Julio Becerra Guerrero

Relatively Weakly Open Sets in Closed Balls of C*-Algebras

Let A be an infinite-dimensional C ∗ -algebra. It is proved that every nonempty relatively weakly open subset of the closed unit ball BA of A has diameter equal to 2. This implies that BA is not dentable, and that there is not any point of continuity for the identity mapping (BA, weak) −→ (BA, norm).

Higher-Order Differential Operators on a Lie Group and Quantization

This talk is devoted mainly to the concept of higher-order polarization on a group, which is introduced in the framework of a Group Approach to Quantization, as a powerful tool to guarantee the irreducibility of quantizations and/or representations of Lie groups in those anomalous cases where the Kostant-Kirilov co-adjoint method or the Borel-Weyl-Bott representation algorithm do not succeed.

Banach Spaces Whose Algebras of Operators are Unitary: A Holomorphic Approach

An element u of a norm-unital Banach algebra A is said to be unitary if u is invertible in A and satisfies � u� = � u −1 � =1 .T he norm-unital Banach algebra A is called unitary if the convex hull of the set of its unitary elements is norm-dense in the closed unit ball of A .I fX is a complex Hilbert space, then the algebra BL(X )o fa ll bounded linear operators on X is unitary by the Russo–Dye theorem. The question of whether this property characterizes complex Hilbert spaces among complex Banach spaces seems to be open. Some partial affirmative answers to this question are proved here. In particular, a complex Banach space X is a Hilbert space if (and only if) BL(X )i sunitary and, for Y equal to X, X ∗ or X ∗∗ , there exists a biholomorphic automorphism of the open unit ball of Y that cannot be extended to a surjective linear isometry on Y .

Quantization on a Lie Group: Higher-Order Polarizations

Much effort has been devoted to the geometrization of quantum mechanics during the second half of this century in an attempt to emulate classical mechanics and classical gravity at mathematical beauty and, why not, to better understand quantum gravity. We wish to report on one particular line of this task, which lies mostly on symmetry grounds and has been developed in the last years trying to accomodate modern aspects of quantum mechanics such as global quantization of systems with non-trivial topology, in particular systems suffering from topological anomalies, and accounting for more general obstructions to the basic rules of local quantization, to be referred to as algebraic anomalies, directly attached to the well-known no-go theorems [1] of the original, standard quantum mechanics. This group approach to quantization (GAQ) [2], in some respect generalizes geometric quantization (GQ), originally developed by Kirillov, Kostant and Souriau [3–5] both as a method of quantization and as a group representation technique, and the more specific representation algorithm of Borel-Weyl-Bott (B-W-B) [6], which essentially applies to finite-dimensional semisimple groups.

Canonical coherent states for the relativistic harmonic oscillator

In this paper there are constructed manifestly covariant relativistic coherent states on the entire complex plane which reproduce others previously introduced on a given SL(2,R) representation, once a change of variables z∈C→zD∈ unit disk is performed. Also introduced are higher‐order, relativistic creation and annihilation operators, a,a°, with canonical commutation relation [a, a°]=1 rather than the covariant one [z, z°]≊ energy and naturally associated with the SL(2,R) group. The canonical (relativistic) coherent states are then defined as eigenstates of a. Finally, a canonical, minimal representation is constructed in configuration space by means of eigenstates of a canonical position operator.

Subspaces of Banach spaces with big slices

We study when diameter two properties pass down to subspaces. We obtain that the slice two property (respectively diameter two property, strong diameter two property) passes down from a Banach space

GENERALIZED CONFORMAL SYMMETRY AND EXTENDED OBJECTS FROM THE FREE PARTICLE

X

Banach spaces whose algebras of operators have a large group of unitary elements

to a subspace

UNFOLDING THE QUANTUM ARNOLD TRANSFORMATION

Y

The Bishop–Phelps–Bollobás Property: a Finite-Dimensional Approach

whenever