Calkin Representations of Unbounded Operator Algebras Acting on Non-Separable Domains

There are constructed representations of unbounded operator algebras which generalize representations of B(H) constructed by J. W. CALKIN and H. BEHNCKE. For a large class of unitary spaces D, each uniformly closed two-sided ideal of the maximal Op*-algebra L⁺ (D) appears as kernel of such a representation. Irreducibility of the representations is characterized in terms of properties of ultrafilters which define the representations.     

On Inner Characterizations of Aõ-Algebras and some Applications to Calkin Algebras and Symmetric Tensor-Algebras

Spacefree characterizations of Aõ-algebras are discussed, and their applications to (unbounded) Calkin algebras and symmetric tensor-algebras are considered. For Calkin algebras, a sufficient condition for being an Aõ-algebra is provided. On symmetric tensor-algebras, Õ-lopologies are constructed and their relations to the well-known topologies are examined.     

Strict Topologies as Topological Algebras

Let X be a completely regular Hausdorff space, C_b(X) the space of all scalar-valued bounded continuous functions on X with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally m-convex.     

Generalized Principal Extensions in Topological Algebras

The ??generalized principal extension?? of a subset of a topological algebra is, by definition, a subset of another topological algebra. Hence, the generalized spectrum of an element, ??local spectrum??, of a topological algebra is thus a subset of a topological algebra, not necessarily of ${{\mathbb C}}$ . We also give a criterion for the continuity of the (generalized) Newburg map (Theorem 3.1).     

Locally Pseudoconvex Inductive Limit of Topological Algebras

Locally pseudoconvex inductive limit of topological algebras (in particular, of locally pseudoconvex F-algebras) is studied. Properties of the quotient algebras of these algebras modulo a closed two-sided ideal and of the direct product of these algebras are presented.     

Universal Extension and Excision for Topological Algebras

We construct a topology on the tensor algebra that makes the multiplication jointly continuous and is universal for algebras with jointly continuous multiplication. This result, together with the methods developed by Cuntz for m-algebras are used to prove the excision theorem of Cuntz and Quillen in the more general case of c-algebras.     

Topological Hopf Algebras and Braided Monoidal Categories

The relation between a monoidal category which has an exact faithful monoidal functor to a category of finite rank projective modules over a Dedekind domain, and the category of continuous modules over a topological bialgebra is discussed. If the monoidal category is braided, the bialgebra is topologically quasitriangular. If the monoidal category is rigid monoidal, the bialgebra is a Hopf algebra.     

On the poles of the resolvent in Calkin algebra

In the present note, we study the problem of lifting poles in Calkin algebra on a separable infinite-dimensional complex Hilbert space . We show by an example that such lifting is not possible in general, and we prove that if zero is a pole of the resolvent of the image of an operator in the Calkin algebra, then there exists a compact operator for which zero is a pole of if and only if the index of is zero on a punctured neighbourhood of zero. Further, a useful characterization of poles in Calkin algebra in terms of essential ascent and descent is provided.

     

Topological centers of certain dual algebras

Let be a Banach algebra with a bounded approximate identity. Let and be, respectively, the topological centers of the algebras and . In this paper, for weakly sequentially complete Banach algebras, in particular for the group and Fourier algebras and , we study the sets , , the relations between them and with several other subspaces of or .

     

Topological results on a certain class of functionals and application

We study in this paper problems of the type $\text{Δu + ¦u¦}{}^{\mathrm{p\; ?\; 1}}\text{u = ?(x)}$, Ω bounded ? $\text{R}$^N, $\text{u = 0¦}{}_{\mathrm{?\Omega }}$, (I) where $\text{?(x)}$ is given and where $\text{p ? (1, (}\text{N + 2}\text{N ? 2}\text{)) (p ? (1, + ∞)}\text{if}\text{N ? 2)}$. Our main result is that (I) has an infinite number of solutions for a residual set of ? in $\text{H}{}^{\mathrm{?1}}\text{(Ω)}$. In particular, for many n ∈ $\text{N}$ there exists an open and dense subset of ? in $\text{H}{}^{\mathrm{?1}}\text{(Ω)}$ such that (I) has n distinct solutions for such an ? This result is to be related to the conjecture developed in [1] of the existence of an infinite number of solutions to (I). The proof relies on a general characterization of level sets for a certain class of functionals, when there are no critical value in a large enough interval. In addition to the study of problem (I), we apply this characterization to give another proof (using, e.g., Brouwer's fixed point theorem) for some classical results about even functionals and saddle points.     

Fourier Algebras on Topological Foundation *-Semigroups.

We introduce the notion of the Fourier and Fourier-Stieltjes algebras of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras.     

拓扑度的一个性质

本建立了拓扑度的一个性质，即拓扑度的容斥性，并提出了一个不动点定理。