Projective Algebras Research Articles

Projective algebra and the calculus of relations

In [1] there were given postulates for an abstract “projective algebra” which, in the words of the authors, represented a “modest beginning for a study of logic with quantifiers from a boolean point of view”. In [5], D. Monk observed that the study initiated in [1] was an initial step in the development of algebraic versions of logic from which have evolved the cylindric and polyadic algebras.Several years prior to the publication of [1], J. C. C. McKinsey [3] presented a set of postulates for the calculus of relations. Following the publication of [1], McKinsey [4] showed that every projective algebra is isomorphic to a subalgebra of a complete atomic projective algebra and thus, in view of the representation given in [1], every projective algebra is isomorphic to a projective algebra of subsets of a direct product, that is, to an algebra of relations.Of course there has since followed an extensive development of projective algebra resulting in the multidimensional cylindric algebras [2]. However, what appears to have been overlooked is the correspondence between the Everett–Ulam axiomatization and that of McKinsey.It is the purpose of this paper to demonstrate the above, that is, we show that given a calculus of relations as defined by McKinsey it is possible to introduce projections and a partial product so that this algebra is a projective algebra and conversely, for a certain class of projective algebras it is possible to define a multiplication so that the resulting algebra is McKinsey's calculus of relations.

Cyclic Banach spaces and reflexive operator algebras

SynopsisLet X be a Banach space and let ℬ be a σ-complete Boolean algebra of projections on X with a cyclic vector. It is shown that there exists a normed Köthe space Lρ, the norm of which has the Fatou property, such that X is linearly homeomorphic to the subspace of Lρ consisting of those functions of absolutely continuous norm and such that, under this homeomorphism, the projections ℬ correspond to operators consisting of multiplication by characteristic functions. This representation theorem for X is used to show that certain operator algebras associated with ℬ are reflexive. As an immediate corollary of the reflexivity result, it is shown that, if T is a scalar type spectral operator whose resolution of the identity has a cyclic vector, then T is reflexive.

Injective and projective zero-dimensional compact universal algebras

Injective and projective zero-dimensional compact universal algebras

A relativity principle in quantum mechanics

Takeuti has studied models of axiomatic set theory in which the “truth values” are elements of a complete Boolean algebra of projections on closed subspaces of a Hilbert space, and has found that the real numbers of such a model can be taken to be self-adjoint operators which can be resolved in terms of projections belonging to the Boolean algebra. It is suggested that this is the mathematical source of the replacement of real quantities by operators in quantizing a classical description, and that quantum theory involves a relativity principle with Takeuti's Boolean algebras serving as reference “frames.”

On Boolean algebras of projections

In spectral theory on Banach spaces, certain more incisive results hold when the underlying space is weakly complete (that is, weakly sequentially complete). The standard proofs rely on the following deep theorem: any bounded linear map from the algebra of all complex continuous functions on a compact Hausdorff space to a weakly complete Banach space is weakly compact. The proof of this result depends in turn on a considerable amount of measure-theoretic machinery (see [4, Section VI.7]). We present here some alternative methods which avoid these technicalities. The results are then used to give an example of a set of projections, each having unit norm, which generate an unbounded Boolean algebra.

A direct sum decomposition for the Brauer group of H-module algebras

Let R be a commutative ring with unit, H a finitely generated projective, commutative and cocommutative Hopf algebra over R. In [3], Long defined a Brauer group BM(R, H) whose elements are equivalence classes of R- Azumaya H-module algebras. In this note, we show that there is a split exact sequence 1 -+ R(R) + BM(R, H) + Gal(R, H) + 1, where Gal(R, H) is the group of Galois H-objects over R, defined in [l], and B(R) is the ordinary Brauer group of R. This sequence generalizes that obtained in [5] for graded algebras, i.e., for H = GR, the dual of the group ring RG.

Quasi-Frobenius-Algebren und lokal vollst�ndige Durchschnitte

Finite projective algebras B over arbitrary commutative rings A are discussed with respect to dualizing B-modules over B and over A. This leads to quasi-Frobenius algebras in a natural way. Several change of rings characterizations of quasi-Frobenius algebras are given. Secondly, locally complete intersections and complete intersections are considered unter the point of view of quasi-Frobenius algebras resp. Frobenius algebras. Characterizations of complete intersections are obtained by using algebraic K-theory and differential modules. Some applications refer to representations of affine curves as idealtheoretic complete intersections.

Eine Charakterisierung der Boole'schen Algebra zu einer co -direkten Zerlegung

In this paper nuclear Boolean Algebras of projections in a locally convex space are considered. This are Boolean Algebras with special continuity properties, which are shared, for instance, by each bounded Boolean Algebra of projections in an ℒ∞-space and by the algebra of each equicontinuos spectral measure in a nuclear space. It will be shown that a ℴ-complete nuclear Boolean Algebra leads to a co-direct sum of locally convex spaces and all the projections of the algebra belong to the complete algebra of projections of this co-direct partition. On the other hand if in a given locally convex space E there exists a nuclear complete Boolean Algebra of projections which has multiplicity one then each equicontinuos Boolean Algebra of projections in E is nuclear.

Boolean algebras of projections

Bade, in (1), studied Boolean algebras of projections on Banach spaces and showed that a σ-complete Boolean algebra of projections on a Banach space enjoys properties formally similar to those of a Boolean algebra of projections on Hilbert space. (His exposition is reproduced in (7: XVII).) Edwards and Ionescu Tulcea showed that the weakly closed algebra generated by a σ-complete Boolean algebra of projections can be represented as a von Neumann algebra; and that the representation isomorphism can be chosen to be norm, weakly, and strongly bicontinuous on bounded sets (8): Bade's results were then seen to follow from their Hilbert space counterparts. I show here that it is natural to relax the condition of σ-completeness to weak relative compactness; indeed, a Boolean algebra of projections has σ-completion if and only if it is weakly relatively compact (Theorem 1). Then, following the derivation of the theorem of Edwards and Ionescu Tulcea from the Vidav characterisation of abstract C*-algebras (see (9)), I give a result (Theorem 2) which, with its corollary, includes (1: 2.7, 2.8, 2.9, 2.10, 3.2, 3.3, 4.5).

A K-theory for the category of projective algebras

A K-theory for the category of projective algebras

ORBITS OF THE GROUP $ \operatorname{GL}\bigl(r,k \lbrack X_1,\dots,X_n\rbrack\bigr)$

In this paper it is shown that the study of projective metabelian Lie algebras of finite rank reduces to a partial solution of Serre's problem on projective modules over polynomial rings. It is also observed that projective commutative-associative algebras of dimension 1 are isomorphic to the ring of polynomials in one variable over the ground field.

Connections between projective geometry and superassociative algebra

This paper establishes a connection between projective geometry and the superassociative algebra of multiplace functions. In Part I a quaternary operation is defined for the points on a line j in a projective plane π relative to a fixed quadrangle and the result of operating on A,B,C,D is denoted by A(B,C,D). It is shown that π is a Pappus plane if and only if this operation is superassociative: (A(B,C,D))(E,F,G)=A(B(E,F,G),C(E,F,G),D(E,F,G)) for any points A,B,C,D,E,F,G on j. Desargues and certain special Desargues planes are also characterized by restrictions of the superassociative law. In Part II projective geometry is developed over superassociative systems.

Some classes of projective Boolean algebras

Denote by P the class of projective Boolean algebras and by K1 the class of those Boolean algebras which are isomorphic to a product of a finite set of algebras each of which is a coproduct of countable algebras. It is well-known that K 1 = P; answering a question of Halmos, we showed in [3] that Kl is a proper subclass of P. This result is improved in this paper in several ways. We define a notion of weak product of Boolean algebras and show that P is closed with respect to countable weak products. Let K (resp. L) be the smallest class of Boolean algebras containing every countable algebra and being closed with respect to coproducts and finite products (resp. coproducts, finite products and countable weak products). We shall show that K~ is a proper subclass of K, K is a proper subclass of L and L is a proper subclass of P. The algebra proving L :k P is the same as constructed for the counterexample in [3]; the algebras working for K1 4: K and K ~: L are much simpler examples proving K 1 # P. After some preliminaries in Chapter 1, we show K 1 :~ K in Chapter 2. Weak products are introduced in Chapter 3. In Chapter 4, we prove that for every A e K there is a B ~ K1 such that A and B have isomorphic completions; this yields K + L . Finally, L~: P is derived in Chapter 5.

Algebras Over Dedekind Domains

The purpose of this paper is two-fold : first, to show that Dedekind domains satisfy a generalization of the Wedderburn-Mal'cev Theorem and, secondly, to classify certain types of finitely generated projective algebras over a Dedekind domain.With respect to the first problem, E. C. Ingraham has shown that a Dedekind domain R is an inertial coefficient ring (IC-ring) if and only if R has zero radical or R is a local Hensel ring.

Remark about the Boolean parts in the postulate-systems of closure, derivative and projective algebras.

Remark about the Boolean parts in the postulate-systems of closure, derivative and projective algebras.

The Separable Closure of Some Commutative Rings

The separable closure of a commutative ring with an arbitrary number of idempotents is defined and its Galois theory studied. Projective separable algebras over the ring are shown to be determined by the ’Galois groupoid’ of the closure. The existence of the closure is demonstrated for certain rings.

On the cohomology of modules over Hopf algebras

Let R be a commutative ring. Define an FH-algebra H to be a Hopf algebra and a Frobenius algebra over R with a Frobenius homomorphism ψ such that ∑ ( h) h (1) ψ( h (2)) = ψ( h) · 1 for all h ϵ H. This is essentially the same as to consider finitely generated projective Hopf algebras with antipode. For modules over FH-algebras we develop a cohomology theory which is a generalization of the cohomology of finite groups. It generalizes also the cohomology of finite-dimensional restricted Lie algebras. In particular the following results are shown. The complete homology can be described in terms of the complete cohomology. There is a cup-product for the complete cohomology and some of the theorems for periodic cohomology of finite groups can be generalized. We also prove a duality theorem which expresses the cohomology of the “dual” of an H-module as the “dual” of the cohomology of the module. The last section provides techniques to describe under certain conditions the cohomology of H by the cohomology of sub- and quotient-algebras of H. In particular we have a generalization of the Hochschild-Serre spectral sequence for the cohomology of groups.

PROJECTIVE METABELIAN LIE ALGEBRAS OF FINITE RANK

In this paper we study the connection between free projective metabelian Lie algebras of finite rank and Serre's problem. We prove that projective metabelian Lie algebras of rank two are free.

On Inflation and Torsion of Amitsur Cohomology

Studies of torsion [2] and inflation [11] of Amitsur cohomology have primarily been concerned with module-finite faithful projective algebras. In this paper, our goal is to consider these topics for more general algebras. The fundamental tool, in case R is a domain with quotient field K, is the functor UK/U (defined in § 1), together with the monomorphic connecting map of Amitsur cohomology H1(S/R, UK/U) → H2(S/R, U) arising from a (commutative) flat R-algebra S and the unit functor U. This map was first considered in [10, Chapter IV, Theorem 1.6], where it was proved to be an isomorphism for certain étale faithfully flat algebras S in case R is an algebraic number ring with trivial Brauer group. In Corollary 1.5 it is shown, in the case of a module-finite faithful projective S over a regular domain R, that the connecting map is the kernel of the canonical homomorphism [8] from H2(S/R, U) to the split Brauer group B(S/R). As the latter map is often injective [8, Corollary 7.7], one might expect instances of vanishing of H1(S/R, UK/U) without assuming both R Noetherian and S module-finite R-projective. Much of § 1 (viz. (1.7)-(1.9)) is devoted to such examples.

Compact Induced Representations

In [15; 16; 17], Horst Leptin introduced what he called generalized group algebras. These Banach *-algebras are formed by letting a locally compact group G act on a Banach *-algebra A both by *-automorphisms and by a cocycle with values in the multiplier algebra, M (A ), of A. We will review the precise construction later, but for now we remark that examples include the group algebra of a group extension, the covariance algebras of quantum field theory, the “projective group algebras” of a group G (that is, for each complex-valued cocycle λ, called a multiplier in the literature, the Banach *-algebra whose nondegenerate *-representations are in bijective correspondence with the λ-projective representations of G), and the twisted group algebras of Edwards and Lewis [8; 9].