The toric Hilbert scheme is a parameter space for all ideals with the same multi-graded Hilbert function as a given toric ideal. Unlike the classical Hilbert scheme, it is unknown whether toric Hilbert schemes are connected. We construct a graph on all the monomial ideals on the scheme, called the flip graph, and prove that the toric Hilbert scheme is connected if and only if the flip graph is ...

We present a general approach to a module frame theory in C∗algebras and Hilbert C∗-modules. The investigations rely on the ideas of geometric dilation to standard Hilbert C∗-modules over unital C∗-algebras that possess orthonormal Hilbert bases, of reconstruction of the frames by projections and by other bounded module operators with suitable ranges. We obtain frame representation and decompos...

A Hilbert C∗-module is a generalisation of a Hilbert space for which the inner product takes its values in a C∗-algebra instead of the complex numbers. We use the bracket product to construct some Hilbert C∗-modules over a group C∗-algebra which is generated by the group of translations associated with a wavelet. We shall investigate bracket products and their Fourier transform in the space of ...

Analogues for Hilbert C∗-modules of classical results of Fourier series theory in Hilbert spaces are considered. Relations between different properties of orthogonal and orthonormal systems for Hilbert C∗-modules are studied with special attention paid on the differences with the well-known Hilbert space situation.

Jim Williams died in 1983. The surviving three authors are pleased to dedicate this paper to his memory. Abstract. Let C denote the category of Hilbert modules which are similar to con-tractive Hilbert modules. It is proved that if H 0 ; H 2 C and if H 1 is similar to an isometric Hilbert module, then the sequence 0 ! H 0 ! H ! H 1 ! 0 splits. Thus the isometric Hilbert modules are projective i...