Projective Algebras Research Articles

The Structure of Galois Algebras

Let B be a ring with 1 and G an automorphism group of B of order n for some integer n. It is shown that if B is a Galois algebra with Galois group G, then B is either a direct sum of central Galois algebras or a direct sum of central Galois algebras and a commutative Galois algebra. Moreover, when G is inner, B is either a direct sum of Azumaya projective group algebras or a direct sum of Azumaya projective group algebras and a commutative Galois algebra. Examples are given for these structures.

Purity and Equational Compactness of Projection Algebras

The notions of purity and equational compactness of universal algebras have been studied by Banaschewski and Nelson. Also, Banaschewski deals with these notions in the special case of G-sets for a group G. In this paper we study these and related concepts in the category PRO of projection algebras, that is in N∞-sets, for the monoid N∞ with the binary operation m.n=min {m,n}. We show that every monomorphism in PRO is pure and hence every equationally compact projection algebra is in fact injective. Then, we introduce the notions of s-purity and s-compactness by which we characterize the retractions and hence equationally compact projection algebras. And, among other results, we show that equationally compact, injective, and complete projection algebras are the same. Finally, we characterize (pure-)essential monomorphisms and construct the Equationally Compact Hulls, equivalently the Injective Hulls, of projection algebras. These results, among other things, generalize the main results of Guili, regarding completeness and s-injectivity in the category PROs of separated projection algebras.

The crossed product theorem for projective Schur algebras

The projective Schur group of a commutative ring was introduced by Lorenz and Opolka. It was revived by Nelis and Van Oystaeyen, and later by Aljadeff and Sonn. In this paper we study the intriguing question that there seems to be no adequate version of the crossed product theorem for the projective Schur group. We present a radical group R(k) (k a field) situated between the Schur group and the projective Schur group, and we prove the crossed product theorem for R(k).

Exponent Reduction for Radical Abelian Algebras

Let k be a field. A radical abelian algebra over k is a crossed product (K/k,α), where K=k(T) is a radical abelian extension of k, T is a subgroup of K* which is finite modulo k*, and α∈H2(G,K*) is represented by a cocycle with values in T. The main result is that if A is a radical abelian algebra over k, and m=exp(A⊗kk(μ)), where μ denotes the group of all roots of unity, then k contains the mth roots of unity. Applications are given to projective Schur division algebras and projective Schur algebras of nilpotent type.

Solutions and Ambiguities of the Structure and Motion Problem for 1DRetinal Vision

In this paper we investigate the geometry and algebra of multiple one-dimensional projections of a two-dimensional environment. This is relevant for one-dimensional cameras, e.g. as used in certain autonomous guided vehicles. It is also relevant for understanding the projection of lines in ordinary vision. A third application is on ordinary vision of vehicles undergoing so called planar motion. The structure and motion problem for such cameras is studied and the two possible minimal cases is solved. The technique of solving these questions reveal interesting ambiguities. It is shown that for each solution with three images there is an ambiguous solution. It is also shown that for each solution for four points there is an ambiguous solution. The connection between these two different types of ambiguities is also given. Although the paper deals with calibrated cameras, it is shown that similar results exist for uncalibrated cameras.

Projective Schur Algebras of Nilpotent Type Are Brauer Equivalent to Radical Algebras

Let k be a field and G a finite group . For any element α ∈ H(G, k∗) (k∗ denotes the units of k , here as a trivial G-module) we can form the twisted group algebra kG. It is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xuσyuτ = xyf(σ, τ)uστ where f : G × G → k∗ is a 2-cocycle representing α. It is easily checked that the 2-cocycle condition satisfied by f insures the associativity of the multiplication in kG. A projective Schur algebra over k is by definition a k-central simple algebra which is a homomorphic image of a twisted group algebra kG for some finite group G and some α ∈ H(G, k∗). Equivalently, a k− central simple algebra A is a projective Schur algebra (over k) if it contains a subgroup Γ of the units of A, which is finite modulo k∗ and spans A as k-vector space. We often use the notation k(Γ). Clearly a projective Schur algebra over k determines an element in the Brauer group Br(k) of the field k . The subgroup in Br(k) generated by (and in fact consisting of) such elements is called the projective Schur group of k and denoted by PS(k). The notions of projective Schur algebra and projective Schur group were introduced in [6] by Lorenz and Opolka. This construction can be viewed as the projective analogue of Schur algebras and the Schur group (denoted by S(k)) of the field k. (See [10]) It is natural to restrict attention to special types of groups G. Our interest here is in k-central simple algebras that are homomorphic images of twisted group algebras kG with G nilpotent (resp. abelian) (see [6]). We call these algebras “projective Schur algebras of nilpotent type (resp. abelian type)”. Clearly, as above, these algebras represent classes in Br(k) and we denote by PNil(k) ( resp. PAb(k)) the subgroup they generate. In [6] it is shown that if k is a number field and n is the number of roots of unity in k then PAb(k) = nBr(k), where nB denotes the n-torsion subgroup of B. More generally, since a symbol algebra of degree n over k is a homomorphic

Morphisms of Relative Hopf Modules, Smash Products, and Duality

The aim of this paper is to extend a result of H.-J. Schneider on endomorphism rings of relative Hopf modules [17, Theorem 3.2] to the case of projective Hopf algebras with bijective antipode, and to show the connection between this extension and several duality theorems for Hopf algebras. A consequence of this generalization is the following: if H is a co-Frobenius Hopf algebra over the field k, A/B is a Hopf-Galois extension and M is a right (A,H)-Hopf module, then, considering the usual ring embeddings

The dual bade theorem in locally convex spaces and reflexivity of a closed unital subalgebra

The results presented in this paper extend a dual version of the reflexivity theorem of W. Bade to locally convex spaces. Dual versión of the Bade theorem in a Banach C(K)-module was firstly discovered in [1]. It is our aim to extend it to a locally convex C(K)-module. As a consequence, it is proven that each unital w* operator topology closed subalgebra of the w* operator topology closed algebra generated by a Boolean algebra of projections is reflexive.

Unification in intuitionistic logic

AbstractWe show that the variety of Heyting algebras has finitary unification type. We also show that the subvariety obtained by adding it De Morgan law is the biggest variety of Heyting algebras having unitary unification type. Proofs make essential use of suitable characterizations (both from the semantic and the syntactic side) of finitely presented projective algebras.

The strong closure of $\sigma$-complete Boolean algebras of projections

A result of T. A. Gillespie implies that the strong operator closure of any abstractly $\sigma $ -complete Boolean algebra of projections in a Banach space X which does not contain a copy of c 0 is Bade complete. It is shown that the same conclusion is valid for another (extensive) class of Banach spaces X, namely those which are weakly compactly generated. As a consequence, it follows that a Boolean algebra of projections in a separable Banach space is abstractly $\sigma $ -complete iff it is abstractly complete. It is also shown that a Banach space X has the property that the strong closure of every abstractly complete Boolean algebra of projections in X is Bade complete iff X does not contain a copy of $\ell ^\infty \!$ .

Criteria for Closedness of Spectral Measures and Completeness of Boolean Algebras of Projections

The intimate connection between spectral measures and σ-complete Boolean algebras of projections in Banach spaces was intensively investigated by W. Bade in the 1950s, [“Linear Operators III: Spectral Operators,” Wiley-Interscience, New York, 1971; Chap. XVII]. It is well known that the most satisfactory situation occurs when the Boolean algebra is actually complete rather than just σ-complete. This makes it desirable to have available criteria (not just in Banach spaces but also in the nonnormable setting) which can be used to determine the completeness of Boolean algebras of projections. Such criteria, which should be effective and applicable in practice, as general as possible and apply to extensive classes of spaces, are presented in this article. A series of examples shows that these criteria are close to optimal.

PROJECTIVE GROUP ALGEBRAS

In this paper we apply a recently proposed algebraic theory of integration to projective group algebras. These structures have received some attention in connection with the compactification of the M theory on noncommutative tori. This turns out to be an interesting field of applications, since the space [Formula: see text] of the equivalence classes of the vector unitary irreducible representations of the group under examination becomes, in the projective case, a prototype of noncommuting spaces. For vector representations the algebraic integration is equivalent to integrate over [Formula: see text]. However, its very definition is related only at the structural properties of the group algebra, therefore it is well defined also in the projective case, where the space [Formula: see text] has no classical meaning. This allows a generalization of the usual group harmonic analysis. Particular attention is given to Abelian groups, which are the relevant ones in the compactification problem, since it is possible, from the previous results, to establish a simple generalization of the ordinary calculus to the associated noncommutative spaces.

On the bicommutant theorem for σ-complete Boolean algebras of projections in Banach spaces

On the bicommutant theorem for σ-complete Boolean algebras of projections in Banach spaces

Embedding the Hopf Automorphism Group into the Brauer Group

AbstractLet H be a faithfully projective Hopf algebra over a commutative ring k. In [8, 9] we defined the Brauer group BQ(k, H) of H and an homomorphism π from Hopf automorphism group AutHopf(H) to BQ(k,H). In this paper, we show that the morphism π can be embedded into an exact sequence.

Projective schur algebras over a field of positive characteristic

If the characteristic of a field K is not zero then the Schur group S(K) = 0. In this paper we ask a similar question for the projective Schur group PS(K) and prove that the subgroup of PS(K) consisting of radical algebras is trivial. This disproves the conjecture that every projective Schur algebra is similar to a radical algebra.

Boolean algebras of projections and resolutions of the identity of scalar-type spectral operators

Let Μ be a Bade complete (or σ-complete) Boolean algebra of projections in a Banach space X. This paper is concerned with the following questions: When is Μ equal to the resolution of the identity (or the strong operator closure of the resolution of the identity) of some scalar-type spectral operator T (with σ(T) ⊆ ℝ) in X? It is shown that if X is separable, then Μ always coincides with such a resolution of the identity. For certain restrictions on Μ some positive results are established in non-separable spaces X. An example is given for which Μ is neither a resolution of the identity nor the strong operator closure of a resolution of the identity.

Boundedness Criteria for Boolean Algebras of Projections

Let E and F be two commuting bounded Boolean algebras of projections on a Banach spaceX. Various geometric conditions onXare established which ensure that the Boolean algebra of projections generated by E and F is also bounded. In particular, this is the case ifXis a Banach lattice or a subspace of ap-concave Banach lattice, where 1⩽p<∞. Several consequences of this boundedness property are also discussed.

The Sequential Closedness of σ-Complete Boolean Algebras of Projections

The Sequential Closedness of σ-Complete Boolean Algebras of Projections

Projective fourier duality and Weyl quantization

The Weyl-Wigner correspondence prescription, which makes large use of Fourier duality, is reexamined from the point of view of Kac algebras, the most general background for noncommutative Fourier analysis allowing for that property. It is shown how the standard Kac structure has to be extended in order to accommodate the physical requirements. An Abelian and a symmetric projective Kac algebras are shown to provide, in close parallel to the standard case, a new dual framework and a well-defined notion of projective Fourier duality for the group of translations on the plane. The Weyl formula arises naturally as an irreducible component of the duality mapping between these projective algebras.

Existence of Bade functionals for complete Boolean algebras of projections in Fréchet spaces

A classical result of W. Bade states that if M is any a-complete Boolean algebra of projections in an arbitrary Banach space X then, for every xo G X, there exists an element x' (called a Bade functional for xo with respect to M) in the dual space X', with the following two properties: (i) M F-* (Mxo, x') is non-negative on M and, (ii) Mxo = 0 whenever M E M satisfies (Mxo, x') = 0. It is shown that a Frechet space X has this property if and only if it does not contain an isomorphic copy of the sequence space w = CN. A Boolean algebra M of selfadjoint projections in a Hilbert space H has the property that for every x0 C H the inner product (Exo, x0), with E C M, is nonnegative and vanishes only if Exo = 0. A satisfactory extension to the Banach space setting of this useful property of the inner product in Hilbert spaces is the following remarkable result of W. Bade, [1, Theorem 3.1]. Theorem 1. Let M be a a-complete Boolean algebra of projections in a Banach space X. Then, for each xo C X, there exists a continuous linear functional x' E X' (called a Bade functional for x0 with respect to M) with the properties (i) (Mxo, x') > 0, for all M C MA, and (ii) if (Mxo, x') = 0 for some M c M, then Mx0 = 0. Theorem 1 fails to be true in the non-normable setting, even in Frechet (locally convex) spaces. Indeed, the Boolean algebra M generated by the co-ordinate projections in the Frechet space w of all complex sequences, equipped with the topology of co-ordinatewise convergence, is u-complete but the element x0 = (1, 1, . . . ) cannot have any Bade functional with respect to M since the continuous dual space w' consists of all complex sequences which have only finitely many non-zero terms. Let M be a a-complete Boolean algebra of projections in a locally convex Hausdorff space X. Then M is said to have Property-(B) if, for every x0 E X, there exists x' E X' satisfying (i) and (ii) of Theorem 1. The space X is said to have the Bade property if every a-complete Boolean algebra of projections in X has Property-(B). Bade's classical theorem above asserts that every Banach space has the Bade property. The above example shows that Frechet spaces in general do Received by the editors March 4, 1996. 1991 Mathematics Subject Classification. Primary 47B15, 46G10, 47C05. (?)1997 American Mathematical Society