Search results for "C*-algebra"

CQ*-algebras and noncommutative measure

2012

In this paper we continue the investigations in [4], [5], [8], [13], [14], [15], and [19], of the structure of quasi *-algebras and extend the results in [1] and [2]. Here, noncommutative Tp-spaces are shown to constitute examples of a class of Banach C*-modules called CQ*-algebras. Moreover, it is shown that any (strongly) *-semisimple proper CQ*-algebra (X ,A), with A a separable C*-algebra, can be represented as a CQ*-algebra of type Tp.

The completion of a C*-algebra with a locally convex topology

2006

There are examples of C*-algebras A that accept a locally convex *-topology t coarser than the given one, such that Ae[t] (the completion of A with respect to t) is a GB*-algebra. The multiplication of A[t] may be or not be jointly continuous. In the second case, Ae[t] may fail being a locally convex *-algebra, but it is a partial *-algebra. In both cases the structure and the representation theory of Ae[t] are investigated. If A[t+] denotes the t-closure of the positive cone A+ of the given C*-algebra A, then the property A[t]+ \cap (−A[t]+) = {0} is decisive for the existence of certain faithful *-representations of the corresponding *-algebra Ae[t].

A note on partial*–algebras and spaces of distributions

2014

Given a rigged Hilbert space (D,H,D'), the spaces D_{loc are considered. It is shown that, if D is a Hilbert *-algebra, D_{loc} carry out a natural structure of partial *-algebra. Furthermore, on D_{loc} it is defined a topology, so that D_{loc} is an interspace. Examples from distributions theory are considered.

Faithfully representable topological *-algebras: some spectral properties

2018

A faithfully representable topological *-algebra (fr*-algebra) A0 is characterized by the fact that it possesses sufficiently many *-representations. Some spectral properties are examined, by constructing a convenient quasi *-algebra A over A0, starting from the order bounded elements of A0.

MR3730338 Reviewed de Jeu, Marcel(NL-LEID-MI); Tomiyama, Jun(J-TOKYM) The closure of ideals of ℓ1(Σ) in its enveloping C∗-algebra. (English summary) …

2018

Given a compact Hausdorff space X and a homeomorphism σ on X, denote by Σ=(X,σ) a topological dynamical system. Then the associated Banach ∗-algebra ℓ1(Σ) is defined as ℓ1(Σ)={a:Z→C(X), ∥a∥:=∑n∈Z∥a(n)∥<∞} with a crossed product–type product (aa′)(n)=∑k∈Za(k)⋅αk(a′(n−k)) and involution a∗(n)=αn(a(−n))¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯, where C(X) denote the space of complex-valued continuous functions on X, and α(f):=f∘σ−1 for f∈C(X). If C∗(Σ) is the enveloping C∗-algebra of ℓ1(Σ), considering a primitive ideal I of ℓ1(Σ), the authors show that there exists a ∗-representation π of ℓ1(Σ) on Hilbert space such that the kernel is I, and that the closure in C∗(Σ) of an ideal of ℓ1(Σ) is an ideal of C∗(Σ).

Locally Convex Quasi *-Algebras and their Representations

2020

This book is a review of the work the authors have done in the past 20 years on the theory of locally convex quasi *-algebras

Operators in rigged Hilbert spaces: toward a spectral analysis

Slight extensions of positive linear functionals: two concrete realizations.

2009

In this paper we show, in full details, some example of slight extensions of a nonclosable positive linear functional ώ defined on a dense *-subalgebra Ao of a given topological *-algebra.

Locally convex quasi $C^*$-normed algebras

2012

Abstract If A 0 [ ‖ ⋅ ‖ 0 ] is a C ∗ -normed algebra and τ a locally convex topology on A 0 making its multiplication separately continuous, then A 0 ˜ [ τ ] (completion of A 0 [ τ ] ) is a locally convex quasi ∗-algebra over A 0 , but it is not necessarily a locally convex quasi ∗-algebra over the C ∗ -algebra A 0 ˜ [ ‖ ⋅ ‖ 0 ] (completion of A 0 [ ‖ ⋅ ‖ 0 ] ). In this article, stimulated by physical examples, we introduce the notion of a locally convex quasi C ∗ -normed algebra, aiming at the investigation of A 0 ˜ [ τ ] ; in particular, we study its structure, ∗-representation theory and functional calculus.

Some classes of topological quasi *-algebras

2001

The completion $\overline{A}[\tau]$ of a locally convex *-algebra $A [ \tau ]$ with not jointly continuous multiplication is a *-vector space with partial multiplication $xy$ defined only for $x$ or $y \in A_{0}$, and it is called a topological quasi *-algebra. In this paper two classes of topological quasi *-algebras called strict CQ$^*$-algebras and HCQ$^*$-algebras are studied. Roughly speaking, a strict CQ$^*$-algebra (resp. HCQ$^*$-algebra) is a Banach (resp. Hilbert) quasi *-algebra containing a C$^*$-algebra endowed with another involution $\sharp$ and C$^*$-norm $\| \|_{\sharp}$. HCQ$^*$-algebras are closely related to left Hilbert algebras. We shall show that a Hilbert space is a H…