In this note we show that an unbounded regular operator t on Hilbert C∗modules over an arbitrary C∗ algebra A has polar decomposition if and only if the closures of the ranges of t and |t| are orthogonally complemented, if and only if the operators t and t∗ have unbounded regular generalized inverses. For a given C∗-algebra A any densely defined A-linear closed operator t between Hilbert C∗-mod...

In this paper we consider C0-group of unitary operators on a Hilbert C*-module E. In particular we show that if A?L(E) be a C*-algebra including K(E) and ?t a C0-group of *-automorphisms on A, such that there is x?E with =1 and ?t (?x,x) = ?x,x t?R, then there is a C0-group ut of unitaries in L(E) such that ?t(a) = ut a ut*.

Evolution algebras are non-associative algebras. In this work we provide an extension of class algebras, in a framework Hilbert spaces, and illustrate the applicability our approach by discussing connection with discrete-time Markov chains infinite countable state space. Specifically, if associate to each possible such process generator evolution algebra structure, then whole dynamics can be de...

We study the complexification of real Hilbert C∗-modules over real C∗-algebras. We give an example of a Hilbert Ac-module that is not the complexification of any Hilbert A-module, where A is a real C∗-algebra.

Given a classical symmetric pair, (G, K), with g = Lie(G), we provide descriptions of the Hilbert series of the algebra of K-invariant vectors in the associated graded algebra of U(g) viewed as a K-representation under restriction of the adjoint representation. The description illuminates a certain stable behavior of the Hilbert series, which is investigated in a case-by-case basis. We note tha...

Let $R$ be a positively graded algebra over field. We say that is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all roots on unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If standard graded, we prove that, under additional hypothesis Koszul or an irreducible $h$-polynomial, algebras coincide with complete i...

Hilbert C∗-modules are useful tools in AW ∗-algebra theory, theory of operator algbras, operator K-theory, group representation theory and theory of operator spaces. The theory of Hilbert C∗-modules is very interesting on its own. In this paper we give fundamentals of the theory of Hilbert C∗-modules and examine some ways in which Hilbert C∗-modules differ from Hilbert spaces.

Let (A,Ao) be a topological quasi *-algebra, which means in particular that Ao is a topological *-algebra, dense in A. Let πo be a *-representation of Ao in some pre-Hilbert space D ⊂ H. Then we present several ways of extending πo, by closure, to some larger quasi *-algebra contained in A, either by Hilbert space operators, or by sesquilinear forms on D. Explicit examples are discussed, both a...

Let H be a full Hilbert bimodule over a C-algebra A. We show that the Cuntz–Pimsner algebra associated to H is exact if and only if A is exact. Using this result, we give alternative proofs for exactness of reduced amalgemated free products of exact C– algebras. In the case that A is a finite–dimensional C–algebra, we also show that the Brown–Voiculescu topological entropy of Bogljubov automorp...