Symmetric topological*-algebras

V. Pták's inequality is valid for every hermitian completeQ locallym-convex (:l.m.c.) algebra. Every algebra of the last kind is, in particular, symmetric. Besides, a (Hausdorff) locallyC ^*-algebra (being always symmetric) with the propertyQ is, within a topological algebraic isomorphism, aC ^*-algebra. Furthermore, a type of Raikov's criterion for symmetry is also valid for non-normed topological^*-algebras. Concerning topological tensor products, one gets that symmetry of the-completed tensor product of two unital Fréchet l.m.c.^*-algebrasE, F ( denotes the projective tensorial topology) is always passed toE, F, while the converse occurs when moreover either ofE, F is commutative.     

Continuity of positive functionals on topological*-algebras

To investigate the continuity of positive functionals on certain topological^*-algebras, we consider general locally convex and non-locally convex topological^*-algebras which need not have identity or continuity of involution. We generalize a number of known results.The research of the first-named author was partially supported by an NSERC grant. The research of the second-named author was carried on when the was a post-doctoral fellow at McMaster University.     

Linear maps between C*-algebras that are *-homomorphisms at a fixed point

Let A and B be C*-algebras. A linear map T : A → B is said to be a *-homomorphism at an element z ∈ A if ab* = z in A implies T (ab*) = T (a)T (b)* = T (z), and c*d = z in A gives T (c*d) = T (c)*T (d) = T (z). Assuming that A is unital, we prove that every linear map T : A → B which is a *-homomorphism at the unit of A is a Jordan *-homomorphism. If A is simple and infinite, then we establish that a linear map T : A → B is a *-homomorphism if and only if T is a *-homomorphism at the unit of A. For a general unital C*-algebra A and a linear map T : A → B, we prove that T is a *-homomorphism if, and only if, T is a *-homomorphism at 0 and at 1. Actually if p is a non-zero projection in A, and T is a ^?-homomorphism at p and at 1 ? p, then we prove that T is a Jordan *-homomorphism. We also study bounded linear maps that are *-homomorphisms at a unitary element in A.     

Universal topological algebra needs closed topological categories

It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.     

Unbounded C*-Seminorms and Biweights on Partial *-Algebras

Unbounded C*-seminorms generated by families of biweights on a partial *-algebra are considered and the admissibility of biweights is characterized in terms of unbounded C*-seminorms they generate. Furthermore, it is shown that, under suitable assumptions, when the family of biweights consists of all those ones which are relatively bounded with respect to a given C*-seminorm q, it can be obtained an expression for q analogous to that one which holds true for the norm of a C*-algebra.     

LINEAR MAPPINGS THAT PRESERVE THE DERIVATIONAL STRUCTURE OF C*-ALGEBRAS

Abstract

Given a C*-algebra A and a suitable set of derivations on A, we consider the algebras A _n of n-differentiable elements of A as described in [B], before passing to an analysis of important classes of bounded linear maps between two such spaces. We show that even in this general framework, all the main features of the theory for the case C^(m)(U) → C ^(p) (V) where U and V are open balls in suitable Banach spaces, are preserved (see for example [A-G-L], [Gu-L], [Ja] and [L]). As part of the theory developed we obtain a non-trivial extension of the Kleinecke-Shirokov theorem in the category of C*-algebras to unbounded partially defined *-derivations. This indicates the existence of a single mathematical principle governing both the non-increasibility of differentiability by continuous homomorphisms and the untenability of the Heisenberg Uncertainty Principle for bounded observables.     

Extension theorems for Invertibility Symbols in Banach algebras

Necessary and sufficient conditions are found for the possibility of an extension of Invertibility Symbols from a dense subalgebra to its closure. It is proved that such an extension is possible if and only if the closure does not have any Kakutani elements. The extension of Invertibility Symbols from a Banach algebra to the matrix-valued case is also considered. Algebras with Polynomial Identities play an important role here. Applications to algebras generated by two idempotents and to algebras generated by singular integral operators are suggested.     

Extension theorems for Fredholm and invertibility symbols

This paper is a continuation of [GK3] where the theory of Invertibility Symbol in Banach algebras was developed. In the present paper we generalize these results for the case when the Invertibility Symbol is defined on a subalgebra of the Banach algebras. The difficulty which arises here in this more general case is connected with the fact that some elements of the subalgebra may have the inverses which do not belong to the subalgebra. This generalization of the theory allows us to study the Fredholm Symbols of linear operators. Applications to subalgebras generated by two idempotents and to algebras generated by singular integral operators are presented.     

Singularly perturbed functional-differential inclusions

Differential inclusions of a retarded type with a small real parameter >0 in part of the derivatives are considered. We prove upper semicontinuity of the map set of solutions at =0⁺ inC[0, 1]×(L ²(0, 1)–weak) topology. In case of constant delay lower semicontinuity inC[0, 1]×(L ¹(0, 1)–strong) is shown.     

Eigenvalue problems with fully discontinuous operators and critical exponents

We construct here more examples of -multivoque Leray-Lions operators that are strongly monotonic and we introduce a new abstract theorem on critical points for multivalued operators available for unbounded domains.     

A bisection algorithm for the numerical Mountain Pass

We propose a constructive proof for the Ambrosetti-Rabinowitz Mountain Pass Theorem providing an algorithm, based on a bisection method, for its implementation. The efficiency of our algorithm, particularly suitable for problems in high dimensions, consists in the low number of flow lines to be computed for its convergence; for this reason it improves the one currently used and proposed by Y.S. Choi and P.J. McKenna in [3]. Susanna Terracini: This work is partially supported by M.I.U.R. project “Metodi Variazionali ed Equazioni Differenziali Nonlineari”.     

On quadratic forms

The present paper is related to the Jordan—von Neumann characterization of inner product spaces, to the Halperin problem concerning quadratic forms, to some results of the present author on quadratic and sesquilinear forms and to recently obtained results of C. T. Ng and of J. Vukman.Dedicated to Professor Otto Haupt with best wishes on his 100th birthday.     

Aur la vacuité du spectre d'un élément d'une algèbre\mathcal{L}\mathcal{F}

We give an example of a complete commutative unitary and semi-simple topological algebra, which is a locally convex inductive limit of an increasing sequence of Fréchet algebras ( algebra), and which contains the field (X) of rational functions; so it contains elements which have empty spectrum and therefore does not contain any character, neither continuous nor non-continuous. This unitary algebra is not a division algebra, so it contains at least one non-trivial maximal ideal; but none of its maximal ideals is closed and they all have infinite codimension. The Gelfand-Mazur Theorem remains therefore unknown for algebras.