Triple Isomorphisms Research Articles

2-Local Lie triple isomorphisms of nest algebras

Let be a nontrivial nest on a complex separable Hilbert space with dim , and Alg be the associated nest algebra. Suppose that is an additive surjective 2-local Lie triple isomorphism. If is not of infinite multiplicity, we prove that δ is of the form for any , where is an isomorphism or the negative of an anti-isomorphism and is a linear map with for all .

Preservers of Triple Transition Pseudo-Probabilities in Connection with Orthogonality Preservers and Surjective Isometries

We prove that every bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW^*-triples automatically preserves orthogonality in both directions. Consequently, each bijection preserving triple transition pseudo-probabilities between the sets of minimal tripotents of two atomic JBW^*-triples is precisely the restriction of a (complex-)linear triple isomorphism between the corresponding JBW^*-triples. This result can be regarded as triple version of the celebrated Wigner theorem for Wigner symmetries on the posets of minimal projections in B(H). We also present a Tingley type theorem by proving that every surjective isometry between the sets of minimal tripotents in two atomic JBW^*-triples admits an extension to a real linear surjective isometry between these two JBW^*-triples. We also show that the class of surjective isometries between the sets of minimal tripotents in two atomic JBW^*-triples is, in general, strictly wider than the set of bijections preserving triple transition pseudo-probabilities.

Endoisomorphisms and character triple isomorphisms

This paper concerns aspects of Clifford Theory of finite groups. In earlier papers, Turull proved that if two finite groups yielded the same element of the Brauer–Clifford group, then there was an endoisomorphism from one group to the other, and furthermore, that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other. The paper adapts the definition of Character Triple Isomorphism so that it involves ordinary and Brauer characters, and it preserves fields of definition, Schur indices, decomposition numbers, and blocks. It is proved that each endoisomorphism yields exactly one character triple isomorphism. Character triple isomorphisms can be composed, restricted, produced by direct sums, extension of fields, and these operations have their parallel for the endoisomorphisms. One goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications.

Notes on centralizing traces and Lie triple isomorphisms on triangular algebras

The aim of the paper is to give a condition on which every centralizing trace of a bilinear map on triangular algebras is commuting. As an application, we give a short proof of a result due to Xiao et al. As another application, we give a description of Lie triple isomorphisms of certain triangular algebras, which improves a result due to Xiao et al.

Centralizing traces and Lie triple isomorphisms on generalized matrix algebras

Let be a generalized matrix algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and is the trace of . We describe the form of satisfying the condition for all . The question of when has the proper form is considered. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism of to be almost standard. As applications we characterize Lie triple isomorphisms of full matrix algebras, of triangular algebras and of certain unital algebras with nontrivial idempotents. Some topics for future research closely related to our current work are proposed at the end of this article.

Centralizing traces and Lie triple isomorphisms on triangular algebras

Let be a triangular algebra over a commutative ring and be the centre of . Suppose that is an -bilinear mapping and that is a trace of . We describe the form of satisfying the condition for all . The question of when has the standard form will be addressed. Using the aforementioned trace function, we establish sufficient conditions for each Lie triple isomorphism on to be almost standard. As applications, we characterize Lie triple isomorphisms of upper triangular matrix algebras and nest algebras. Some further research topics related to current work are proposed at the end of this article.

Completely 1-complemented subspaces of Schatten spaces

We consider the Schatten spaces S p S^p in the framework of operator space theory and for any 1 ≤ p ≠ 2 > ∞ 1\leq p\not =2>\infty , we characterize the completely 1 1 -complemented subspaces of S p S^p . They turn out to be the direct sums of spaces of the form S p ( H , K ) S^p(H,K) , where H , K H,K are Hilbert spaces. This result is related to some previous work of Arazy and Friedman giving a description of all 1 1 -complemented subspaces of S p S^p in terms of the Cartan factors of types 1–4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L p L^p -spaces. Also we show that for any n ≥ 2 n\geq 2 , there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2 n 2n which is not completely isometric, and we investigate L p L^p -versions of such isomorphisms.

Von neumann regularity and quadratic conorms in JB*-triples and C*-algebras

We revise the notion of von Neumann regularity in JB*-triples by finding a new character- isation in terms of the range of the quadratic operator Q(a). We introduce the quadratic conorm of an element a in a JB*-triple as the minimum reduced modulus of the mapping Q(a). It is shown that the quadratic conorm of a coincides with the infimum of the squares of the points in the triple spectrum of a. It is established that a contractive bijection between JBW*-triples is a triple isomorphism if, and only if, it preserves quadratic conorms. The continuity of the quadratic conorm and the generalized inverse are discussed. Some applications to C*-algebras and von Neumann algebras are also studied.

Multiplicative Jordan triple isomorphisms on the self-adjoint elements of von Neumann algebras

In this paper we consider multiplicative Jordan triple isomorphisms between the sets of self-adjoint elements (respectively the sets of positive elements) of von Neumann algebras. These transformations are the bijective maps which satisfy the equality ϕ ( ABA ) = ϕ ( A ) ϕ ( B ) ϕ ( A ) on their domains. We show that all those transformations originate from linear ∗-algebra isomorphisms and linear ∗-algebra antiisomorphisms in the case when the underlying von Neumann algebras do not have commutative direct summands. An application of our results concerning non-linear maps which preserve the absolute value of products is also presented.

Isometries between C*-algebras

Let A and B be C*-algebras and let T be a linear isometry from A into B . We show that there is a largest projection p in B^{**} such that T(\cdot)p : A \longrightarrow B^{**} is a Jordan triple homomorphism and T(a b^* c + c b^* a) p= T(a) T(b)^* T(c) p + T(c) T(b)^* T(a) p for all a , b , c in A . When A is abelian, we have \|T(a)p\|=\|a\| for all a in A . It follows that a (possibly non-surjective) linear isometry between any C*-algebras reduces locally to a Jordan triple isomorphism, by a projection.

Functional Identities in Jordan Algebras: Associating Traces and Lie Triple Isomorphisms

Let J be a prime nondegenerate Jordan algebra. The problem of describing the form of a symmetric biadditive map B : J × J → J satisfying (B(a, a) · b) · a = B(a, a) · (b · a) for all a, b ∈ J is discussed. As an application, Lie triple isomorphisms of Jordan algebras are considered.

On isomorphisms of standard operator algebras

The aim of this paper is to show that between standard operator algebras every bijective map with a certain multiplicativity property related to Jordan triple isomorphisms of associative rings is automatically additive.

On Additive Isomorphisms of Prime Rings Preserving Polynomials

We study additive isomorphisms of prime rings preserving a multilinear polynomial of degree ≥2. Our main theorem generalizes a number of results obtained for Jordan, n-Jordan, Lie, and Lie triple isomorphisms of prime rings.

Order isomorphisms and triple isomorphisms of operator ideals and their reflexivity

In this paper we give the general form of order isomorphisms and that of triple isomorphisms of standard operator algebras. For any Schatten- von Neumann class of compact operators as well as for the whole operator algebra, it is proved that the group of all of its order automorphisms is reflexive in the algebra of all linear (not necessarily continuous) selfmaps of the underlying operator algebra. A similar reflexivity result concerning triple automorphisms of an arbitrary operator ideal is also presented.

Jordan and Jordan Triple Isomorphisms of Rings

Jordan and Jordan Triple Isomorphisms of Rings