On a characterization of commutativity for $$C^*$$-algebras via gyrogroup operations

We show that a unital \(C^*\)-algebra is commutative if and only if the gyrogroup of the set of positive invertible elements is in fact a group.     

Orthogonality preserving property for pairs of operators on Hilbert $$C^*$$ C ∗ -modules

Aequationes mathematicae - We investigate the orthogonality preserving property for pairs of operators on inner product $$C^*$$ -modules. Employing the fact that the $$C^*$$ -valued inner product...     

Orthogonal-gradings on $$H^*$$-algebras

We study set-gradings on proper \(H^*\)-algebras A, which are compatible with the involution and the inner product of A, that will be called orthogonal-gradings. If A is an arbitrary \(H^*\)-algebra with a fine grading, we obtain a (fine) orthogonal-graded version of the main structure theorem for proper arbitrary \(H^*\)-algebras. If A is associative, we show that any fine orthogonal-grading is either a group-grading or a (non-group grading) Peirce decomposition of A respect to a family of orthogonal projections. If A is alternative, we prove that any fine orthogonal-grading is either a fine orthogonal-grading of a (proper) associative \(H^*\)-algebra, or a \({\mathbb Z}_2^3\)-grading of the complex octonions \({\mathbb O}\) or a non-group grading which is a refinement of the Peirce decomposition of \({\mathbb O}\) respect to its family of orthogonal projections. Finally, we also show that any orthogonal-grading on the real octonion division algebra is necessarily a group-grading.     

Orthogonality in Classes

 Let denote the von Neumann–Schatten class, its norm and let be an elementary operator defined by . We shall characterize those operators which are orthogonal to the range of in the sense that for all . The main results of this paper are: If and (i) if A, C, respectively B, D are commuting normal operators with , or (ii) if A, B are contractions and , then is orthogonal to the range of if and only of S is in the kernel of . Furthermore, in both cases, the algebraic direct sum satisfies . (Received 9 February 2000; in revised form 21 February, 2001)     

Orthogonality in multiway classifications

We point out the difference between weak orthogonality and strict orthogonality of two factors when all other factors (if any) have been eliminated. Necessary and sufficient conditions for weak orthogonality are obtained, and a simple method of computing the sum of squares for such designs is given. An example of a situation with weak orthogonality is given. It is shown that under some mild conditions a connected equireplicate row-column-treatment design with weak orthogonality for every pair of factors must be of Latin-square type.     

Orthogonality in multiobjective optimization

Properties of nonlinear multiobjective problems implied by the Karush-Kuhn-Tucker necessary conditions are investigated. It is shown that trajectories of Lagrange multipliers corresponding to the components of the vector cost function are orthogonal to the corresponding trajectories of vector deviations in the balance space (to the balance set for Pareto solutions).     

Orthogonality via Transforms

Let e(x, t) = ∑p_n(x)tⁿ be the generating function of a polynomial sequence, and the transform of multiplication by x relative to e(x, t). We show that the sequence p_n(x) is orthogonal precisely when is a t-variable, i.e., maps K[t] into itself and increases degree by 1. We also show how transform techniques can shed light on the recursion relations and differential equations for p_n(x).     

Nonlinear Triple Product $$A^{*}B + B^{*}A$$ for Derivations on $$\ast$$ -Algebras

Mathematical Notes - Let $$\mathcal{A}$$ be a prime $$\ast$$ -algebra. In this paper, assuming that $$\Phi:\mathcal{A}\to\mathcal{A}$$ satisfies $$\Phi(A\diamond B \diamond C)=\Phi(A)\diamond B...     

Orthogonality of matrices

Let X be a real finite-dimensional normed space with unit sphere S_X and let L(X) be the space of linear operators from X into itself. It is proved that X is an inner product space if and only if for A,C∈L(X)

     

Complements on Diminnie Orthogonality

A new orthogonality relation for normed linear spaces is introduced by C. R. DIMINNIE in [10]. Some interesting properties of such orthogonality and its relationship with Birkhoff orthogonality are studied in the above paper. The first part of this paper begins with a geometrical interpretation of Diminnie-orthogonality which allows us to obtain some other properties of such orthogonality. The second part deals with relationships between Diminnie orthogonality and some other known orthogonalities.     

A Hilbert $$C^*$$ -Module with Extremal Properties

Functional Analysis and Its Applications - We construct an example of a Hilbert $$C^*$$ -module which shows that Troitsky’s theorem on the geometric essence of $$ {\mathcal A} $$ -compact...     

Orthogonality of Jack polynomials in superspace

Jack polynomials in superspace, orthogonal with respect to a “combinatorial” scalar product, are constructed. They are shown to coincide with the Jack polynomials in superspace, orthogonal with respect to an “analytical” scalar product, introduced in [P. Desrosiers, L. Lapointe, P. Mathieu, Jack polynomials in superspace, Comm. Math. Phys. 242 (2003) 331-360] as eigenfunctions of a supersymmetric quantum mechanical many-body problem. The results of this article rely on generalizing (to include an extra parameter) the theory of classical symmetric functions in superspace developed recently in [P. Desrosiers, L. Lapointe, P. Mathieu, Classical symmetric functions in superspace, J. Algebraic Combin. 24 (2006) 209-238].     

Orthogonality correction in the conjugate-gradient method

This paper describes a simple algorithm for calculating the carryover term β in the conjugate-gradient method. The proposed algorithm incorporates an orthogonality correction as well as an automatic restart. Its performance is compared with alternate β forms reported, using five test functions and two cases of parameter estimation.     

Orthogonality and Hecke operators

In this article we analyze orthogonality relations between old forms and the connection to the theory of Hecke operators.     

2-Local {*}-Lie isomorphisms of operator algebras

Let \({H}\) be a complex Hilbert space of dimension greater than \({3}\). We show that every surjective 2-local \({*}\)-Lie isomorphism \({\Phi}\) of \({B(H)}\) has the form \({\Phi=\Psi+\tau}\), where \({\Psi}\) is a \({*}\)-isomorphism or the negative of a \({*}\)-anti-isomorphism of \({B(H)}\), and \({\tau}\) is a homogeneous map from \({B(H)}\) into \({\mathbb{C}I}\) vanishing on every sum of commutators.     

Compact Operators and Uniform Structures in Hilbert $$C^*$$ -Modules

Functional Analysis and Its Applications - Quite recently a criterion for the $$\mathcal{A}$$ -compactness of an ajointable operator $$F\colon {\mathcal M} \to\mathcal{N}$$ between Hilbert $$C^*$$...     

The Index Semigroup and K-Groups of Graph-Groupoid C^{*}-Algebras

In this paper, we consider the relation between index theory and $K$ -theory induced by directed graphs. In particular, we study index-morphism on finite trees, and classify the set of finite trees in terms of our index-morphism. Such a morphism generate certain semigroup, called the index semigroup. From the index semigroup, we find a ple, interesting connection between semigroup-elements and $K$ -group computations of groupoid $C^{*}$ -algebras generated by graphs. In conclusion, we show that the pure combinatorial data of graphs completely characterize and classify the elements of the index semigroup (or equivalently, graph-index on finite trees), Watatani’s Jones index on groupoid $C^{*}$ -algebras generated by finite trees, and $K$ -group computations of certain $C^{*}$ -algebras.