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.

scholarly journals Jordan Subalgebras of Banach algebras

1978 ◽  
Vol 21 (2) ◽  
pp. 103-110 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Hilbert Space ◽  
Jordan Algebra ◽  
Banach Algebras ◽  
Jordan Algebras ◽  
Adjoint Operators

We recall that a JC-algebra (Størmer (3)) is a norm closed Jordan algebra of self-adjoint operators on a Hilbert space. Recently, Alfsen, Shultz, and Størmer (1) have introduced a class of abstract normed Jordan algebras called JB-algebras, and have proved that every special JB-algebra is isometrically isomorphic to a JC-algebra. We show that this result brings to a satisfactory conclusion the discussion in (2) of certain wedges W in Banach algebras and their related Jordan algebras W–W, and leads to two characterisations of the bicontinuously isomorphic images of JC-algebras.

scholarly journals An Algorithm to Compute the Canonical Basis of an Irreducible Module Over a Quantized Enveloping Algebra

2003 ◽  
Vol 6 ◽  
pp. 105-118 ◽  
Author(s):  
Willem A. de Graaf
Keyword(s):  
Tensor Product ◽  
Lie Algebra ◽  
Canonical Basis ◽  
Irreducible Module ◽  
Semisimple Lie Algebra ◽  
Canonical Bases ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebra

AbstractThe paper describes an algorithm to compute the canonical basis of an irreducible module over a quantized enveloping algebra of a finite-dimensional semisimple Lie algebra. The algorithm works for any module that is constructed as a submodule of a tensor product of modules with known canonical bases.

Hilbert space methods in the theory of Jordan algebras. I

1975 ◽  
Vol 78 (2) ◽  
pp. 293-300 ◽  
Author(s):  
C. Viola Devapakkiam
Keyword(s):  
Hilbert Space ◽  
Classification Problem ◽  
Jordan Algebras ◽  
Inner Product ◽  
Uniqueness Problem ◽  
Associative Algebras ◽  
Infinite Dimensional

In this paper, we study the structure of certain infinite-dimensional Jordan algebras admitting an inner product. These algebras, called J*-algebras in the sequel, have already been considered in (4) in connexion with the norm uniqueness problem for non-associative algebras. We deal here with the structure and classification of these algebras. Existence of self-adjoint idempotents plays a central role in the classification problem.

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.

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.

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 .

WHAT TYPE OF DYNAMICS ARISE IN E0-DILATIONS OF COMMUTING QUANTUM MARKOV SEMIGROUPS?

2008 ◽  
Vol 11 (03) ◽  
pp. 393-403 ◽  
Author(s):  
ORR MOSHE SHALIT
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Type I ◽  
Markov Semigroups ◽  
Finite Dimensional

Let H be a separable Hilbert space, and let ϕ and θ be two strongly commuting CP0-semigroups on B(H). In a previous paper we constructed a Hilbert space K ⊇ H and two (strongly) commuting E0-semigroups α and β such that [Formula: see text] for all s, t ≥ 0 and all A ∈ B(K). In this note we prove that if ϕ is not an automorphism semigroup, then the semigroup α (given by the above mentioned construction) is cocycle conjugate to the minimal *-endomorphic dilation of ϕ, and that if ϕ is an automorphism semigroup, then α is also an automorphism semigroup. In particular, we conclude that if ϕ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional), then α is a type I E0-semigroup.

Non-Isomorphic Non-Hyperfinite Factors

1969 ◽  
Vol 21 ◽  
pp. 1293-1308 ◽  
Author(s):  
Wai-Mee Ching
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Crossed Product ◽  
Type Ii ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Type Iii ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.

scholarly journals A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190604 ◽  
Author(s):  
Gerd Niestegge
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Jordan Algebras ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Dimensional Version

Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.

scholarly journals On Maximal Abelian Subrings of Factors of Type II

1960 ◽  
Vol 12 ◽  
pp. 289-296 ◽  
Author(s):  
Lajos Pukánszky
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Type I ◽  
Type Ii ◽  
Bounded Operators ◽  
Unitary Space ◽  
Infinite Dimensional ◽  
Complete Knowledge ◽  
Inner Automorphism

Although we possess a fairly complete knowledge of the abelian subrings of rings of operators in a Hilbert space which are algebraically isomorphic to the ring of all bounded operators of a finite or infinite dimensional unitary space, that is of factors of Type I, very little is known of abelian subrings of factors of Type II1. In (1), Dixmier investigated several properties of maximal abelian subrings of factors of Type II. It turned out that their structure differs essentially from that of maximal abelian subrings of factors of Type I. He showed the existence of maximal abelian subrings in approximately finite factors, possessing the property that every inner*-automorphism carrying this subring into itself is necessarily implemented by a unitary operator of this subring. These maximal abelian subrings are called singular. In addition, he constructed a IIi factor containing two singular abelian subrings which cannot be connected by an inner *automorphism of this ring.

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