Diagonal Algebras Research Articles

Quasi-diagonal C∗-algebras

We consider perturbations of C ∗-algebras by compact operators. We show that if A is a separable liminal algebra of operators on a separable Hilbert space, then it is a subalgebra of a compact perturbation of a block diagonal algebra.

An effective fixed-point theorem in intuitionistic diagonalizable algebras

Within the technical frame supplied by the algebraic variety of diagonalizable algebras, defined by R. Magari in [2], we prove the following:

The uniqueness of the fixed-point in every diagonalizable algebra

It is well known that, in Peano arithmetic, there exists a formulaTheor (x) which numerates the set of theorems. By Godel's and Lob's results, we have that Open image in new window Therefore, the considered «equations» admit, up to provable equivalence, only one solution.

On the autological character of diagonalizable algebras

Let ℱ be the first order theory of diagonalizable algebras. We define a bijection ϕ from the atomic formulas of ℱ (identities) to the open formulas of ℱ. ϕ is an algebraic analogous of ≒. We prove that ϕ, ϕ-1 preserve the validity.

For everyn, then-freely generated algebra is not functionally free in the equational class of diagonalizable algebras

This paper is devoted to the algebraization of theories in which, as in Peano arithmetic, there is a formula,Theor(x), numerating the set of theorems, and satisfying Hilbert-Bernays derivability conditions. In particular, we study the diagonalizable algebras, which are been introduced by R. Magari in [6], [7]. We prove that for every natural numbern, then-freely generated algebraF n is not functionally free in the equational class of diagonalizable algebras; we also prove that the diagonalizable algebra of Peano arithmetic is not an element of the equational class generated byF n .

Representation and duality theory for diagonalizable algebras

The duality theory established byHalmos in [2] for boolean hemimorphism applies of course to the diagonalizable algebra, because ντν is an hemimorphism. For commodity in working on diagonalizable algebras we recall the basic facts and give the characteristic conditions on the dual of ντν.

On the equational class of diagonalizable algebras

It is well-known that, in Peano arithmetic, there exists a formulaTheor(x) which numerates the set of theorems and that this formula satisfies Hilbert-Bernays derivability conditions. RecentlyR. Magari has suggested an algebraization of the properties ofTheor, introducing the concept of diagonalizable algebra (see [7]): of course this algebraization can be applied to all these theories in which there exists a predicate with analogous properties. In this paper, by means of methods of universal algebra, we study the equational class of diagonalizable algebras, proving, among other things, that the set of identities satisfied byTheor which are consequences of the known ones is decidable.

The fixed-point theorem for diagonalizable algebras

The fixed-point theorem for diagonalizable algebras

Cohomology and weight systems for nilpotent Lie algebras

1. This paper announces results concerning the cohomology groups H*(N, N) where A* is in a certain class of finite-dimensional nilpotent Lie algebras over a field k and T is an abelian Lie algebra faithfully represented as a maximal diagonalizable algebra of derivations of N; we shall refer to such an iV as a T-algebra. The additional hypotheses to be placed on the pair N, T are inspired by the case when J is a Cartan subalgebra and T+N=B is a Borel subalgebra of a complex semisimple Lie algebra. In that case Kostant has shown [2] that H%N, N)=0 for i^.2 and the authors applied this result in [3] to conclude that H*(B, B) = 0. (A similar argument shows H*(P9 P ) = 0 for P parabolic.) Here we are concerned with the relations between the vanishing ofH%N, N), especially for i=2 , and the structure of the algebras N. Let W denote the set of weights of T in N. If dirn(r)=dim(7\T/A)=m then the subset of W arising from the induced representation of T on N/N has precisely m elements, say {a1? • • • , am}. Every a e Wthen has a unique representation a = 2 ^a* with each c? a nonnegative integer and ct 0. For such an a we call the sum (in Z) 2 i the height of a and denote it by |a|. For a in W, denote by Aa the weight space for a in N. DEFINITION. A T-algebra is called positive if (i) dim(70=dim(A7iV), (ii) N is graded by the heights of the weights, i.e., if N(j) = Q)\al=j Na then [N(j)9N(k)]cN(j+k). REMARK. Condition (ii) is superfluous in characteristic 0. However, in characteristic p>0 it has such consequences as N=0 for r > ( / > l ) d i m ( r ) .

Addition and correction to the paper "Diagonal algebras"

Addition and correction to the paper "Diagonal algebras"

Nonfinitizability of classes of representable polyadic algebras

The notion of polyadic algebra was introduced by Halmos to reflect algebraically the predicate logic without equality. Later Halmos enriched the study with the introduction of the notion of equality. These algebras are very closely related to the cylindric algebras of Tarski. The notion of diagonal free cylindric algebra predates that of cylindric algebra and is also due to Tarski. The theory of diagonal free algebras forms an important fragment of the theories of polyadic and cylindric algebras.

Remarks on the reduction theory of von Neumann algebras

In [7, Problem 4, p. 3.55], S. Sakai proposed the following question: If any component of a measurable field -yi-*M('y) of von Neumann algebras over (r, bt) is isomorphic to some fixed von Neumann algebra oZo with separable predual; where r means a locally compact Hausdorff space and IA a positive Radon measure on r, is it true that the direct integral f M< l(,y)d,u('y) is isomorphic to the tensor product a?0 Mo of the associated diagonal algebra'(=L(r, I) and So? If r satisfies the second countability axiom, then one can easily settle the above problem affirmatively as an application of [1, Proposition 4, p. 187]. However, in the case that the countability assumption of r is dropped, J. Dixmier proposed the similar problem in [1, p. 175]. Namely, roughly speaking, if zyl-yj(,y) is a measurable operator field over (r, IA) for each iFI and there exists a family of bounded operators {xi}irr on a Hilbert space and an isometric operator u(,y) for each y such that u('y)xiu('y)--=yi(y) for every yr and iCI, does there exist an isometry which transforms 1 Xxi to f ED yi(,y)du('y) for every

Diagonal algebras

Diagonal algebras

Remarks on diagonal and generalized diagonal algebras

Remarks on diagonal and generalized diagonal algebras

A complement to ``On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras''

A complement to ``On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras''

On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras

On the unitary equivalence among the components of decompositions of representations of involutive Banach algebras and the associated diagonal algebras