scholarly journals On Pre-Hilbert Noncommutative Jordan Algebras Satisfying

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-11
Author(s):  
Mohamed Benslimane ◽  
Abdelhadi Moutassim
Keyword(s):  
Vector Space ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Alternative Algebra ◽  
Jordan Algebras ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
First Case ◽  
Hilbert Norm ◽  
Complex Powers

Let be a real or complex algebra. Assuming that a vector space is endowed with a pre-Hilbert norm satisfying for all . We prove that is finite dimensional in the following cases. (1) is a real weakly alternative algebra without divisors of zero. (2) is a complex powers associative algebra. (3) is a complex flexible algebraic algebra. (4) is a complex Jordan algebra. In the first case is isomorphic to or and is isomorphic to in the last three cases. These last cases permit us to show that if is a complex pre-Hilbert noncommutative Jordan algebra satisfying for all , then is finite dimensional and is isomorphic to . Moreover, we give an example of an infinite-dimensional real pre-Hilbert Jordan algebra with divisors of zero and satisfying for all .

Noetherian Banach Jordan pairs

2001 ◽  
Vol 130 (1) ◽  
pp. 25-36 ◽  
Author(s):  
N. BOUDI ◽  
H. MARHNINE ◽  
C. ZARHOUTI ◽  
A. FERNANDEZ LOPEZ ◽  
E. GARCIA RUS
Keyword(s):  
Recent Work ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jacobson Radical ◽  
Alternative Algebra ◽  
Chain Condition ◽  
Finite Capacity ◽  
Jordan Pair ◽  
Finite Dimensional ◽  
Ascending Chain Condition

An associative or alternative algebra A is Noetherian if it satisfies the ascending chain condition on left ideals. Sinclair and Tullo [21] showed that a complex Noetherian Banach associative algebra is finite dimensional. This result was extended by Benslimane and Boudi [5] to the alternative case.For a Jordan algebra J or a Jordan pair V, the suitable Noetherian condition is the ascending chain condition on inner ideals. In a recent work Benslimane and Boudi [6] proved that a complex Noetherian Banach Jordan algebra is finite dimensional.Here we show the following results:(i) the Jacobson radical of a Noetherian Banach Jordan pair is finite dimensional;(ii) nondegenerate Noetherian Banach Jordan pairs have finite capacity;(iii) complex Noetherian Banach Jordan pairs are finite dimensional.

ON SEMITRANSITIVE JORDAN ALGEBRAS OF MATRICES

2011 ◽  
Vol 10 (02) ◽  
pp. 319-333 ◽  
Author(s):  
J. BERNIK ◽  
R. DRNOVŠEK ◽  
D. KOKOL BUKOVŠEK ◽  
T. KOŠIR ◽  
M. OMLADIČ ◽  
...  
Keyword(s):  
Vector Space ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Linear Operators ◽  
Closed Field ◽  
Toeplitz Algebra ◽  
Nilpotent Operator ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Unital Associative Algebra

A set [Formula: see text] of linear operators on a vector space is said to be semitransitive if, given nonzero vectors x, y, there exists [Formula: see text] such that either Ax = y or Ay = x. In this paper we consider semitransitive Jordan algebras of operators on a finite-dimensional vector space over an algebraically closed field of characteristic not two. Two of our main results are: (1) Every irreducible semitransitive Jordan algebra is actually transitive. (2) Every semitransitive Jordan algebra contains, up to simultaneous similarity, the upper triangular Toeplitz algebra, i.e. the unital (associative) algebra generated by a nilpotent operator of maximal index.

On Homomorphic Images of Special Jordan Algebras

1954 ◽  
Vol 6 ◽  
pp. 253-264 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Vector Space ◽  
Linear Algebra ◽  
Associative Algebra ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Special Jordan Algebra ◽  
Image Position

A linear algebra is called a Jordan algebra if it satisfies the identities(1) ab = ba, (a2b) a = a2(ba).It is well known that a linear algebra S over a field of characteristic different from two is a Jordan algebra if there is an isomorphism a → a of the vector-space underlying S into the vector-space of some associative algebra A such that1,where the dot denotes the multiplication in A. Such an algebra S is called a special Jordan algebra.

Nodal Non-Commutative Jordan Algebras

1960 ◽  
Vol 12 ◽  
pp. 488-492 ◽  
Author(s):  
Louis. A. Kokoris
Keyword(s):  
Associative Algebra ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Finite Dimensional

A finite dimensional power-associative algebra 𝒰 with a unity element 1 over a field J is called a nodal algebra by Schafer (7) if every element of 𝒰 has the form α1 + z where α is in J, z is nilpotent, and if 𝒰 does not have the form 𝒰 = ℐ1 + n with n a nil subalgebra of 𝒰. An algebra SI is called a non-commutative Jordan algebra if 𝒰 is flexible and 𝒰+ is a Jordan algebra. Some examples of nodal non-commutative Jordan algebras were given in (5) and it was proved in (6) that if 𝒰 is a simple nodal noncommutative Jordan algebra of characteristic not 2, then 𝒰+ is associative. In this paper we describe all simple nodal non-commutative Jordan algebras of characteristic not 2.

Malcev–Poisson–Jordan algebras

2016 ◽  
Vol 15 (09) ◽  
pp. 1650159
Author(s):  
Malika Ait Ben Haddou ◽  
Saïd Benayadi ◽  
Said Boulmane
Keyword(s):  
Vector Space ◽  
Lie Algebras ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Leibniz Rule ◽  
Malcev Algebra ◽  
Jordan Structure ◽  
Alternative Algebras

Malcev–Poisson–Jordan algebra (MPJ-algebra) is defined to be a vector space endowed with a Malcev bracket and a Jordan structure which are satisfying the Leibniz rule. We describe such algebras in terms of a single bilinear operation, this class strictly contains alternative algebras. For a given Malcev algebra [Formula: see text], it is interesting to classify the Jordan structure ∘ on the underlying vector space of [Formula: see text] such that [Formula: see text] is an MPJ-algebra (∘ is called an MPJ-structure on Malcev algebra [Formula: see text]. In this paper we explicitly give all MPJ-structures on some interesting classes of Malcev algebras. Further, we introduce the concept of pseudo-Euclidean MPJ-algebras (PEMPJ-algebras) and we show how one can construct new interesting quadratic Lie algebras and pseudo-Euclidean Malcev (non-Lie) algebras from PEMPJ-algebras. Finally, we give inductive descriptions of nilpotent PEMPJ-algebras.

scholarly journals On a Semigroup Problem II

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1392
Author(s):  
Viorel Nitica ◽  
Andrew Torok
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Infinite Dimensional ◽  
Finite Dimensional

We consider the following semigroup problem: is the closure of a semigroup S in a topological vector space X a group when S does not lie on “one side” of any closed hyperplane of X? Whereas for finite dimensional spaces, the answer is positive, we give a new example of infinite dimensional spaces where the answer is negative.

scholarly journals Some Remarks on Symmetric and Frobenius Algebras

1960 ◽  
Vol 16 ◽  
pp. 65-71 ◽  
Author(s):  
J. P. Jans
Keyword(s):  
Vector Space ◽  
Space Dimension ◽  
Symmetric Algebra ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Symmetric Algebras ◽  
Finite Dimensional ◽  
Frobenius Algebras ◽  
Finite Dimensional Case

In [5] we defined the concepts of Frobenius and symmetric algebra for algebras of infinite vector space dimension over a field. It was shown there that with the introduction of a topology and the judicious use of the terms continuous and closed, many of the classical theorems of Nakayama [7, 8] on Frobenius and symmetric algebras could be generalized to the infinite dimensional case. In this paper we shall be concerned with showing certain algebras are (or are not) Frobenius or symmetric. In Section 3, we shall see that an algebra can be symmetric or Frobenius in “many ways”. This is a problem which did not arise in the finite dimensional case.

A Generalization of Degree Two Simple Finite-Dimensional Noncommutative Jordan Algebras

1980 ◽  
Vol 32 (2) ◽  
pp. 480-493
Author(s):  
Mary Ellen Conlon
Keyword(s):  
Vector Space ◽  
Simple Algebra ◽  
Jordan Algebras ◽  
Prime Characteristic ◽  
Nilpotent Elements ◽  
Finite Dimensional ◽  
Image Position ◽  
Algebra Over A Field

Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose1Then is noncommutative Jordan.The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.

Hilbert space methods in the theory of Jordan algebras. II

1976 ◽  
Vol 79 (2) ◽  
pp. 307-319 ◽  
Author(s):  
C. Viola Devapakkiam ◽  
P. S. Rema
Keyword(s):  
Hilbert Space ◽  
Jordan Algebra ◽  
Canonical Basis ◽  
Classification Problem ◽  
Jordan Algebras ◽  
Type I ◽  
Type Ii ◽  
Canonical Bases ◽  
Finite Dimensional ◽  
Jordan Isomorphism

In this paper we consider the classification problem for separable special simple J*-algebras (cf. (8)). We show, using a result of Ancochea, that if is the (finite-dimensional) Jordan algebra of all complex n × n matrices and ø a Jordan isomorphism of onto a special J*-algebra J then An can be given the structure of an H*-algebra such that ø is a *-preserving isomorphism of the J*-algebra onto J. This result enables us to construct explicitly a canonical basis for a finite-dimensional simple special J*-algebra isomorphic to a Jordan algebra of type I from which we also obtain canonical bases for special simple finite-dimensional J*-algebras isomorphic to Jordan algebras of type II and III.

Products of idempotent endomorphisms of an independence algebra of infinite rank

1993 ◽  
Vol 114 (2) ◽  
pp. 303-319 ◽  
Author(s):  
John Fountain ◽  
Andrew Lewin
Keyword(s):  
Vector Space ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Common Generalization ◽  
Finite Dimensional Vector Space ◽  
Infinite Rank ◽  
Finite Dimensional ◽  
Linear Maps

AbstractIn 1966, J. M. Howie characterized the self-maps of a set which can be written as a product (under composition) of idempotent self-maps of the same set. In 1967, J. A. Erdos considered the analogous question for linear maps of a finite dimensional vector space and in 1985, Reynolds and Sullivan solved the problem for linear maps of an infinite dimensional vector space. Using the concept of independence algebra, the authors gave a common generalization of the results of Howie and Erdos for the cases of finite sets and finite dimensional vector spaces. In the present paper we introduce strong independence algebras and provide a common generalization of the results of Howie and Reynolds and Sullivan for the cases of infinite sets and infinite dimensional vector spaces.

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