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It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.

We discuss properties of pseudospectrum and condition spectrum of an element in a complex unital Banach algebra and its $\epsilon$-perturbation. Several results are proved about linear maps preserving pseudospectrum/ condition spectrum. These include the following: (1) Let $A, B$ be complex unital Banach algebras and $\epsilon$ is positive. Let $\Phi: A\rightarrow B$ be an $\epsilon$-pseudospectrum preserving linear onto map. Then $\Phi$ preserves spectrum. If $A$ and $B$ are uniform algebras, then, $\Phi$ is an isometric isomorphism. (2) Let $A, B$ be uniform algebras and $\epsilon \in (0,1)$. Let $\Phi:A\rightarrow B$ be an $\epsilon$-condition spectrum preserving linear map. Then $\Phi$ is an $\epsilon^{'}$-almost multiplicative map, where $\epsilon, \epsilon^{'}$ tend to zero simultaneously.

n this paper, we introduce the notion of $\phi$-weakly compatible mapping for a pair of mappings. A fixed point theorem for two pairs of $\phi$-weakly compatible mappings satisfying a rational type contraction in a normed space is also established. Subsequently we use our result to find existence of solutions of variational inequalities.

We show that a regular von Neumann $Q$-$m$-convex Frechet algebra is of finite dimension. We also show that a regular von Neumann $m$-convex Frechet algebra is a projective limit of finite dimensional algebras. Finally, we prove that a bilateral $Q$-$F$-algebra is a regular von Neumann algebra if and only if it is isomorphic to a finite product of algebras which are also fields.

This is a synthetic presentation of several barrelledness notions, in locally convex algebras. These are characterized, as in locally convex spaces, via (algebra) seminorms. This approach reveals a new notion of barrelledness. The latter shows to be what is needed to have meaningful statements in locally uniformly convex algebras.

In this work, given a unital Banach algebra $\A$ and $a\in \A$ such that $a$ has a Moore-Penrose inverse $a^\dagger$, it will be characterized when $aa^\dagger-a^\dagger a$ is invertible. A particular subset of this class of objects will also be studied. In addition, perturbations of this class of elements will be studied. Finally, the Banach space operator case will be also considered., 14 pages, original research article

Conditions such that a locally $k$-convex inductive limit of a sequence of $k_n$-normed algebras is a locally $m$-($k$-convex) algebra, are given. It is shown that every locally pseudoconvex inductive limit $E$ of a sequence of commutative locally $m$-pseudoconvex algebras is a commutative locally\break $m$-pseudoconvex algebra if the multiplication in $E$ is jointly continuous.

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb C$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb R)$ with $\det M>0$ is a product of two real exponential matrices., 9 pages

Weighted EP Banach space operators and Banach algebra elements are characterized using different kinds of factorizations. The results presented extend well-known characterizations of (weighted) EP matrices, (weighted) EP Hilbert space operators and (weighted) EP $C^*$-algebra elements., 14 pages, original research article. arXiv admin note: text overlap with arXiv:1401.0469, arXiv:1306.4093

We review our recent formulation of Colombeau type algebras as Hausdorff sequence spaces with ultranorms, defined by sequences of exponential weights. We extend previous results and give new perspectives related to echelon type spaces, possible generalisations, asymptotic algebras, concepts of association, and applications thereof.

It is shown that every continuous homomorphism of Arens-Michael algebras can be obtained as the limit of a morphism of certain projective systems consisting of Fr\'{e}chet algebras. Based on this we prove that a complemented subalgebra of an uncountable product of Fr\'{e}chet algebras is topologically isomorphic to the product of Fr\'{e}chet algebras. These results are used to characterize injective objects of the category of locally convex topological vector spaces. Dually, it is shown that a complemented subspace of an uncountable direct sum of Banach spaces is topologically isomorphic to the direct sum of ({\bf LB})-spaces. This result is used to characterize projective objects of the above category., Comment: 25 pages

For an arbitrary normed algebra with an approximate identity, we introduce some notions of invertibility of a net in this algebra with respect to the given approximate identity. By means of that, we show a Wiener-Tauberian type theorem in that general situation. An application yields an abstract Wiener-Tauberian theorem for a class of commutative Banach algebras, which implies the classical Wiener-Tauberian theorem for locally compact Abelian groups.

We will find the $p\alpha-$dual for $X_{T}$, where $X$ is one of the spaces $c$, $c_{0}$, $\ell_{\infty}$ and $T$ is a triangle matrix. This will be achieved in two ways: firstly, under some conditions for the inverse matrix $S$ of $T$ and secondly, for arbitrary triangles $T$.

Let $H$ be an infinite dimensional complex Hilbert space and let $\phi$ be a surjective linear map on $B(H)$ with $\phi(I)-I\in{\mathcal{K}}(H)$, where $\mathcal{K}(H)$ denotes the closed ideal of all compact operators on $H$. If $\phi$ preserves the set of upper semi-Weyl operators and the set of all normal eigenvalues in both directions, then $\phi$ is an automorphism of the algebra $B(H)$. Also the relation between the linear maps preserving the set of upper semi-Weyl operators and the linear maps preserving the set of left invertible operators is considered.

A topological algebra A is said to be fundamental if there exists b > 1 such that for every sequence(xn ) in A, (xn ) is Cauchy whenever the sequence bn (xn − xn-1 ) tends to zero as n → ∞. Let A be a complex unital fundamental F-algebra with bounded elements such that A* separates the points on A. Then we prove that the spectrum σ(a) of every element a ∈ A is nonempty compact. Moreover, if A is a division algebra,then A is isomorphic to the complex numbers ℂ. This result is a generalization of Gelfand-Mazur theoremfor a large class of F-algebras, containing both locally bounded algebras and locally convex algebras withbounded elements.

In this paper, we introduce the new notions of $\phi$-biflatness, $\phi$-biprojectivity, $\phi$-Johnson amenability and $\phi$-Johnson contractibility for Banach algebras, where $\phi$ is a non-zero homomorphism from a Banach algebra $A$ into $\mathbb{C}$. We show that a Banach algebra $A$ is $\phi$-Johnson amenable if and only if it is $\phi$-inner amenable and $\phi$-biflat. Also we show that $\phi$-Johnson amenability is equivalent with the existence of left and right $\phi$-means for $A$. We give some examples to show differences between these new notions and the classical ones. Finally, we show that ${L^{1}(G)}$ is $\phi$-biflat if and only if $G$ is an amenable group and $A(G)$ is $\phi$-biprojective if and only if $G$ is a discrete group.

This paper is a continuation of [2], [9] and [10]. Its aim is to characterize non-removable ideals in the class ${\cal M}$ of all multiplicatively convex bornological algebras. Let $A$ be in ${\cal M}$. We show that an ideal $I\subset A$ is ${\cal M}$-non-removable if and only if it consists of joint m-bounding elements (D\'ef. 9). Such an ideal doesn't consist, in general, of joint bounding elements (D\'ef. 8). This shows that the concept of ${\cal M}$-non-removable ideals is of relative character (D\'ef. 4).

We show that a unital involutive Banach algebra, with identity of norm one and continuous involution, is a C*-algebra, with the given involution and norm, if every continuous linear functional attaining its norm at the identity is positive., 3 pages. Some typos corrected. Final version, to appear in Ann. Funct. Anal

The uniform norm on a uniform normed Q-algebra is the only uniform Q-algebra norm on it. The uniform norm on a regular uniform normed Q-algebra with unit is the only uniform norm on it. Let A be a uniform topological algebra whose spectrum M (A) is equicontinuous, then A is a uniform normed algebra. Let A be a regular semisimple commutative Banach algebra, then every algebra norm on A is a Q-algebra norm on A., Comment: This paper is accepted for publication in Annals of the Alexandru Ioan Cuza University - Mathematics

Let $S$ and $T$ be continuous linear operators on an ultraprime Banach algebra $A$. We show that if $S$, $T$, and $ST$ are close to satisfy the derivation identity on $A$, then either $S$ or $T$ approaches to zero. If $T$ is close to satisfy the derivation identity and $[T(a),a]$ is near the centre of $A$ for each $a \in A$, then either $T$ approaches to zero or $A$ is nearly commutative. Further, we give quantitative estimates of these phenomena.

$m$-infrabarrelledness, in the context of locally convex algebras, is considered to prove results previously obtained for barrelled algebras. Thus, any unital commutative $m$-infrabarrelled advertibly complete and pseudo-complete locally $m$-convex algebra with bounded elements has the $Q$-property; hence, it is functionally continuous (: all characters are continuous). In the framework of commutative $GB^{\ast }$-algebras with jointly continuous multiplication and bounded elements, the notions {\em $m$-infrabarrelled algebra} and {\em $C^{\ast }$-algebra} coincide. In unital uniform locally $m$-convex algebras, $m$-infrabarrelledness is equivalent to the Banach algebra structure, modulo pseudo-completeness. Moreover, $m$-infrabarrelledness for locally $A$-convex algebras (in particular, $A$-normed ones) is also examined.

Let $G$ be a locally compact group, $\omega$ be a weight function on $G$ and $0

The tensor product of two proper Hausdorff locally m-convex H*-algebras with continuous involution, endowed with the projective tensor product topology, along with its completion, are algebras of the same type with the factors. Under appropriate conditions, a canonical orthogonal basis is provided in the completion of the tensor product algebra. Based on this, the minimal closed 2-sided ideals are determined, yielding, in turn, the second Wedderburn structure theorem. Copyright © 2012 Rocky Mountain Mathematics Consortium.