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We consider pseudoquotient extensions of positive linear functionals on a commutative Banach algebra $\mathcal{A}$ and give conditions under which the constructed space of pseudoquotients can be identified with all Radon measures on the structure space $\hat{\mathcal{A}}$.

Let $A$ and $B$ be subalgebras of $C(X)$ and $C(Y)$, respectively, for some topological spaces $X$ and $Y$. An arbitrary map $T: A\rightarrow B$ is said to be multiplicatively range-preserving if for every $f,g\in A$, $(fg)(X)=(TfTg)(Y)$, and $T$ is said to be separating if $TfTg=0$ whenever $fg=0$. For a given metric space $X$ and $\alpha\in (0,1]$, let Lip$_c(X,\alpha)$ be the algebra of all complex-valued functions on $X$ satisfying the Lipschitz condition of order $\alpha$ on each compact subset of $X$. In this note we first investigate the general form of multiplicatively range-preserving maps from $C(X)$ onto $C(Y)$ for realcompact spaces $X$ and $Y$ (not necessarily compact or locally compact) and then we consider such preserving maps from Lip$_c(X, \alpha)$ onto Lip$_c(Y,\beta)$ for metric spaces $X$ and $Y$ and $\alpha, \beta\in (0,1]$. We show that in both cases multiplicatively range-preserving maps are weighted composition operators which induce homeomorphisms between $X$ and $Y$. We also give a description of a linear separating map $T: A\rightarrow C(Y)$, where $A$ is either $C(X)$ for a normal space $X$ or Lip$_c(X,\alpha)$ for a metric space $X$ and $0

Topologically $Q$-algebras, a very convenient generalization of (topological) $Q$-algebras, have been recently considered by {\em A. Najmi} and independently by {\em H. Arizmendi et al.}. Here we extend certain basic results of {\em A. Najmi}. Within the same vein of ideas, we introduce a new class of topological algebras, the {\em ``Cauchy topologically $Q$-algebras"} (Cauchy $tQ$-algebras), that yields a new characterization of the standard advertibly complete algebras, as defined by {\em S. Warner}.

Using Fréchet algebraic technique, we show the existence of a nuclear Fréchet space without basis, thus providing yet another proof (of a different flavor) of a negative answer to a well known problem of Grothendieck from 1955. Using Fefferman's construction (which is based on complex-variable technique) of a $C^\infty$-function on the unit circle with certain properties, we give much simpler, transparent, and "natural" examples of restriction spaces without bases of nuclear Fréchet spaces of $C^\infty$-functions; these latter spaces, being classical objects of study, have attracted some attention because of their relevance to the theories of PDE and complex dynamical systems, and harmonic analysis. In particular, the restriction space $A^\infty(E)$, being a quotient algebra of the algebra $A^\infty(\Gamma)$, is the central one to other examples; the algebras $A^\infty$ had played a crucial role in solving a well-known problem of Kahane and Katznelson in the negative.

Properties of higher characters are developed and applied to symmetric products and Frobenius algebras. A `constructive' proof of the Gel'fand-Kolmogorov theorem is given. Generalisations of that theorem and the Nullstellensatz to symmetric products are discussed.Applications to the theory of multi-symmetric functions are also discussed. It is proved that the first three characters determine the Jordan algebra associated to a Frobenius algebra and as a corollary one obtains the theorem of Hoehnke and Johnson that a finite group is determined by the first three characters of its regular representation., To appear in Russian Math Surveys

In this short note we prove that the locally convex inductive limit of commutative locally m-convex algebras is already locally m-convex whenever multiplication is continuous. A topological algebra is an (associative) algebra endowed with a vector space topology such that multiplication is jointly continuous. If one considers inductive limits (in the category of topological vector spaces) of topological algebras multiplication may fail to be continuous since it is a bilinear map on the product which is certainly not linear, and forming inductive limits only respects the linear structure (although this is quite clear, the dierence between linearity and bilinearity in this context has been overlooked several times in the literature). The most important category of topological algebras is that of locally m-convex (l.m.c.) algebras having a 0-neighbourhood basis consisting of m-convex sets (i.e. absolutely convex sets B which are multiplicative in the sense that B 2 B). This class was introduced by Michael [9] and Arens [2] since it allows generalizations and applications of the theory of Banach algebras. Since their work a number of authors studied the question whether the inductive limit (in the category of locally convex spaces) of l.m.c. algebras is again l.m.c., see e.g. [1, 3, 4, 5, 7, 8, 10]. This is indeed true for countable inductive limits of normed algebras [1, 4]. On the other hand, Warner [10] gave an example of an inductive limit of metrizable l.m.c. algebras which is not l.m.c., and in [4] it is shown that [

The purpose of this paper is to introduce a local cohomology theory in the unital commutative Banach algebras context and to describe a connection between the local cohomology functor and direct limit of hom functors.

We show that a unitary commutative locally convex algebra, with a continuous product which is a Baire space and in which entire functions operate is actually m-convex. Whence, as a consequence, the same result of Mitiagin, Rolewicz and Zelazko, in commutative B0-algebras. It is known that entire functions operate in complete m-convex algebras [1]. In [3] Mitiagin, Rolewicz and Zelazko show that a unitary commutative B0-algebra in which all entire functions operate is necessarily m-convex. Their proof is quite long and more or less technical. They use particular properties of B0-algebras, a Baire argument and the polarisation formula. Here we show that any unitary commutative locally convex algebra, with a continuous product which is a Baire space and in which all entire functions operate is actually m-convex. The proof is short, direct and selfcontained. A locally convex algebra (A; ), l. c. a. in brief, is an algebra over a eld K (K = R or C) with a Hausdor locally-convex topology for which the product is separately continuous. If the product is continuous in two variables, (A; ) is said to be with continuous product. A l. c. a. (A; ) is said to bem-convex (l. m. c. a.) if the origin 0 admits a fundamental system of idempotent neighbourhoods ([2]). An

Every element 6 of a radical Banach algebra & satisfies l iπv^ ||6||=0. We are concerned here with the existence of a lower bound for the rate of decrease of the sequence (||δ||) under various assumptions over b and ^ , when b is not nilpotent and & commutative. If the nilpotents are dense in & then for every sequence (λn) of positive reals there exists a nonnilpotent b e & such that lim inf*-«, ||6||/ϋ»=0. A stronger result holds if & possesses furthermore a bounded approximate identity. On the other direction if & has no nilpotent element and if some element of & which is not a divisor of zero acts compactly on & then there exists a sequence (2n) of positive reals such that \immfn-+0O\\b n\\lλn— + °o for every nonzero be&. Also there exists universal lower bounds for the rate of decrease of ll