We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = ∑ ann −s−ū, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hil...

If A is a unital C∗-algebra and if H is a complex Hilbert space, then the set SH(A) of all unital completely positive linear maps from A to the algebra B(H) of continuous linear operators on H is an operator-valued, or generalised, state space of A. The usual state space of A occurs with the one-dimensional Hilbert space C. The structure of the extreme points of generalised state spaces was det...

In this notes unbounded regular operators on Hilbert C∗-modules over arbitrary C∗-algebras are discussed. A densely defined operator t possesses a densely defined adjoint operator if the graph of t is an orthogonal summand. Moreover, for a densely defined operator t the graph of t is orthogonally complemented if and only if t is regular. For a given C∗-algebra A any densely defined A-linear clo...

A 2-Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we deene a 2-Hilbert space to be an abelian category enriched over Hilb with a-structure, conjugate-linear on the hom-sets, satisfying hf g; hi = hg; f hi = hf; hg i. We also deene monoidal, braided monoidal, and symmetric monoidal versions of 2-Hilbert spaces, which we call 2-H...

Here the reduction problem is studied in an algebraic structure called dependence space. We characterize the reducts by the means of dense families of dependence spaces. Dependence spaces defined by indiscernibility relations are also considered. We show how we can determine dense families of dependence spaces induced by indiscernibility relations by applying indiscernibility matrices. We also ...

In this notes unbounded regular operators on Hilbert C∗-modules over arbitrary C∗-algebras are discussed. A densely defined closed operator t possesses a densely defined adjoint operator if the graph of t is an orthogonal summand. Moreover, for a densely defined closed operator t the graph of t is orthogonally complemented if and only if t is regular. For a given C*-algebra A any densely define...

We will show that Feichtinger’s algebra S0(R ) (also known as modulation space M 0 (R ) and well defined for general locally compact Abelian groups) is a natural Hilbert module for quantum tori and that a result by V. Losert can be interpreted as implying that the construction of such Hilbert modules gives Feichtinger’s algebra, if one considers the category of time-frequency invariant homogeno...

Beweistheorie, or “Proof Theory,” was the phrase that David Hilbert used to describe the program by which he hoped to secure the foundations of mathematics. Set forth in the early 1920’s, his plan was to represent mathematical reasoning by formal deductive systems, and show, using safe, “finitary,” methods, that such reasoning could never lead to contradiction. This particular goal was shown by...

We construct geometrically the generating fields of a W algebra which acts irreducibly on the direct sum of the cohomology rings of the Hilbert schemes X [n] of n points on a projective surface X for all n ≥ 0. We compute explicitly the commutators among the Fourier components of the generating fields of the W algebra, and identify this algebra with a W1+∞-type algebra. A precise formula of cer...

We construct geometrically a W algebra which acts irreducibly on the direct sum of the cohomology rings of the Hilbert schemes X [n] of n points on a projective surface X for all n ≥ 0. We compute explicitly the commutators among a set of linear basis elements of the W algebra, and identify this algebra with a W1+∞-type algebra. A precise formula of certain Chern character operators, which is e...