie algebroids in Bastiani calculus.
problem Define and study first-order locally convex Lie algebroids.
method Define sheaves of Lie algebroid forms and morphisms, prove category structure, study representations and cohomology.
result First-order locally convex Lie algebroids form a category and have applications in Lie II theorems.
Study relative commutants in von Neumann algebras using contraction notions.
problem Understanding relative commutants in group and tracial crossed product von Neumann algebras.
method Introducing contraction notions to study relative commutants.
result Results applied to negatively curved groups and SL(d, Z).
Reformulates divergence map for Turaev cobracket in non-commutative geometry.
problem Algebraic description of Turaev cobracket on surfaces.
method Non-commutative geometry, flat connection, associative algebras, Lie operad.
result Algebraic description of Turaev cobracket on surfaces.
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.
Presented an algebra structure for a specific geometric surface.
problem Understanding algebraic structures of geometric surfaces.
method Explicit presentation of Kauffman bracket skein algebra.
result Explicit algebraic structure for a 5-punctured sphere.
Probabilistic theory counts intersections in Riemannian spaces.
problem Counting intersections in Riemannian homogeneous spaces.
method Introduces probabilistic intersection ring HE(M), a graded commutative and associative real Banach algebra. result Probabilistic intersection ring structure defined for spheres, real projective space, and complex projective space.
We prove a Frobenius theorem for Banach distributions on manifolds that are modelled over locally convex spaces. Moreover, we recall how Frobenius theorems can be applied to infinite-dimensional Lie groups and obtain, that given a Lie subalgebra of the Lie algebra of a Lie group that is modelled over a locally convex s…
Unified approach to decomposing commutative n-ary superalgebras with skew-symmetric forms.
problem Decomposing commutative n-ary superalgebras with skew-symmetric invariant forms.
method Unified approach using derived bracket formalism and inductive orthogonal sums and generalized double extensions.
result Any commutative n-ary superalgebra with a skew-symmetric invariant form can be obtained by inductive orthogonal sums and generalized double extensions.
We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduc…
A super Lie group is a group whose operations are G∞ mappings in the sense of Rogers. Thus the underlying supermanifold possesses an atlas whose transition functions are G∞ functions. Moreover the images of our charts are open subsets of a graded infinite-dimensional Banach space since our space of …
This paper extends Riemannian geometry concepts to Hom-ρ-commutative algebras.
problem Extending Riemannian geometry concepts to Hom-ρ-commutative algebras. method Recalling Hom-ρ-commutative algebras, developing metric, connection, torsion, curvature, and differential operators. result Established differential calculus and symplectic/Poisson structures on Hom-ρ-commutative algebras. New calculus framework for vector bundles with metrics.
problem Developing calculus for vector bundles with fiber metrics.
method Adapting differential calculus to graded commutative algebras and focusing on diole and triole algebras.
result Triole algebra provides a suitable environment for vector bundle calculus with fiber metrics.
Develops Morse theory for commuting gradient-like vector fields.
problem Understanding the algebraic structure of the infrared data.
method Formal analogue of Morse theory for commuting vector fields, constructing L-infinity algebra and Maurer-Cartan elements.
result The algebraic formalism is similar to the algebra of the infrared of Gaiotto, Moore, and Witten.
Study differential and integral calculus on noncommutative C*-algebras.
problem Develop calculus on noncommutative spaces.
method Formal smooth structure on nonpure states of C*-algebras.
result Prove Stokes' theorem in both commutative and noncommutative settings.
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.
Covariance is shown as a commutator in random variable calculus.
problem Expressing covariance as a commutator of operators.
method Demonstrated through commutator identities involving expectations and products of functions.
result Revealed the underlying Lie algebraic structure in efficient influence curve calculus.