General Theory of Banach Algebra. By C.E. Rickart. Pp. 394. 79s. 1960. (Van Nostrand, London)

Structure of gelfand-mazur algebras

We give a generalization of the Stone–Weierstrass property for subalgebras of C (X), with X a completely regular Hausdorff space. In particular, we study in this paper some subalgebras of C0(X), with X a locally compact Hausdorff space, provided with weighted norm topology. By using the Stone–Weierstrass property, we then describe the ideal structure of these algebras. (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generalization of the B*-algebra (C0(X),|| ||_infinity)

http://www.math.chalmers.se/Math/Grundutb/CTH/mve021/1112/conference/slides/mathieu.pdf

C∗-Segal algebras with order unit

We study subalgebras of Cb(X) equipped with topologies that generalize both the uniform and the strict topology. In particular, we study the Stone–Weierstrass property and describe the ideal structure of these algebras.

/pdf/generalization-of-the-topological-algebra-c-b-x-beta-4okqkrsw8z.pdf

Generalization of the topological algebra $(C_b (X), \beta )$

We develop the theory of Segal algebras of commutative C∗-algebras, with an emphasis on the functional representation. Our main results extend the Gelfand– Naimark Theorem. As an application, we describe faithful principal ideals of C∗-algebras. A key ingredient in our approach is the use of Nachbin algebras to generalize the Gelfand representation theory.

/pdf/on-dense-ideals-of-c-algebras-and-generalizations-of-the-2yqfdvduih.pdf

On dense ideals of C* -algebras and generalizations of the Gelfand-Naimark Theorem

The ideal structure of the algebra C(X, A) has been studied in many papers under various topological assumptions on the space X and the algebra A. In this paper we shall study the case where I is a completely regular topological space and A is a locally convex star algebra. In such case the structure of closed (proper) ideals can be described not only by using points of X and some family of closed ideals of A , as usual, but also by using points of the carrier space à.{A) of A and some family of closed ideals of C{X, A) depending on those points and also by using different kind of slice ideals of C(X,A).

/pdf/on-the-ideal-structure-of-algebras-of-lmc-algebra-valued-uo9gk87jja.pdf

On the ideal structure of algebras of LMC-algebra valued functions

Let X be a topological space, a cover of X and Cb(X, ; ) the alge- bra of all -valued continuous functions on X, which are bounded on every S ∈ . Necessary and sufficient conditions for a subalgebra A of Cb(X, ; ) to be dense in Cb(X, ; ) in the topology τ of -convergence and in the -strict topology β on Cb(X, ; ) are given. Also, necessary and sufficient conditions for the completeness of the topological algebras (Cb(X, ; ) ,τ ) and (Cb(X, ; ) ,β ) are given. 1 Introduction Let X be a topological space and K one of the fields R of real numbers or C of complex numbers. The algebra Cb(X,K) of all K-valued continuous and bounded functions on X is one of the most studied objects in modern analysis. Usually it has been equipped either with the topology τX of uniform convergence or with the strict topology βX . It is well-known that the structure of (Cb(X) ,τ X ) is quite complicated. For exam- ple, its Gelfand space (the set of all maximal regular ideals equipped with the relative weak*-topology) is homeomorphic to the Stone-Cech compactification β(X )o fX. Another difficulty is based on the fact that (Cb(X) ,τ X ) does not satisfy the Stone-Weierstrass prop- erty, i.e., a pointseparating subalgebra (if K = C, then a self-adjoint subalgebra), which is bounded away from zero, is not necessarily uniformly dense in Cb(X). In fact, in order that a subalgebra is dense in (Cb(X) ,τ X ) one has to replace the property separation of points by separation of zero-sets. On the other hand, (Cb(X) ,β X) is in some sense easier to handle than (Cb(X) ,τ X ). For example, its Gelfand space is homeomorphic to X and it also satisfies the Stone-Weierstrass property. However, (Cb(X) ,β X ) is not necessarily complete whereas (Cb(X) ,τ X ) always has this property. The structure of Cb(X), when equipped with the uniform or with the strict topology, has been generalized in the literature in several ways. In (6) and (7) proper subalgebras of Cb(X) and their Stone-Weierstrass type properties and ideal structures were developed. In the present paper, Stone-Weierstrass properties as well as completeness properties of algebras of continuous functions containing also unbounded functions (in both topologies described above) are considered. These results are important by the remark made in (5). Namely, Stone-Weierstrass property is closely related to the Gelfand representations of several kinds of topological algebras. In particular, the Gelfand representations of locally A-convex and uniformly locally A-convex algebras can be described by means of such function algebras (see (4)). Also, the results of this paper are needed in (3), where a description of the ideal structure of algebras containing continuous unbounded functions is given.

Jorma Arhippainen

Papers

Generalization of the B*-algebra (C0(X),|| ||_infinity)

Generalization of the topological algebra $(C_b (X), \beta )$

On dense ideals of C* -algebras and generalizations of the Gelfand-Naimark Theorem

On the ideal structure of algebras of LMC-algebra valued functions

Stone-weierstrass type theorems for algebras containing continuous unbounded functions