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Truncation and Duality Results for Hopf Image Algebras

Teodor Banica (2014)

Bulletin of the Polish Academy of Sciences. Mathematics

Associated to an Hadamard matrix H ∈ M N ( ℂ ) is the spectral measure μ ∈ [0,N] of the corresponding Hopf image algebra, A = C(G) with G ⊂ S ⁺ N . We study a certain family of discrete measures μ r ∈ [ 0 , N ] , coming from the idempotent state theory of G, which converge in Cesàro limit to μ. Our main result is a duality formula of type ∫ 0 N ( x / N ) p d μ r ( x ) = ∫ 0 N ( x / N ) r d ν p ( x ) , where μ r , ν r are the truncations of the spectral measures μ,ν associated to H , H t . We also prove, using these truncations μ r , ν r , that for any deformed Fourier matrix H = F M ⊗ Q F N we have μ = ν.

Twisted spectral triples and covariant differential calculi

Ulrich Krähmer, Elmar Wagner (2011)

Banach Center Publications

Connes and Moscovici recently studied "twisted" spectral triples (A,H,D) in which the commutators [D,a] are replaced by D∘a - σ(a)∘D, where σ is a second representation of A on H. The aim of this note is to point out that this yields representations of arbitrary covariant differential calculi over Hopf algebras in the sense of Woronowicz. For compact quantum groups, H can be completed to a Hilbert space and the calculus is given by bounded operators. At the end, we discuss an explicit example of...

Two characterizations of automorphisms on B(X)

Peter Šemrl (1993)

Studia Mathematica

Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.

Two complete and minimal systems associated with the zeros of the Riemann zeta function

Jean-François Burnol (2004)

Journal de Théorie des Nombres de Bordeaux

We link together three themes which had remained separated so far: the Hilbert space properties of the Riemann zeros, the “dual Poisson formula” of Duffin-Weinberger (also named by us co-Poisson formula), and the “Sonine spaces” of entire functions defined and studied by de Branges. We determine in which (extended) Sonine spaces the zeros define a complete, or minimal, system. We obtain some general results dealing with the distribution of the zeros of the de-Branges-Sonine entire functions. We...

Two Families of Self-adjoint Indecomposable Operators in an Orthomodular Space

Carla Barrios Rodríguez (2008)

Annales mathématiques Blaise Pascal

Orthomodular spaces are the counterpart of Hilbert spaces for fields other than ℝ or ℂ . Both share numerous properties, foremost among them is the validity of the Projection theorem. Nevertheless in the study of bounded linear operators which started in [3], there appeared striking differences with the classical theory. In fact, in this paper we shall construct, on the canonical non-archimedean orthomodular space E of [5], two infinite families of self-adjoint bounded linear operators having no...

Two geometric constants for operators acting on a separable Banach space.

E. Martín Peinador, E. Induráin, A. Plans Sanz de Bremond, A. A. Rodes Usan (1988)

Revista Matemática de la Universidad Complutense de Madrid

The main result of this paper is the following: A separable Banach space X is reflexive if and only if the infimum of the Gelfand numbers of any bounded linear operator defined on X can be computed by means of just one sequence on nested, closed, finite codimensional subspaces with null intersection.

Two mappings related to semi-inner products and their applications in geometry of normed linear spaces

Sever Silvestru Dragomir, Jaromír J. Koliha (2000)

Applications of Mathematics

In this paper we introduce two mappings associated with the lower and upper semi-inner product ( · , · ) i and ( · , · ) s and with semi-inner products [ · , · ] (in the sense of Lumer) which generate the norm of a real normed linear space, and study properties of monotonicity and boundedness of these mappings. We give a refinement of the Schwarz inequality, applications to the Birkhoff orthogonality, to smoothness of normed linear spaces as well as to the characterization of best approximants.

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