In this paper we calculate some geometric constants for Morrey spaces and small Morrey spaces, namely generalized Von Neumann-Jordan constant, modified Von Neumann-Jordan constants, and Zbáganu constant. All these constants measure the uniformly nonsquareness of the spaces. We obtain that their values are the same as the value of Von Neumann-Jordan constant for Morrey spaces and small Morrey spaces.
In this paper, we calculate four geometric constants for discrete Morrey spaces. The constants are generalized von Neumann-Jordan constant, modified von Neumann-Jordan constant, von Neumann-Jordan type constant, and Zbäganu constant. The four constants measure uniformly nonsquareness of the above spaces. We obtain that the value of each of the four constants for the above spaces is two, which means that the spaces are NOT uniformly nonsquare.
In this article, we compute Von Neumann-Jordan constant, James constant, and Dunkl-Williams constant for small Morrey spaces. Our approach can also be seen as an alternative way in computing the three constants for the (classical) Morrey spaces. In addition, we prove constructively that Morrey spaces are not uniformly non-octahedral.
In this paper we calculate three geometric constants, namely the von Neumann-Jordan constant, the James constant, and the Dunkl-Williams constant, for Morrey spaces and discrete Morrey spaces. These constants measure uniformly nonsquareness of the associated spaces. We obtain that the three constants are the same as those for $L^1$ and $L^\infty$ spaces.
In this note we prove that the $n$-th Von Neumann-Jordan constant and the $n$-th James constant for discrete Morrey spaces $\ell^p_q$ where $1\le p<q<\infty$ are both equal to $n$. This result tells us that the discrete Morrey spaces are not uniformly non-$\ell^1$, and hence they are not uniformly $n$-convex.
There are discussed some modifications of n-th von Neumann-Jordan constant considered by M. Kato, Y. Takahashi and K. Hashimoto. We study, among others, upper, lower, upper modified and lower modified n-th von Neumann-Jordan constant and relationships between them. There are characterized uniformly non-l_{n}^{1} Banach spaces in terms of the upper modified n-th von Neumann-Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces L^{p} and l^{p} (1\leq p\leq oo).
In this article, we show constructively that Morrey spaces are not uniformly non-$\ell^1_n$ for any $n\ge 2$. This result is sharper than those previously obtained in \cite{GKSS, MG}, which show that Morrey spaces are not uniformly non-square and also not uniformly non-octahedral. We also discuss the $n$-th James constant $C_{\rm J}^{(n)}(X)$ and the $n$-th Von Neumann-Jordan constant $C_{\rm NJ}^{(n)}(X)$ for a Banach space $X$, and obtain that both constants for any Morrey space $\mathcal{M}^p_q(\mathbb{R}^d)$ with $1\le p<q<\infty$ are equal to $n$.
A von Neumann-Jordan Constant of Non-Normable Metrics
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam（胡志明市科学大学数学与计算机科学学院）；Vietnam National University, Ho Chi Minh City, Vietnam（胡志明市越南国立大学）；Faculty of Mathematics, College of Natural Sciences, Can Tho University, Can Tho City, Vietnam（芹苳市芹苳大学自然科学学院数学系）
We characterize completely bounded normal Jordan $*$-homomorphisms acting on von Neumann algebras. We also characterize completely positive isometries acting on noncommutative $\mathrm{L}^p$-spaces.
It is known that any normed vector space which satisfies the parallelogram law is actually an inner product space. For finite dimensional normed vector spaces over R, we formulate an approximate version of this theorem: if a space approximately satisfies the parallelogram law, then it has a near isometry with Euclidean space. In other words, a small von Neumann Jordan constant E + 1 for X yields a small Banach-Mazur distance with R^n, d(X, R^n) < 1 + B_n E + O(E^2). Finally, we examine how this estimate worsens as the dimension, n, of X increases, with the conclusion that B_n grows quadratical…
We show that the Weyl symbol of a Born-Jordan operator is in the same class as the Born-Jordan symbol, when Hörmander symbols and certain types of modulation spaces are used as symbol classes. We use these properties to carry over continuity and Schatten-von Neumann properties to the Born-Jordan calculus.
It is established that every (not necessarily linear) 2-local $^*$-homomorphism from a von Neumann algebra into a C$^*$-algebra is linear and a $^*$-homomorphism. In the setting of (not necessarily linear) 2-local $^*$-homomorphism from a compact C$^*$-algebra we prove that the same conclusion remains valid. We also prove that every 2-local Jordan $^*$-homomorphism from a JBW$^*$-algebra into a JB$^*$-algebra is linear and a Jordan $^*$-homomorphism.
A Fan-Theobald-von Neumann system is a triple $(V,W,λ)$, where $V$ and $W$ are real inner product spaces and $λ:V \to W$ is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In the previous paper (arXiv:1902.06640) we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions…
In this paper, we study the structure of operators in a type $\mathrm{I}_{n}$ von Neumann algebra $\mathscr{A}$. Inspired by the Jordan canonical form theorem, our main motivation is to figure out the relation between the structure of an operator $A$ in $\mathscr{A}$ and the property that a bounded maximal abelian set of idempotents contained in the relative commutant $\{A\}^{\prime} \cap\mathscr{A}$ is unique up to similarity. Furthermore, we classify this class of operators with the property by $K$-theory for Banach algebras. Some views and techniques are from von Neumann's reduction theory.
We investigate the situation when a normal positive linear unital map on a semifinite von Neumann algebra leaving the trace invariant does not change fixed quantum Renyi's entropy of the density of a normal state. It is also shown that such a map does not change the entropy of any density if and only if it is a Jordan *-isomorphism on the algebra.
Let M be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec-Pelczynski subsequence decomposition of bounded sequences in L^p[0,1] to the case of the Haagerup L^p-spaces (1\le p<\infty). In particular, we prove that if (ϕ_n)_n is a bounded sequence in the predual M_* of M, then there exist a subsequence (ϕ_{n_k})_k of (ϕ_n)_n, a decomposition ϕ_{n_k}= y_k+ z_k such that {y_k, k\ge 1} is relatively weaklycompact and the support projections s(z_k)\downarrow_k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexi…
Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial state $τ$ and let $Δ$ denote the associated Fuglede-Kadison determinant. In this paper, we characterize all unital bijective maps $ϕ$ on the set of invertible positive elements in $\mathscr{R}$ which satisfy $$Δ(ϕ(A)+ϕ(B)) = Δ(A+B).$$ We show that any such map originates from a $τ$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either $*$-automorphism or $*$-anti-automorphism in the more restrictive case of finite factors). In establishing the aforementioned result, we make crucial use of the solutions to the equat…
A synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective "synaptic," borrowed from biology, is meant to suggest that such an algebra coherently "ties together" the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice. Prototypic examples of synaptic algebras are the special Jordan algebra of all self-adjoint elements in a von Neumann algebra, the…
Given a complex Hilbert space H and the von Neumann algebra L(H) of all bounded linear operators on H, we study the Grassmann manifold M of all projections in L(H) that have a fixed finite rank r. We take the Jordan-Banach triple theory approach which allows us to define a natural Levi-Civita connection on M. We identify the geodesics, compute the Riemann distance and prove some properties of M
We define a notion of nonassociative $\mathrm{L}^p$-space associated to a $\mathrm{JBW}^*$-algebra (Jordan von Neumann algebra) equipped with a normal faithful state $φ$. In the particular case of $\mathrm{JW}^*$-algebras underlying von Neumann algebras, we connect these spaces to a complex interpolation theorem of Ricard and Xu on noncommutative $\mathrm{L}^p$-spaces. We also make the link with the nonassociative $\mathrm{L}^p$-spaces of Iochum associated to $\mathrm{JBW}$-algebras and the investigation of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces. More prec…
The $Λ$-sequence spaces $Λ_p$ for $1< p\leq\infty$ and its generalization $Λ_{\hat{p}}$ for $1<\hat{p}<\infty$, $\hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $Λ_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $Λ$-sequence space $Λ_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$. Hence it follows that sequence spaces $Λ_p$ and $Λ_{\hat{p}}$ possesses the uniform Opial property, property $(β)$ of Rolewicz and weak uniform normal …
