In this note we settle some technical questions concerning finite rank quasi-free Hilbert modules and develop some useful machinery. In particular, we provide a method for determining when two such modules are unitarily equivalent. Along the way we obtain representations for module maps and study how to determine the underlying holomorphic structure on such modules.

Each matrix representation : G ?! GL n (K) of a nite group G over a eld K induces an action of G on the module A n over the polynomial algebra A = Kx 1 ; ; x n ]. The graded A-submodule M() of A n generated by the orbit of (x 1 ; ; x n) is studied. A decomposition of M() into generic modules is given. Relations between the numerical invariants of and those of M(), the later being eeciently comp...

We use the fixed point alternative theorem to prove that, under suitable conditions, every A-valued approximate isometry on a Hilbert C∗-module over the C∗-algebra A can be approximated by a unique A-valued isometry. AMS subject classifications: Primary 39B52, 39B82, 46B04, 46L08; Secondary 47H10

We show that the unit ball of a full Hilbert C∗-module is sequentially compact in a certain weak topology if and only if the underlying C∗-algebra is finite dimensional. This provides an answer to the question posed in J. Chmieliński et al [Perturbation of the Wigner equation in inner product C∗-modules, J. Math. Phys. 49 (2008), no. 3, 033519].

In this paper, the principal tool to describe transversal polymatroids with Gorenstein base ring is polyhedral geometry, especially the Danilov−Stanley theorem for the characterization of canonical module. Also, we compute the a − invariant and the Hilbert series of base ring associated to this class of transversal polymatroids.