M-convex Algebras Research Articles

Spectral operation in locally convex algebras

We show that if A is a spectrally bounded algebra, then all functions operate spectrally on A if and only if SpAx is finite for every x ∈ A. We also prove that if A is a commutative Q-l.m.c.a, then all functions operate spectrally on A if and only if A/RadA is algebraic. Furthermore, if A is a semi-simple commutative Q-l.m.c.a. which is a Baire space, all functions operate spectrally on A if and only if it is isomorphic to Cn for some n ∈ N. A structure result concerning semi-simple commutative complete m-convex algebras of countable dimension is also given.

Corrigendum and Addendum to “Advertibly complete locally m-convex two-sided algebras”

This corrigendum and addendum is written to correct the statement and proof of the main result of El Kinani et al. (Rend Circ Mat Palermo 56(2):185–197, 2007). We add Proposition 2.1 which states that every locally m-convex probilateral algebra is actually normal. Thus, in all results of El Kinani et al. (Rend Circ Mat Palermo 56(2):185–197, 2007), where the algebras considered are bilateral, the normality hypothesis is redundant.

On maximal ideals of codimension one in m-convex algebras

In Studia Math. 58:291–298, 1976, W. Żelazko gives a characterization of complex commutative complete unital m-convex algebras in which all maximal ideals are of codimension one. In these algebras every element has a bounded spectrum.

On smooth modules

In this note we introduce the notion of smooth module to extend results from homology theory of Banach algebras to the locally convex category. A complete locally convex module over an m-convex algebra is shown to be smooth if and only if it is topologically isomorphic to a reduced inverse limit of Banach modules over Banach algebras. Stability properties for smoothness are discussed and conditions under which an arbitrary locally convex module is rendered smooth are given.

Criteria of existence of bounded approximate identities in topological algebras

Some results and criteria of existence concerning bounded approximate identities in Banach algebras are extended to the topological algebras setting. We thereby prove that the bidual of a commutative locally C*-algebra with either of the two Arens products is a unital commutative algebra, and that a quasinormable Frechet m-convex algebra has a left (resp. right) bounded approximate identity if and only if it can be represented as an inverse limit of Banach algebras each of which has a left (resp. right) bounded approximate identity.

Integral representation for powers of elements in locally m-convex algebras and applications

This paper proves a new integral representation theorem for powers of elements in locally m-convex algebras. It is shown that the conditions handled in this theorem are satisfied for any sequentially complete locally m-convex Q-algebra. Various applications of this result concerning the spectrum, the spectral radius or the numerical radius are given.

Results of Vidav-Palmer Type in m-Convex Algebras

Using the notion of a normal cone, we give results of Vidav-Palmer type in terms of hermitian, quasi-hermitian and uniformly quasi-hermitian elements. Consequences are also given.

Inductive limits of locally m-convex algebras

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 [

Structure m-convexe dans l'espace à poids $L_\Omega ^p\left( R^n\right) $

We consider the space {\bf $L_\Omega ^p\left( R^n\right) ,$ }$1\leq p < +\infty ,$ where $\Omega $ is a family of weights. We give a necessary and sufficient condition, on $\Omega ,$ for {\bf $L_\Omega ^p\left( R^n\right) $ } to be locally m-convex algebra. Fundamental properties of this algebra are also obtained.

Topological algebras with ascending or descending chain condition

We examine some topological algebras with ascending or descending chain condition. We prove that a commutative noetherian F-algebra is necessarily a Q-algebra. We characterize noetherian F-algebras which are Q-algebras among those whose left ideals are closed. We show that any commutative artinian m-convex algebra is finite dimensional.

Entire functions and {$m$}-convex structure in commutative Baire algebras

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

On locally pseudoconvex square algebras

Let $A$ be an algebra over the field of complex numbers with a (Hausdorff) topology given by a family $\Cal{Q}= \{q_\lambda |\lambda\in\Lambda\}$ of square preserving $r_\lambda$-homogeneous seminorms ($r_\lambda \in (0,1]$). We shall show that $(A,T(\Cal{Q}))$ is a locally m-convex algebra. Furthermore we shall show that $A$ is commutative.

Differentiation in modules over topological ∗-algebras

In recent years, there has been a systematic development of the theory of A-bundles, that is, vector bundles where the field of coefficients is replaced by a topological algebra A, usually an involutive one. The above extension is justified by its applications in pure mathematics (differential topology [15]) as well as in relativistic quantum mechanics, via, e.g., the Serre-Swan theorem [22]. Bundles with coefficients in a commutative unital Banach algebra, or yet a C*-algebra, were considered by T. K. Kandelaki [6, 71, K. Fujii [S], and A. S. MiSEenko and Yu. P. Solov’ev [ 15, 161. Furthermore, A. Mallios studied A-bundles, with A being a locally m-convex (*-)algebra with unit [l&12]. Note also that certain results are still valid for more general topological algebras [ 13, Sect. 53, or even topological rings [21]. The results already obtained are mainly within the framework of topological A-bundles. The basic difficulty in the development of a theory of differential A-bundles is the lack of a suitable differential calculus in the libres, the latter being topological modules over A. One might, of course, use the existing methods of differentiation in the underlying topological vector spaces (cf. for instance [18]). However, besides the intrinsic weaknesses of these methods (regarding the higher order chain rule, continuity of differentiable maps, differentiability of the composition map and evaluation map, which are necessary for differential geometric considerations on bundles), the additional algebraic operations (scalar multiplication by elements of A, involution) should be taken into account, because they are essentially involved in the structure of A-bundles (cf. [ 11, 183). For the same reason (incompatibility with involution), the 255 0022-247X/92 55.00

Continuité des Caractères Dans les Algèbres de Fréchet à Bases

AbstractIn [1] and [2] T. Husain and J. Liang show the following results: RI. Every character on a Fréchet algebra with a Schauder basis (xi)i≧1 such that: (1) XiXj = XjXi = Xj if i ≦ j, (2) Pi(Xi) ≠ 0 and Pi(xi+1) = 0 (where (pi)i≧1 is a denumerable family of semi-norms defining the topology of the algebra) is continuous. R2. Every character on a Fréchet algebra with orthogonal and unconditional Schauder basis is continuous. The proofs of these last results are very long and introduce complex calculation without aid of spectral theory of locally ra-convex algebras. We give here short proofs of these results with aid of a characterization of elements of the spectrum in locally m-convex algebras with values of characters.

On locally m-convex algebras of type ES

On locally m-convex algebras of type ES

On Strong Spectrally Bounded LMC*-Algebras

A locally m-convex algebra E ≡ (E, (pα); α ∈ A) is called “strong spectrally bounded” if suppα (x) < ∞, for every x ∈ E. A locally m-convex C*-algebra of the last kind always accepts a faithful representation, say φ, with a continuous inverse, such that ‖‖φ(x)‖‖ = suppα, for al x ∈ E. Particularly, in case E is moreover barrelled, φ is also continuous, E becoming thus a *-subalgebra of a concrete C*-algebra ℒ︁(H), H a Hilbert space.

The automorphisms of convolution algebras of C∞ functions

We find the automorphisms and the spectra of several different topological convolution algebras of C ∞-functions on the real line. Starting with the convolution algebra of compactly supported C ∞-functions, equipped with the usual LF-topology, we define a corresponding convolution algebra of C ∞-functions of arbitrarily fast exponential decay at ∞; and convolution algebras of a given finite degree r of exponential decay at ∞. These algebras may be described topologically as “hyper Schwartz spaces.” With a natural Frechet topology, which we define, they get a structure as locally m-convex algebras. The continuous automorphisms and spectra of these algebras are described completely. We show that the algebra of C ∞-functions of infinitly fast exponential decay at ∞, H J , on the one hand, and the algebra of C ∞-functions of only a finite degree e −r¦x¦ decay at ∞, J r 0, on the other hand, have quite different automorphisms, although H J = ∩ r J r 0. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of J r 0, and that this representation does not extend to a representation on the Banach algebra L 1( R ).

Multiplicative Functionals on Frechet Algebras with Bases

Let A denote a complex (or real) Fréchet algebra (i.e. a complete metrizable locally m-convex algebra, see [2] or [3]). It is known [2] that the topology of such an algebra can be defined by an increasing sequence [qn] (i.e. qn(x) ≦ qn+i(x) for all x £ A and n ≧ 1) of submultiplicative (i.e. qn(xy) ≦ qn(x)qn(y) for all x, 3/ Ç ^4 and for each n ≧ I) seminorms.

On numerical ranges of operators on locally convex spaces

Numerical range theory for linear operators on normed linear spaces and for elements of normed algebras is now firmly established and the main results of this study are conveniently presented by Bonsall and Duncan in (1971) and (1973). An extension of the spatial numerical range for a class of operators on locally convex spaces was outlined by Moore in (1969) and (1969a), and an extension of the algebra numerical range for elements of locally m-convex algebras was presented by Giles and Koehler (1973). It is our aim in this paper to contribute further to Moore's work by extending the concept of spatial numerical range to a wider class of operators on locally convex spaces.

The spectral radius in a locally convex algebra

In (1), Allan introduced the concept of the spectrum of an element of a locally convex algebra, and developed a spectral theory for pseudo-complete algebras. In a commutative Banach algebra the spectral radius is a continuous seminorm, and so it is natural to investigate continuity properties of the spectral radius in various classes of locally convex algebra. Continuity is too strong a condition to be expected in any general case, and an interesting property to investigate appears to be lower semi-continuity. We shall show easily that in a commutative pseudo-complete locally m-convex algebra (in the sense of (4)) the spectral radius is lower semi-continuous. We shall then exhibit a commutative complete metrizable algebra in which lower semi-continuity fails to hold.