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In the present note we examine possible extensions of Runge, Mergelyan and Arakelian Theorems, when the uniform approximation is meant with respect to the metric ρ of a metrizable compactification (S, ρ) of the complex plane C.

In this work, we introduce the concept of *-operator frame, which is a generalization of *-frames in Hilbert pro-C*-modules, and we establish some results. We also study the tensor product of *-operator frame for Hilbert pro-C*-modules. [ABSTRACT FROM AUTHOR]

For a Banach algebra A and a character φ on A, essential φ -amenability and essential character amenability have been introduced. These concepts will be generalized for Fréchet algebras. In this paper, some of the previous results about essential φ -amenability and essential character amenability for the case of Banach algebras will be verified for Fréchet algebras. Finally, we give some examples related to all of the results of this work. [ABSTRACT FROM AUTHOR]

We prove that if A is an ordinary and advertibly complete l. m. c. a. each element of which has a convex spectrum, then A modulo its Jacobson radical is isomorphic to C . We obtain the same conclusion for l.A.c.a and l.u.A-c.a. A purely algebraic version is also given. In the involutive case, the same conclusion, for an involutive ordinary Arens-Michael algebra, is obtained only under the convexity hypothesis on the spectrum of each normal element. Finally, if the algebra is additionally hermitian, it suffices to assume that the spectrum of each unitary element is convex. [ABSTRACT FROM AUTHOR]

Let A be a locally C ∗ -algebra and S(A) be the family of continuous C * -seminorms and let E be a Hilbert A-module. We prove that every dynamical system of unitary operators on E defines a dynamical system of automorphisms on the compact operators on E and show that under certain conditions, the converse is true. We define a generalized derivation on E and prove that if E is a full Hilbert A-module and δ : E → E is a bounded generalized derivation, then δp : Ep → Ep is a generalized derivation on the Hilbert module Ep over the C * -algebra Ap for each p ∈ S(A). [ABSTRACT FROM AUTHOR]

A smooth envelope of a topological algebra is introduced, and the following result is announced: the smooth envelope of a given subalgebra A in C( M) coincides with C( M) if and only if A has the same tangent bundle as M. [ABSTRACT FROM AUTHOR]

We study closedness of the range, adjointability and generalized invertibility of modular operators between Hilbert modules over locally C"-algebras of coefficients. Our investigations and the recent results of M. Frank reveal a number of equivalence properties of the category of Hilbert modules over locally C"-algebras which characterize precisely the inverse limit of C"-algebras of the C"-algebra of compact operators. [ABSTRACT FROM AUTHOR]

In this paper we investigate multinormed W*-algebras in terms of the central topologies of W*-algebras. The main result asserts that each multinormed W*-algebra can be realized as a local von Neumann algebra on a certain domain in a Hilbert space. Moreover, it admits the predual (unique up to an isometry), which is the ℓ1-normed space. In the normed case the assertion is reduced to the known Sakai theorem. [ABSTRACT FROM AUTHOR]

The notion of the completely dissipative maps on σ-C*-algebras is introduced. We show that a completely dissipative map induces a representation of σ-C*-algebra. The classical Stinespring's dilation-type theorem is extended to a more general setting. [ABSTRACT FROM AUTHOR]

We investigate the extent to which the study of quasimultipliers can be made beyond Banach algebras. We will focus mainly on the class of F-algebras, in particular on complete k-normed algebras, 0 < k ≤ 1, not necessarily locally convex. We include a few counterexamples to demonstrate that some of our results do not carry over to general F-algebras. The bilinearity and joint continuity of quasimultipliers on an F-algebra A are obtained under the assumption of strong factorability. Further, we establish several properties of the strict and quasistrict topologies on the algebra QM(A) of quasimultipliers of a complete k-normed algebra A having a minimal ultra-approximate identity. [ABSTRACT FROM AUTHOR]

We show that two continuous inverse limit actions α and β of a locally compact group G on two pro- C *-algebras A and B are stably outer conjugate if and only if there is a full Hilbert A-module E and a continuous action u of G on E such that E and E *(the dual module of E) are countably generated in M( E)(the multiplier module of E), respectively M( E *) and the pair ( E, u) implements a strong Morita equivalence between α and β. This is a generalization of a result of F. Combes [Proc. London Math. Soc. 49(1984), 289–306]. [ABSTRACT FROM AUTHOR]

Let A be a locally m-convex Fréchet algebra. We give a necessary and sufficient condition for a cyclic Fréchet A-module X = A+/I to be strictly flat, generalizing thereby a criterion of Helemskii and Sheinberg. To this end, we introduce a notion of "locally bounded approximate identity" (a locally b.a.i. for short), and we show that X is strictly flat if and only if the ideal I has a right locally b.a.i. Next we apply this result to amenable algebras and show that a locally m-convex Fréchet algebra A is amenable if and only if A is isomorphic to a reduced inverse limit of amenable Banach algebras. We also extend a number of characterizations of amenability obtained by Johnson and by Helemskii and Sheinberg to the setting of locally m-convex Fréchet algebras. As a corollary, we show that Connes and Haagerup's theorem on amenable C*-algebras and Sheinberg's theorem on amenable uniform algebras hold in the Fréchet algebra case. We also show that a quasinormable locally m-convex Fréchet algebra has a locally b.a.i. if and only if it has a b.a.i. On the other hand, we give an example of a commutative, locally m-convex Fréchet-Montel algebra which has a locally b.a.i., but does not have a b.a.i. [ABSTRACT FROM AUTHOR]

If E is a l.m.c.*-algebra with a b.a.i., ℰ(E), ℛ(E) denote the enveloping algebra and the space of representations of E respectively, while ℬ(E) stands for the non-zero extreme points of the continuous positive linear forms on E. Thus, for suitable l.m.c.*-algebras E, F and an admissible topology on E ⊗ F, ℰ(E F) is given by the completed υ-tensor product of ⊗(E), ⊗(F) (where υ is the projective tensorial l.m.c.C*-topology), while ℛ(E F) by the cartesian product of ℛ(E), ℛ(F). An analogous decomposition of ℬ(E F) is not valid in general. [ABSTRACT FROM AUTHOR]