scholarly journals On compact action in JB-algebras

1983 ◽  
Vol 26 (3) ◽  
pp. 353-360 ◽  
Author(s):  
L. J. Bunce
Keyword(s):  
Banach Space ◽  
Jordan Algebra ◽  
Dual Space ◽  
Image Position

A real Jordan algebra which is also a Banach space with a norm which satisfiesfor each pair a, b of elements, is said to be a JB-algebra. A JB-algebra which is also a Banach dual space is said to be a JBW-algebra.

On the centres of hereditary JBW-subalgebras of a JBW-algebra

1979 ◽  
Vol 85 (2) ◽  
pp. 317-324 ◽  
Author(s):  
C. M. Edwards
Keyword(s):  
Banach Space ◽  
General Theory ◽  
Jordan Algebra ◽  
Identity Element ◽  
Image Position

A JB-algebra A is a real Jordan algebra, which is also a Banach space, the norm in which satisfies the conditions thatandfor all elements a and b in A. It follows from (1.1) and (l.2) thatfor all elements a and b in A. When the JB-algebra A possesses an identity element then A is said to be a unital JB-algebra and (1.2) is equivalent to the condition thatfor all elements a and b in A. For the general theory of JB-algebras the reader is referred to (2), (3), (7) and (10).

JV-algebras

1980 ◽  
Vol 87 (1) ◽  
pp. 47-50 ◽  
Author(s):  
J. Martinez Moreno
Keyword(s):  
Banach Space ◽  
Jordan Algebra ◽  
Complex Banach Space ◽  
C Algebra ◽  
Image Position

Let J be a complex Banach space and a complex Jordan algebra equipped with an algebra involution *. Then J is a Jordan C*-algebra if the following conditions are satisfied:(Ua is defined on page 3).

Complex strict and uniform convexity and hyponormal operators

1984 ◽  
Vol 96 (3) ◽  
pp. 483-493 ◽  
Author(s):  
Kirsti Mattila
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Numerical Range ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Uniform Convexity ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Image Position

Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. The (spatial) numerical range of an operator TεB(X) is defined as the setIf V(T) ⊂ ℝ, then T is called hermitian. More about numerical ranges may be found in [8] and [9].

scholarly journals Hermitian Operators on Banach Jordan Algebras

1979 ◽  
Vol 22 (2) ◽  
pp. 169-180 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Banach Space ◽  
Jordan Algebra ◽  
Numerical Range ◽  
Jordan Algebras ◽  
Image Position

In this note, we examine some of the properties of Hermitian operators on complex unital Banach Jordan algebras, that is, those operators with real numerical range. Recall that a unital Banach Jordan algebra J, is a (real or complex) Jordan algebra with product a ˚ b, having a unit 1, and a norm ∥·∥, such that J, with norm ∥·∥, is a Banach space, ∥1∥ = 1, and, for all a and b in j,

Semifinite traces on JBW-algebras

1983 ◽  
Vol 93 (3) ◽  
pp. 503-509 ◽  
Author(s):  
W. P. C. King
Keyword(s):  
Banach Space ◽  
Jordan Algebra ◽  
Dual Space

A JB-algebra is a real Jordan algebra A which is also a Banach space and whose norm and multiplication satisfy the two following conditions(i) ∥a2∥ = ∥a∥2,(ii) ∥a2 − b2∥ ≤ max{∥a2∥, ∥b2∥},for all elements a and b in A. A JB-algebra which is also a Banach dual space is called a JBW-algebra. The properties of JB-algebras and JBW-algebras can be found in (3), (4), (8) and (15).

scholarly journals On the numerical radius of an element of a normed algebra

1974 ◽  
Vol 15 (1) ◽  
pp. 90-93
Author(s):  
GH. Mocanu
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Complex Field ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Normed Algebra ◽  
Numerical Radius ◽  
Image Position

Let A be a unital normed algebra over the complex field ℂ, A' the dual space of A, i.e., the Banach space of all continuous linear functionals on A, and let S be the set of all states on A, i.e.,

SOME RESULTS OF THE -APPROXIMATION PROPERTY FOR BANACH SPACES

2018 ◽  
Vol 61 (03) ◽  
pp. 545-555 ◽  
Author(s):  
JU MYUNG KIM
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Dual Space ◽  
Approximation Property ◽  
Compact Operators ◽  
Operator Ideal ◽  
Formula Group ◽  
Image Position ◽  
The Ideal

AbstractGiven a Banach operator ideal $\mathcal A$, we investigate the approximation property related to the ideal of $\mathcal A$-compact operators, $\mathcal K_{\mathcal A}$-AP. We prove that a Banach space X has the $\mathcal K_{\mathcal A}$-AP if and only if there exists a λ ≥ 1 such that for every Banach space Y and every R ∈ $\mathcal K_{\mathcal A}$(Y, X), $$ \begin{equation} R \in \overline {\{SR : S \in \mathcal F(X, X), \|SR\|_{\mathcal K_{\mathcal A}} \leq \lambda \|R\|_{\mathcal K_{\mathcal A}}\}}^{\tau_{c}}. \end{equation} $$ For a surjective, maximal and right-accessible Banach operator ideal $\mathcal A$, we prove that a Banach space X has the $\mathcal K_{(\mathcal A^{{\rm adj}})^{{\rm dual}}}$-AP if the dual space of X has the $\mathcal K_{\mathcal A}$-AP.

Nonlinear random equations with maximal monotone operators in Banach spaces

1985 ◽  
Vol 98 (3) ◽  
pp. 529-532 ◽  
Author(s):  
Dimitrios Kravvaritis
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Real Banach Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Measurable Space ◽  
Random Operator ◽  
Image Position ◽  
Monotone Type

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such thatRandom equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

The structure of type I JBW-algebras

1981 ◽  
Vol 90 (3) ◽  
pp. 477-482 ◽  
Author(s):  
P. J. Stagey
Keyword(s):  
Jordan Algebra ◽  
Dual Space ◽  
Cardinal Number ◽  
Complex Hilbert Space ◽  
Type I ◽  
Direct Sum ◽  
Cardinal Numbers ◽  
Weakly Closed ◽  
Image Position ◽  
Adjoint Operators

A complete normed real Jordan algebra A with 1 is said to be a JB-algebra if and only if, for all a, b ∈ A,where o denotes the Jordan product on A. If A is a Banach dual space then A is known as a JBW-algebra. In this case, by Theorem 3·9 of (7),where Aex is an exceptional Jordan algebra and Asp is Jordan isomorphic to a JW-algebra (i.e. a weakly closed Jordan algebra of self-adjoint operators on a complex Hilbert space). Theorem 3–9 of (7) gives a description of Aex and so in this paper attention will be restricted to JW-algebras. Such an algebra A is said to be of type I if there exists an idempotent e in A with central cover equal to 1 such that e is abelian i.e. such that eAe is commutative as a set of operators. (See § 7 of (11) for further details.) A type I algebra A is said to be of type In, where n is some cardinal number, if there exists an orthogonal family {eα: α ∈ Λ} of n abelian projections in A such that Σαeα = 1 and such that, for each α ≠ β, there exists a symmetry sαβ in A with eβ = sαβeαsαβ. (A symmetry is an element sαβ of A such that ). By theorems 15 and 16 of (11) any type I JW-algebra A can be uniquely decomposed into a direct sum of type In(α) JW-algebras for suitable cardinal numbers n(α). Thus in order to investigate the structure of type I JW-algebras it is sufficient to consider type In JW-algebras.

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

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