Algebraic Reflexivity Research Articles

Kakutani's automorphism theorem for vector-valued bounded holomorphic functions

We give descriptions for algebra homomorphisms and automorphisms of vector-valued bounded holomorphic functions defined on the open unit disc. The characterizations derived allow us to establish the algebraic reflexivity properties of the group of automorphisms on these spaces.

Algebraic Reflexivity of Non-Canonical Isometries on Lipschitz Spaces

Let Lip([0,1]) be the Banach space of all Lipschitz complex-valued functions f on [0,1], equipped with one of the norms: fσ=|f(0)|+f′L∞ or fm=max|f(0)|,f′L∞, where ·L∞ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of Lip([0,1]). Namely, we prove that every approximate local isometry of Lip([0,1]) can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of Lip([0,1]). Additionally, some complete characterizations of such reflections and projections are stated.

Algebraic reflexivity of diameter-preserving linear bijections between $C(X)$-spaces

We prove that if X and Y are first countable compact Hausdorff spaces, then the set of all diameter-preserving linear bijections from C(X) to C(Y ) is algebraically reflexive.

Algebraic Reflexivity of isometries on spaces of continuously differentiable functions

Algebraic Reflexivity of isometries on spaces of continuously differentiable functions

Approximate local isometries of Banach algebras of differentiable functions

Let X and Y be compact subsets of R such that X and Y coincide with the closures of their interiors. For any n∈N, let C(n)(X) be the Banach algebra of all n-times continuously differentiable complex-valued functions f on X, with the norm ‖f‖C=maxx∈X⁡(∑k=0n(|f(k)(x)|/k!)). We prove that every approximate local isometry of C(n)(X) to C(n)(Y) is an isometric linear algebra monomorphism multiplied by a fixed n-times continuously differentiable unimodular function. This description allows us to establish the algebraic and 2-algebraic reflexivity of the set of linear isometries of C(n)(X) onto C(n)(Y). Furthermore, this algebraic reflexivity becomes topological whenever X and Y are compact intervals of R. Another application of our main result shows that the sets of isometric reflections and generalized bi-circular projections of C(n)(X) are topologically and 2-topologically reflexive.

Algebraic reflexivity of isometry groups of algebras of Lipschitz maps

We study groups of surjective linear isometries on Banach algebras of Lipschitz maps with values in some unital C⁎-algebras. In this paper, these spaces are endowed with the sum norm. For the case where the C⁎-algebras are commutative whose groups of all surjective isometries are algebraically reflexive, we prove that the group of all surjective isometries on the corresponding Banach algebra of Lipschitz maps are algebraically reflexive. We also prove that the group of unital surjective isometries between matrix-valued Lipschitz algebras are reflexive.

Algebraic reflexivity of sets of bounded linear operators on absolutely continuous function spaces

Algebraic reflexivity of sets of bounded linear operators on absolutely continuous function spaces

Range-compatible homomorphisms on matrix spaces

Let K be a (commutative) field, and U and V be finite-dimensional vector spaces over K. Let S be a linear subspace of the space L(U,V) of all linear operators from U to V. A map F:S→V is called range-compatible when F(s)∈Ims for all s∈S. Obvious examples of such maps are the evaluation maps s↦s(x), with x∈U.In this article, we classify all the range-compatible group homomorphisms on S provided that codimL(U,V)S≤2dim⁡V−3, unless K has cardinality 2 and codimL(U,V)S=2dim⁡V−3. Under those assumptions, it is shown that the linear range-compatible maps are the evaluation maps, and the above upper-bound on the codimension of S is optimal for this result to hold.As an application, we obtain new sufficient conditions for the algebraic reflexivity of an operator space and, with the above conditions on the codimension of S, we give an explicit description of the range-restricting and range-preserving homomorphisms on S.

On algebraic reflexivity of sets of surjective isometries between spaces of weak⁎ continuous functions

We establish the algebraic reflexivity of sets of surjective isometries between Banach spaces of weak⁎ or weakly continuous functions. We also derive some results on the algebraic structure of the isometry group of WC⁎(X,E⁎).

A generalization of expander graphs and local reflexivity of uniform Roe algebras

We introduce a generalization of expander graphs, which is called a weak expander sequence. It is proved that a uniform Roe algebra of a weak expander sequence is not locally reflexive. It follows that uniform Roe algebras of expander graphs are not exact. We introduce the notion of a generalized box space to discuss box spaces and expander sequences in a unified framework. Key tools for the proof are amenable traces and measured groupoids associated with generalized box spaces.

Algebraic reflexivity of isometry groups and automorphism groups of some operator structures

We establish the algebraic reflexivity of three isometry groups of operator structures: the group of all surjective isometries on the unitary group, the group of all surjective isometries on the set of all positive invertible operators equipped with the Thompson metric, and the group of all surjective isometries on the general linear group of B(H), the operator algebra over a complex infinite dimensional separable Hilbert space H. We show that those isometry groups coincide with certain groups of automorphisms of corresponding structures and hence we also obtain the reflexivity of some automorphism groups.

Operator algebras from the discrete Heisenberg semigroup

AbstractWe study reflexivity and structural properties of operator algebras generated by representations of the discrete Heisenberg semigroup. We show that the left regular representation of this semigroup gives rise to a semi-simple reflexive algebra. We exhibit an example of a representation that gives rise to a non-reflexive algebra. En route, we establish reflexivity results for subspaces of $H^{\infty}(\mathbb{T})\otimes\mathcal{B}(\mathcal{H})$.

Reflexivity of operator algebras of finite split strict multiplicity

We study the invariant subspaces of abelian operator algebras of finite split strict mul- tiplicity. We give sufficient conditions for the reflexivity and hereditary reflexivity of these alge- bras.

Wandering vectors and the reflexivity of free semigroup algebras

A free semigroup algebra S is the weak-operator-closed (non-self-adjoint) operator algebra generated by n isometries with pairwise orthogonal ranges. A unit vector x is said to be wandering for S if the set of images of x under non-commuting words in the generators of S is orthonormal. We establish the following dichotomy: either a free semigroup algebra has a wandering vector, or it is a von Neumann algebra. Consequences include that every free semigroup algebra is reflexive, and that certain free semigroup algebras are hyper-reflexive with a very small hyper-reflexivity constant.

Algebraic reflexivity of sets of bounded operators on vector valued Lipschitz functions

In this paper we establish algebraic reflexivity properties of subsets of bounded linear operators acting on spaces of vector valued Lipschitz functions. We also derive a representation for the generalized bi-circular projections on these spaces.

Algebraic reflexivity of the isometry group of some spaces of Lipschitz functions

We show that the isometry groups of Lip ( X , d ) and lip ( X , d α ) with α ∈ ( 0 , 1 ) , for a compact metric space ( X , d ) , are algebraically reflexive. We also prove that the sets of isometric reflections and generalized bi-circular projections on such spaces are algebraically reflexive. In order to achieve this, we characterize generalized bi-circular projections on these spaces.

Algebraic reflexivity for semigroups of operators

Algebraic reflexivity of sets and semigroups of linear transformations are studied in this paper. Some new examples of algebraically reflexive sets and semigroups of linear transforma- tions are given. Using known results on algebraically orbit reflexive linear transformations, those linear transformations on a complex Banach space that are determined by their invariant subsets are characterized.

Algebraic reflexivity of some subsets of the isometry group

Let X be a compact first countable space. In this paper we show that the set of isometries of C ( X ) that are involutions is algebraically reflexive. As a consequence of a recent work of Botelho and Jamison this leads to the conclusion that the set of generalized bi-circular projections on C ( X ) is also algebraically reflexive. We also consider these questions for the space C ( X , E ) where E is a uniformly convex Banach space.

Algebraic reflexivity of C(X,E) and Cambern's theorem

Algebraic reflexivity of C(X,E) and Cambern's theorem

A note on the algebraic reflexivity of the group of surjective isometries of [formula omitted

This short note deals with question of algebraic reflexivity of the group of surjective isometries of the space of compact operators K ( C ( K ) ) for a compact set K. We show that when K is a countable metric space the group of surjective isometries is algebraically reflexive.