Geometry of oblique projections

Let A be a unital C*-algebra. Denote by P the space of selfadjoint projections of A. We study the relationship between P and the spaces of projections P_a determined by the different involutions #_a induced by positive invertible elements a in A. The maps f_p: P \to P_a sending p to the unique q in P_a with the same range as p and \Omega_a: P_a \to P sending q to the unitary part of the polar decomposition of the symmetry 2q-1 are shown to be diffeomorphisms. We characterize the pairs of idempotents q, r in A with |q-r|<1 such that there exists a positive element a in A verifying that q, r are in P_a. In this case q and r can be joined by an unique short geodesic along the space of idempotents Q of A.Comment: 25 pages, Latex, to appear in Studia Mathematic

Projective spaces of a C*-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebra A with a fixed projection p. The resulting space P(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group of A. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean - in Schwarz-Zaks terminology) are considered, allowing a comparison among P(p), the Grassmann manifold of A and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection e = 2p-1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.Comment: 26 pages, Late

The Schur-Horn theorem for operators and frames with prescribed norms and frame operator

Let $\mathcal H$ be a Hilbert space. Given a bounded positive definite operator $S$ on $\mathcal H$, and a bounded sequence $\mathbf{c} = \{c_k \}_{k \in \mathbb N}$ of non negative real numbers, the pair $(S, \mathbf{c})$ is frame admissible, if there exists a frame $\{f_k \}_{k \in \mathbb{N}}$ on $\mathcal H$ with frame operator $S$, such that $\|f_k \|^2 = c_k$, $k \in \mathbb {N}$. We relate the existence of such frames with the Schur-Horn theorem of majorization, and give a reformulation of the extended version of Schur-Horn theorem, due to A. Neumann. We use it to get necessary conditions (and to generalize known sufficient conditions) for a pair $(S, \mathbf{c})$, to be frame admissible.Comment: To appear in Illinois Journal of Mat

Nullspaces and frames

In this paper we give new characterizations of Riesz and conditional Riesz frames in terms of the properties of the nullspace of their synthesis operators. On the other hand, we also study the oblique dual frames whose coefficients in the reconstruction formula minimize different weighted norms.Comment: 16 page

Biochemical and structural characterization of a novel arginine kinase from the spider <i>Polybetes pythagoricus</i>

Energy buffering systems are key for homeostasis during variations in energy supply. Spiders are the most important predators for insects and therefore key in terrestrial ecosystems. From biomedical interest, spiders are important for their venoms and as a source of potent allergens, such as arginine kinase (AK, EC 2.7.3.3). AK is an enzyme crucial for energy metabolism, keeping the pool of phosphagens in invertebrates, and also an allergen for humans. In this work, we studied AK from the Argentininan spider Polybetes pythagoricus (PpAK), from its complementary DNA to the crystal structure. The PpAK cDNA from muscle was cloned, and it is comprised of 1068 nucleotides that encode a 384-amino acids protein, similar to other invertebrate AKs. The apparent Michaelis-Menten kinetic constant (Km) was 1.7 mM with a kcat of 75 s-1. Two crystal structures are presented, the apoPvAK and PpAK bound to arginine, both in the open conformation with the active site lid (residues 310-320) completely disordered. The guanidino group binding site in the apo structure appears to be organized to accept the arginine substrate. Finally, these results contribute to knowledge of mechanistic details of the function of arginine kinase.Instituto de Investigaciones Bioquímicas de La Plat

Biochemical and structural characterization of a novel arginine kinase from the spider <i>Polybetes pythagoricus</i>

Energy buffering systems are key for homeostasis during variations in energy supply. Spiders are the most important predators for insects and therefore key in terrestrial ecosystems. From biomedical interest, spiders are important for their venoms and as a source of potent allergens, such as arginine kinase (AK, EC 2.7.3.3). AK is an enzyme crucial for energy metabolism, keeping the pool of phosphagens in invertebrates, and also an allergen for humans. In this work, we studied AK from the Argentininan spider Polybetes pythagoricus (PpAK), from its complementary DNA to the crystal structure. The PpAK cDNA from muscle was cloned, and it is comprised of 1068 nucleotides that encode a 384-amino acids protein, similar to other invertebrate AKs. The apparent Michaelis-Menten kinetic constant (Km) was 1.7 mM with a kcat of 75 s-1. Two crystal structures are presented, the apoPvAK and PpAK bound to arginine, both in the open conformation with the active site lid (residues 310-320) completely disordered. The guanidino group binding site in the apo structure appears to be organized to accept the arginine substrate. Finally, these results contribute to knowledge of mechanistic details of the function of arginine kinase.Instituto de Investigaciones Bioquímicas de La Plat