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Julio Becerra Guerrero - One of the best experts on this subject based on the ideXlab platform.

  • Relatively Weakly Open Convex Combinations of Slices and Scattered C$$^*$$-Algebras
    Mediterranean Journal of Mathematics, 2020
    Co-Authors: Julio Becerra Guerrero, Francisco J. Fernández-polo
    Abstract:

    We prove that given a locally compact Hausdorff space K and a compact C$$^*$$-algebra $$\mathcal {A}$$, the C$$^*$$-algebra $$C(K, \mathcal {A})$$ satisfies $$(\hbox {P1})$$, namely that every convex combination of slices of the Closed Unit Ball is a relatively weakly open subset of the Closed Unit Ball, if and only if K is scattered and $$\mathcal {A}$$ is some $$c_0$$-sum of finite-dimensional C$$^*$$-algebras. To obtain a similar characterization in the setting of general C$$^*$$-algebras, we consider a weaker property $$(\overline{\hbox {P1}})$$, namely For every convex combination of slicesCof the Unit Ball of a Banach spaceXand$$x\in C$$, there exists a relatively weakly open setWcontainingx, such that$$W\subseteq \overline{C}$$. We prove that a C$$^*$$-algebra has property $$(\overline{\hbox {P1}})$$ if and only if it is scattered with finite-dimensional irreducible representations. We obtain some stability results for property $$(\overline{\hbox {P1}})$$. For instance, this property passes down from Banach spaces to its Closed ideals. As a consequence, we prove that an $$L_1$$-predual Banach space contains no isomorphic copy of $$\ell _1$$ if and only if it has property $$(\overline{\hbox {P1}})$$.

  • Relatively weakly open sets in Closed Balls of Banach spaces, and the centralizer
    Mathematische Zeitschrift, 2008
    Co-Authors: Julio Becerra Guerrero, Angel Rodríguez-palacios
    Abstract:

    We prove that, if the centralizer of a Banach space X is infinite-dimensional, then every nonempty relatively weakly open subset of the Closed Unit Ball of X has diameter equal to 2. This result, together with a suitable refinement also proven in the paper, contains (and improves in some cases) previously known facts for C*-algebras, JB*-triples, spaces of vector valued continuous functions, and spaces of operators.

  • Relatively weakly open sets in Closed Balls of Banach spaces, and real JB ∗ -triples of finite rank
    Mathematische Annalen, 2004
    Co-Authors: Julio Becerra Guerrero, Antonio M. Peralta, Ginés López Pérez, Angel Rodríguez-palacios
    Abstract:

    We prove that, given a real JB*-triple X, there exists a nonempty relatively weakly open subset of the Closed Unit Ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real JB*-triples whose Banach spaces are isomorphic to Hilbert spaces. Such real JB*-triples are also characterized in two different purely algebraic ways.

  • RELATIVELY WEAKLY OPEN SETS IN Closed BallS OF $C^*$-ALGEBRAS
    Journal of the London Mathematical Society, 2003
    Co-Authors: Julio Becerra Guerrero, Ginés López Pérez, Angel Rodríguez-palacios
    Abstract:

    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).

  • Transitivity of the norm on Banach spaces having a Jordan structure
    Manuscripta Mathematica, 2000
    Co-Authors: Angel Rodríguez Palacios, Julio Becerra Guerrero
    Abstract:

    We study transitivity conditions on the norm of JB -triples, C-algebras, JB - algebras, and their preduals. We show that, for the predual X of a JBW -triple, each one of the following conditions i) and ii) implies that X is a Hilbert space. i) The Closed Unit Ball of X has some extreme point and the norm of X is convex transitive. ii) The set of all extreme points of the Closed Unit Ball of X is non rare in the Unit sphere of X. These results are applied to obtain partial affirmative answers to the open problem whether every JB -triple with transitive norm is a Hilbert space. We extend to arbitrary C-algebras previously known characterizations of transitivity (20) and convex transitivity (36) of the norm on commutative C-algebras. Moreover, we prove that the Calkin algebra has convex transitive norm. We also prove that, if X is a JB -algebra, and if either the norm of X is convex transitive or X has a predual with convex transitive norm, then X is associative. As a consequence, a JB -algebra with almost transitive norm is isomorphic to the field of real numbers.

Angel Rodríguez-palacios - One of the best experts on this subject based on the ideXlab platform.

Antonio M. Peralta - One of the best experts on this subject based on the ideXlab platform.

  • absolutely compatible pairs in a von neumann algebra
    Electronic Journal of Linear Algebra, 2019
    Co-Authors: Nabin K Jana, Anil Kumar Karn, Antonio M. Peralta
    Abstract:

    Let a, b be elements in a Unital C∗-algebra with 0 ≤ a, b ≤ I. The element a is absolutely compatible with b if |a − b| + |I − a − b| = I. In this note, some technical characterizations of absolutely compatible pairs in an arbitrary von Neumann algebra are found. These characterizations are applied to measure how far are two absolute compatible positive elements in the Closed Unit Ball from being mutually orthogonal or commuting. In the case of 2 by 2 matrices, the results admit a geometric interpretation. Namely, non-commutative matrices of the form a = ( t α ) and b = ( x β ) with x, t ∈ (0, 1)\{ 1 }, |α|2 < t(1 − t) α¯ 1 − t β 1 − x 2 and |β|2 < x(1 − x), are absolutely compatible if, and only if, the corresponding point b = (x, lRe(β), S'm(β)) in R3 lies in the ellipsoid Ea = {x ∈ R3 : d2(x, a) + d2(x, a,) = 1}, where d2 denotes the Euclidean distance in R3, and the elements a and a, are (t, lRe(α), S'm(α)) and (1 − t, −lRe(α), −S'm(α)), respectively. The description of absolutely compatible pairs of positive 2 by 2 matrices is applied to determine absolutely compatible pairs of positive elements in the Closed Unit Ball of Mn.

  • The $\lambda$-function in the space of trace class operators
    arXiv: Operator Algebras, 2018
    Co-Authors: Antonio M. Peralta
    Abstract:

    Let $C_1(H)$ denote the space of all trace class operators on an arbitrary complex Hilbert space $H$. We prove that $C_1(H)$ satisfies the $\lambda$-property, and we determine the form of the $\lambda$-function of Aron and Lohman on the Closed Unit Ball of $C_1(H)$ by showing that $$\lambda (a) = \frac{1 - \|a\|_1 + 2 \|a\|_{\infty}}{2},$$ for every $a$ in ${C_1(H)}$ with $\|a\|_1 \leq 1$. This is a non-commutative extension of the formula established by Aron and Lohman for $\ell_1$.

  • Approximation and convex decomposition by extremals and the $\lambda$-function in JBW*-triples
    arXiv: Operator Algebras, 2014
    Co-Authors: Fatmah B. Jamjoom, Antonio M. Peralta, Akhlaq A. Siddiqui, Haifa M. Tahlawi
    Abstract:

    We establish new estimates to compute the $\lambda$-function of Aron and Lohman on the Unit Ball of a JB$^*$-triple. It is established that for every Brown-Pedersen quasi-invertible element $a$ in a JB$^*$-triple $E$ we have $$\hbox{dist} (a, \mathfrak{E} (E_1)) = \max \left\{ 1- m_q (a) , \|a\|-1\right\},$$ where $\mathfrak{E} (E_1)$ denotes the set of extreme points of the Closed Unit Ball $E_1$ of $E$. It is proved that $\lambda (a) = \frac{1+m_q (a)}{2},$ for every Brown-Pedersen quasi-invertible element $a$ in $E_1$, where $m_q (a)$ is the square root of the quadratic conorm of $a$. For an element $a$ in $E_1$ which is not Brown-Pedersen quasi-invertible we can only estimate that $\lambda (a)\leq \frac12 (1-\alpha_q (a)).$ A complete description of the $\lambda$-function on the Closed Unit Ball of every JBW$^*$-triple is also provided, and as a consequence, we prove that every JBW$^*$-triple satisfies the uniform $\lambda$-property.

  • On the facial structure of the Unit Ball in the dual space of a JB*-triple
    Mathematische Annalen, 2010
    Co-Authors: Francisco J. Fernández-polo, Antonio M. Peralta
    Abstract:

    It is shown that every proper weak* Closed face of the Closed Unit Ball \({E_1^*}\) in the dual space of a JB*-triple E coincides with set of all elements in the Unit sphere of E* attaining their norm at a unique compact tripotent in E**. In particular every proper weak* Closed face of the Closed Unit Ball \({E_1^*}\) is weak*-semi-exposed. This result provides an affirmative answer to a conjecture posed over 20 years ago.

  • Relatively weakly open sets in Closed Balls of Banach spaces, and real JB ∗ -triples of finite rank
    Mathematische Annalen, 2004
    Co-Authors: Julio Becerra Guerrero, Antonio M. Peralta, Ginés López Pérez, Angel Rodríguez-palacios
    Abstract:

    We prove that, given a real JB*-triple X, there exists a nonempty relatively weakly open subset of the Closed Unit Ball of X with diameter less than 2 (if and) only if the Banach space of X is isomorphic to a Hilbert space. Moreover we give the structure of real JB*-triples whose Banach spaces are isomorphic to Hilbert spaces. Such real JB*-triples are also characterized in two different purely algebraic ways.

Francisco J. Fernández-polo - One of the best experts on this subject based on the ideXlab platform.

  • Relatively Weakly Open Convex Combinations of Slices and Scattered C$$^*$$-Algebras
    Mediterranean Journal of Mathematics, 2020
    Co-Authors: Julio Becerra Guerrero, Francisco J. Fernández-polo
    Abstract:

    We prove that given a locally compact Hausdorff space K and a compact C$$^*$$-algebra $$\mathcal {A}$$, the C$$^*$$-algebra $$C(K, \mathcal {A})$$ satisfies $$(\hbox {P1})$$, namely that every convex combination of slices of the Closed Unit Ball is a relatively weakly open subset of the Closed Unit Ball, if and only if K is scattered and $$\mathcal {A}$$ is some $$c_0$$-sum of finite-dimensional C$$^*$$-algebras. To obtain a similar characterization in the setting of general C$$^*$$-algebras, we consider a weaker property $$(\overline{\hbox {P1}})$$, namely For every convex combination of slicesCof the Unit Ball of a Banach spaceXand$$x\in C$$, there exists a relatively weakly open setWcontainingx, such that$$W\subseteq \overline{C}$$. We prove that a C$$^*$$-algebra has property $$(\overline{\hbox {P1}})$$ if and only if it is scattered with finite-dimensional irreducible representations. We obtain some stability results for property $$(\overline{\hbox {P1}})$$. For instance, this property passes down from Banach spaces to its Closed ideals. As a consequence, we prove that an $$L_1$$-predual Banach space contains no isomorphic copy of $$\ell _1$$ if and only if it has property $$(\overline{\hbox {P1}})$$.

  • The minimax principle and related topics in the Jordan setting
    arXiv: Functional Analysis, 2019
    Co-Authors: Francisco J. Fernández-polo
    Abstract:

    We prove a minimax principle for weakly compact JB$^*$-triples characterizing geometrically the singular values of an element. Among the consequences of this principle we present a Weyl inequality on the perturbation of the singular values and a Cauchy-Poincar\'e (interlacing) theorem. We also obtain a version of the Ky Fan maximum principle in the setting of weakly compact JB$^*$-triples. We study perturbations of the spectral resolutions showing that small perturbations of an element produces small perturbations of the corresponding spectral resolutions. As a consequence we obtain that weakly compact JB$^*$-triples satisfy the property that perturbations of a convex combination of elements in the Closed Unit Ball coincide with a convex combination of perturbations of the elements also in the Closed Unit Ball. All these results hold true when particularized to weakly compact JB$^*$-algebras.

  • On the facial structure of the Unit Ball in the dual space of a JB*-triple
    Mathematische Annalen, 2010
    Co-Authors: Francisco J. Fernández-polo, Antonio M. Peralta
    Abstract:

    It is shown that every proper weak* Closed face of the Closed Unit Ball \({E_1^*}\) in the dual space of a JB*-triple E coincides with set of all elements in the Unit sphere of E* attaining their norm at a unique compact tripotent in E**. In particular every proper weak* Closed face of the Closed Unit Ball \({E_1^*}\) is weak*-semi-exposed. This result provides an affirmative answer to a conjecture posed over 20 years ago.

John N. Mcdonald - One of the best experts on this subject based on the ideXlab platform.

  • polynomial extreme elements of the Closed Unit Ball of h b
    Complex Variables and Elliptic Equations, 1994
    Co-Authors: John N. Mcdonald
    Abstract:

    Properties of homogeneous polynomials which are extreme points of the Closed Unit Ball of the space H ∞(B) of bounded analytic functions on the open Unit Ball in C n are studied. A complete characterization is given in the case of degree 2 and applied to the study of holomorphic functions on B having positive real part. A sufficient condition for extremality is also given.

  • Polynomial extreme elements of the Closed Unit Ball of H ∞(B)
    Complex Variables Theory and Application: An International Journal, 1994
    Co-Authors: John N. Mcdonald
    Abstract:

    Properties of homogeneous polynomials which are extreme points of the Closed Unit Ball of the space H ∞(B) of bounded analytic functions on the open Unit Ball in C n are studied. A complete characterization is given in the case of degree 2 and applied to the study of holomorphic functions on B having positive real part. A sufficient condition for extremality is also given.