We give a new Banach module characterization of W *-modules, also known as selfdual Hilbert C *-modules over a von Neumann algebra. This leads to a generalization of the notion, and the theory, of W *-modules, to the setting where the operator algebras are σ-weakly closed algebras of operators on a Hilbert space. That is, we find the appropriate weak* topology variant of our earlier notion of r...

We study a noncommutative (operator space) version of the ‘boundary’, and in particular the Shilov boundary, of a function space. The main idea is that Hilbert C∗−modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We include some general notes on the ‘commutative case’ of some of the topics we discuss, coming in part from j...

In this paper we show that the fundamental group !T of the von Neumann algebra 2(Foo) of a free (noncommutative) group with infinitely many generators is lR+ \ {O}. This extends the result of Voiculescu who previously proved [26,27] that Q+ \ {O} is contained in !T(2(Foo)). This solves a classical problem in the harmonic analysis of the free group F 00. In particular, it follows that there exis...

In this paper, we will study some connections between Hilbert algebras and binary block-codes.With these codes, we can eassy obtain orders which determine suplimentary properties on these algebras. We will try to emphasize how, using binary block-codes, we can provide examples of classes of Hilbert algebras with some properties, in our case, classes of semisimple Hilbert algebras and classes of...

In this paper we show that the fundamental group !T of the von Neumann algebra 2(Foo) of a free (noncommutative) group with infinitely many generators is lR+ \ {O}. This extends the result of Voiculescu who previously proved [26,27] that Q+ \ {O} is contained in !T(2(Foo)). This solves a classical problem in the harmonic analysis of the free group F 00. In particular, it follows that there exis...

In this paper, we introduce a notion of strongly quasi-local algebras. They are defined for each discrete metric space with bounded geometry, and sit between the Roe algebra quasilocal algebra. We show that algebras coarse invariants, hence encoding geometric information underlying spaces. prove geometry which admits embedding into Hilbert space, inclusion induces an isomorphism in $K$-theory.

An e-ring is a pair (R,E) consisting of an associative ring R with unity 1 together with a subset E ⊆ R of elements, called effects, with properties suggested by the so-called effect operators on a Hilbert space. We establish the basic properties of e-rings, investigate commutative e-rings called c-rings, relate certain c-rings called b-rings to Boolean algebras, and prove a structure theorem f...

We give an elementary proof of a result which characterizes onto *-isomorphisms of the algebra BL(H) of all the bounded linear operators on a Hilbert space H. A known proof of this result (Arveson, 1976) relies on the theory of irreducible representations of C∗-algebras, whereas the proof given by us is based on elementary properties of operators on a Hilbert space which can be found in any int...

The theory of spectra for Banach algebras is outlined, including the Gelfand-Nǎımark theorem for commutative B∗-algebras. Resolutions of the identity are introduced, with examples; finally, we prove the spectral theorem for bounded normal operators on a Hilbert space, and conclude with some applications.