It is well known that a function K : Ω × Ω → L(Y) (where L(Y) is the set of all bounded linear operators on a Hilbert space Y) being (1) a positive kernel in the sense of Aronszajn (i.e. ∑N i,j=1〈K(ωi, ωj)yj , yi〉 ≥ 0 for all ω1, . . . , ωN ∈ Ω, y1, . . . , yN ∈ Y, and N = 1, 2, . . . ) is equivalent to (2) K being the reproducing kernel for a reproducing kernel Hilbert space H(K), and (3) K ha...

Let H be a Hilbert C-module over a matrix algebra A. It is proved that any function T : H → H which preserves the absolute value of the (generalized) inner product is of the form Tf = φ(f)Uf (f ∈ H), where φ is a phase-function and U is an A-linear isometry. The result gives a natural extension of Wigner’s classical unitary-antiunitary theorem for Hilbert modules. 1991 Physics and Astronomy Cla...

In this work we examine generalized Connes-Lott models, with C ⊕ C as finite algebra, over the two-sphere. The Hilbert space of the continuum spectral triple is taken as the space of sections of a twisted spinor bundle, allowing for nontrivial topological structure (magnetic monopoles). The finitely generated projective module over the full algebra is also taken as topologically non-trivial, wh...

In type A, the q, t-Fuß-Catalan numbers can be defined as a bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with non-negative integer coefficients. We prove the conjectures for ...

The Hilbert polynomial can be defined axiomatically as a group homomorphism on the Grothendieck group K(X) of a projective variety X, satisfying certain properties. The Chern polynomial can be similarly defined. We therefore define these rather abstract notions to try and find a nice description of this relationship. Introduction Let X = P be a complex projective variety over an algebraically c...

In this paper, first, we introduce the new concept of 2-inner product on Banach modules over a $C^*$-algebra. Next,  we present the concept of 2-linear operators over a $C^*$-algebra. Our result improve  the main result of the paper  Z. Lewandowska.  In the final of this paper, we define the notions 2-adjointable mappings between 2-pre Hilbert C*-modules and prove supperstability of them ...

A W-algebra action is constructed via Hecke transformations on the equivariant Borel–Moore homology of Hilbert scheme points a nonreduced plane in three-dimensional affine space. The resulting W-module then identified to vacuum module. construction based generalization ADHM as well W-action moduli space instantons by Schiffmann and Vasserot.

We establish the existence and uniqueness of finite free resolutions and their attendant Betti numbers for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory. For example,...

Let G be a finite abelian group. A number field K is called a Hilbert-Speiser field of type G if every tame G-Galois extension L/K has a normal integral basis, i.e., the ring of integers OL is free as an OKG-module. Let Cp denote the cyclic group of prime order p. We show that if p ≥ 7 (or p = 5 and extra conditions are met) and K is totally real with K/Q ramified at p, then K is not Hilbert-Sp...

In type A, the q, t-Fuß–Catalan numbers can be defined as the bigraded Hilbert series of a module associated to the symmetric group. We generalize this construction to (finite) complex reflection groups and, based on computer experiments, we exhibit several conjectured algebraic and combinatorial properties of these polynomials with nonnegative integer coefficients. We prove the conjectures for...