We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior $$ \sigma -compact subsets and, relying on this classification, we study properties corresponding $$C^*$$ -algebras in terms frame theory and {\mathscr A} operators Hilbert -modules; some pathological examples are constructed.

We show that every (continuous) faithful product system admits a (continuous) faithful nondegenerate representation. For Hilbert spaces this is equivalent to Arveson’s result that every Arveson system comes from an E0–semigroup. We point out that for Hilbert modules this is not so. As applications we show a C–algebra version of a result for von Neumann algebras due to Arveson and Kishimoto, and...

In this paper, we study approximate duals of $g$-frames and fusion frames in Hilbert $C^ast-$modules. We get some relations between approximate duals of $g$-frames and biorthogonal Bessel sequences, and using these relations, some results for approximate duals of modular Riesz bases and fusion frames are obtained. Moreover, we generalize the concept of $Q-$approximate duality of $g$-frames and ...

Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C∗-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott’s classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neu...

the concept of soft sets, introduced by molodtsov [20] is a mathematicaltool for dealing with uncertainties, that is free from the difficultiesthat have troubled the traditional theoretical approaches. in this paper, weapply the notion of the soft sets of molodtsov to the theory of hilbert algebras.the notion of soft hilbert (abysmal and deductive) algebras, soft subalgebras,soft abysms and sof...

We determine the ideal structure of the Toeplitz C∗-algebra on the bidisk 0 Introduction A large part of doing research in mathematics is asking the right question. Posing a timely question can trigger thought and provoke insights, even when the matter has no good resolution. That is what happened in the case at hand. After devoting much time to the study of Hilbert modules and quotient Hilbert...

We initiate a study of Hilbert modules over the polynomial algebra A = C[z1, . . . , zd] that are obtained by completing A with respect to an inner product having certain natural properties. A standard Hilbert module is a finite multiplicity version of one of these. Standard Hilbert modules occupy a position analogous to that of free modules of finite rank in commutative algebra, and their quot...

The purpose of this paper is to build a new bridge between category theory and generalized probability known as noncommutative or quantum probability, which was originated mathematical framework for theory, in terms states linear functional defined on algebras. We clarify that algebras can be considered matrix the notions state algebra turns out conceptual generalization measures sets discrete ...