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 ∞, $\text{H J}$, on the one hand, and the algebra of C^∞-functions of only a finite degree $\text{e}{}^{\mathrm{?r¦x¦}}$ decay at ∞, $\text{J}$_r⁰, on the other hand, have quite different automorphisms, although $\text{H J}$ = ∩_r$\text{J}$_r⁰. As an application, we show that the conformal group is canonically represented as the full group of automorphisms of $\text{J}$_r⁰, and that this representation does not extend to a representation on the Banach algebra L¹($\text{R}$).     

Fuzzy subobjects in a category and the theory of C-sets

In this paper we shall define the concept of a fuzzy subobject of an object in arbitrary categories. This concept is generated by the representation theorem of fuzzy sets. By using fuzzy subobjects one can include most of the fuzzy concepts defined in the literature, such as: fuzzy groups, fuzzy relations and fuzzy convex sets. In the second part of the paper we shall define a new concept; that of a $\text{C}$-set. This concept will generalize that of a fuzzy set and we shall also prove that $\text{C}$-sets can be represented by some sets of functors. More precisely, $\text{C}$-sets form a category which can be represented by a category of functors. The utility of $\text{C}$-sets resides in the fact that one can replace “ordering” by the more general concept of a morphism in category. The new representation of $\text{C}$-sets is weaker than that of fuzzy sets.     

Fuzzy topological spaces and fuzzy compactness

It is the purpose of this paper to go somewhat deeper into the structure of fuzzy topological spaces. In doing so we found we had to alter the definition of a fuzzy topology used up to now. We shall also introduce two functors $\text{}\text{?}$gw and $\text{}\text{?}$gi which will allow us to see more clearly the connection between fuzzy topological spaces and topological spaces. Finally we shall introduce the concept of fuzzy compactness as the generalization of compactness in topology. It will be shown in a following publication that contrary to the results obtained up to now, the Tychonoff-product theorem is safeguarded with fuzzy compactness.     

The relationship between a commutative Banach algebra and its maximal ideal space

Our main result is an extension of a theorem due to Novodvorskii and Taylor; we give some special cases. Let A be a commutative Banach algebra with identity, and let Δ be its maximal ideal space. Let B be a Banach algebra with identity; let B^?1 denote the invertible group in B and id B denote the set of idempotents in B. Let [$\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] denote the set of path components of $\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$, and [Δ, B^?1] denote the set of homotopy classes of continuous maps of Δ into B^?1. We prove that the Gelfand transform on A induces a bijection of [$\text{(A}\text{}\text{?}\text{bo B)}{}^{\mathrm{?1}}$] onto [Δ, B^?1], and extend this result to prove a theorem of Davie. We show that the Gelfand transform induces a bijection of [$\text{id}\text{(A}\text{}\text{?}\text{bo B)}$] onto [Δ, id B], and investigate consequences of this result for specific examples of the Banach algebra B.     

On the generators of semisimple lie algebras

It will be shown that given any element X in a simple Lie algebra $\text{Q}$ over C, there exists a Y ∈ $\text{Q}$ such that the Lie algebra generated by X and Y is $\text{Q}$. The result is extended to the real semisimple Lie algebras. In some sense the main theorem of this paper can be regarded as an extension of Morozov-Jacobson theorem concerning three dimensional simple Lie algebras (see the remark at the end of Sec. 4). A new property of a special class of regular elements, known as the cyclic elements, is given.     

Codes projectifs a deux ou trois poids associfs aux hyperquadriques d'une geometrie finie

The notion of closure structures of finite rank is introduced and several closure structures familiar from algebra and logic are shown to be of finite rank. The following theorem is established: Let k be any natural number and $\text{C}$ any closure structure of rank k + 2. If B is a finite base (generating set) of $\text{C}$and D is an irredundant (independent) base of $\text{C}$ such that |B| + k < |D|, then there is an irredundant base E of $\text{C}$ such that |B| < |E| < |B| + k.     

Prime ideals and automatic continuity problems for Banach algebras

Conditions are given for Banach algebras $\text{U}$ and commutative Banach algebras $\text{B}$ which insure that every homomorphism v from $\text{U}$ into $\text{B}$ is continuous. Similar results are obtained for derivations which either map the algebra $\text{U}$ into itself or map the algebra into a suitable $\text{U}$-module.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.     

Reflexive algebras with finite width lattices: Tensor products,cohomology, compact perturbations

Reflexive algebras play a central role in the study of general operator algebras. For a reflexive algebra the associated invariant subspace lattice has structural importance analogous to that of the algebraic commutant in the study of ¹-algebras. Tomita's tensor product commutation theorem can be restated in the form Alg($\text{L}$₁ ? $\text{L}$₂) = Alg $\text{L}$₁ ? Alg $\text{L}$₂, where each $\text{L}$_i is a reflexive ortho-lattice. This same formula is proved (for n-fold tensor products) in the setting when each $\text{L}$_i is a nest. Thus, in particular, a tensor product of nest algebras is again a reflexive algebra. Lance has shown that the Hochschild cohomology of nest algebras vanishes; modifications of his arguments show that cohomology vanishes for arbitrary CSL algebras whose lattices are generated by finitely many independent nests. This appears to be the strongest possible result in this direction. The class of irreducible tridiagonal algebras with finite-width commutative lattices is investigated and it is shown that these algebras have nontrivial first cohomology. Finally, it is shown that if $\text{L}$ is a finite-width commutative subspace lattice and $\text{K}$ is the set of compact operators then the quasitriangular algebra Alg $\text{L}$ + $\text{K}$ is closed in the norm topology. This extends to arbitrary finite-width CSL algebras a result obtained for nest algebras by Fall, Arveson, and Muhly.     

Compact G-Fuzzy topological spaces

Considering complete Boolean algebras $\text{G}$ as sets of truth values a new concept of compactness—so-called probabilistic compactness — is introduced to $\text{G}$-fuzzy topological spaces. The aim of this paper is to show that the most important theorems of the theory of ordinary compact spaces remain true; e.g. probabilistic compactness is preserved under projective limits, every probabilistic compact space has an unique $\text{G}$-fuzzy uniformity being compatible with the underlying $\text{G}$-fuzzy topology, etc. Finally using the selection theorem due to Kuratowski and Ryll-Nardzewski a non-trivial example of a probabilistic compact space is given.     

Analyticity and flows in von Neumann algebras

Let $\text{B}$ be a von Neumann algebra, let {α_t}_tεR be an ultraweakly continuous one-parameter group of ¹-automorphisms of $\text{B}$, and let $\text{U}$ be the set of all A such that for each ? in $\text{B}$₁, the function t → ?(α_t(A)) lies in H^∞($\text{R}$. Then $\text{U}$ is an ultraweakly closed subalgebra of $\text{B}$ containing the identity which is proper and non-self-adjoint if {α_t}_tεR is not trivial. In this paper, a systematic investigation into the structure theory of $\text{U}$ is begun. Two of the more note-worthy developments are these. First of all, conditions under which $\text{U}$ is a subdiagonal algebra in $\text{B}$, in the sense of Arveson, are determined. The analysis provides a common perspective from which to view a large number of hitherto unrelated algebras. Second, the invariant subspace structure of $\text{U}$ is determined and conditions under which $\text{U}$ is a reductive subalgebra of $\text{B}$ are found. These results are then used to produce examples where $\text{U}$ is a proper, non-self-adjoint, reductive subalgebra of $\text{B}$. The examples do not answer the reductive algebra question, however, because although ultraweakly closed, the subalgebras are weakly dense in $\text{B}$.     

Spectral measures and the Bade reflexivity theorem

Let $\text{B}$ be a strongly equicontinuous Boolean algebra of projections on the quasi-complete locally convex space X and assume that the space L(X) of continuous linear operators on X is sequentially complete for the strong operator topology. Methods of integration with respect to spectral measures are used to show that the closed algebra generated by $\text{B}$ in L(X) consists precisely of those continuous linear operators on X which leave invariant each closed $\text{B}$-invariant subspace of X.     

Interpolation by type I factors and the flip automorphism

Given two von Neumann algebras $\text{A}$ ?$\text{B}$ we study the relation between the existence of an interpolating type I factor $\text{F}$, namely $\text{A}$ ?$\text{F}$ ?$\text{B}$, the implementability of the flip automorphism of $\text{A}$ ? $\text{A}$ by a unitary in $\text{B}$ ? $\text{B}$, and the statistical independence of $\text{A}$ and $\text{B}$′ ($\text{A}$ and $\text{B}$′ generate a $\text{W}$¹-tensor product). As an application in Q.F.T. we derive in a natural way a structure theorem of Buchholz for the von Neumann algebras of local observables associated to free fields.     

Descriptions of equivalence of O-connectivity

We first study some properties of the subspace, and investigate into the relationship of separation between a fuzzy topological space (fts) and its subspace. Then we obtain the equivalence conditions for O-connectivity. The results on O-connectivity and separation are very similar to those in general topology. Finally we discuss the relationship of connectivity between an O-connected set A in the fts (X, ω ($\text{T}$)) induced by the crisp topological space (X, $\text{T}$) and the crisp set A₀ (=supp A) in (X, $\text{T}$).     

Integration theory for Hammerstein operators

We develop in this article a strong nonlinear integral and obtain a Riesz-type theorem (utilizing this integral) for the class of (nonlinear) Hammerstein operators. The integral is extended to the class $\text{M}$_E($\text{B}$) of E-valued totally $\text{B}$-measurable functions and convergence theorems are studied. Then an exchange of information is carried out between the operators and the corresponding set functions; for example, the implication of the operator being compact or unconditionally summing is drawn. In the latter case it is shown that the representing set function is analogous to strongly bounded set functions. A vast body of literature exists for both of these concepts.     

Remarks on contact relations on Boolean algebras

The paradigmatic example for a contact relation is that of having nondisjoint closures for regions in n-dimensional space. Generally, a contact relation is a binary relation on a Boolean algebra reflecting some of the properties of this example. We prove in this paper a topological and an algebraic representation theorem for contact algebras and solve two problems due to Düntsch and Winter and to Dimov and Vakarelov, respectively.     

Unbounded operators generated by a given spectral measure

Let T be a closed densely-defined operator on a Banach space X and let E(·) be a spectral measure whose range $\text{E}$ is a complete Boolean algebra of projections in X. Then T is of the form ∝f(λ) dE(λ) if and only if T commutes with $\text{E}$ and leaves invariant every invariant subspace of $\text{E}$.     

More game-theoretic properties of boolean algebras

The following infinite game $\text{G}$ was investigated in [5]: Let B be a Boolean algebra. Two players, White and Black, take turns to choose successively a sequence