scholarly journals Numerical ranges of powers of hermitian elements

1986 ◽  
Vol 28 (1) ◽  
pp. 37-45 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Previous Knowledge ◽  
Line Segments ◽  
Straight Line ◽  
Image Position ◽  
Unital Banach Algebra

An element k of a unital Banach algebra A is said to be Hermitian if its numerical rangeis contained in ℝ; equivalently, ∥eitk∥ = 1(t ∈ ℝ)—see Bonsall and Duncan [3] and [4]. Here we find the largest possible extent of V(kn), n ∈ ℕ, given V(k) ⊆ [−1, 1], and so ∥k∥ ≤ 1: previous knowledge is in Bollobás [2] and Crabb, Duncan and McGregor [7]. The largest possible sets all occur in a single example. Surprisingly, they all have straight line segments in their boundaries. The example is in [2] and [7], but here we give A. Browder's construction from [5], partly published in [6]. We are grateful to him for a copy of [5], and for discussions which led to the present work. We are also grateful to J. Duncan for useful discussions.

On Banach algebra elements of thin numerical range

1979 ◽  
Vol 86 (1) ◽  
pp. 71-83 ◽  
Author(s):  
R. R. Smith
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Natural Generalization ◽  
The Real ◽  
C Algebra ◽  
Image Position ◽  
Unital Banach Algebra ◽  
Real Subspace

Among the elements of a complex unital Banach algebra the real subspace of hermitian elements deserves special attention. This forms the natural generalization of the set of self-adjoint elements in a C*-algebra and exhibits many of the same properties. Two equivalent definitions may be given: if W(h) ⊂ , where W(h) denotes the numerical range of h (7), or if ║eiλh║ = 1 for all λ ∈ . In this paper some related subsets are introduced and studied. For δ ≥ 0, an element is said to be a member of if the conditionis satisfied. These may be termed the elements of thin numerical range if δ is small.

scholarly journals Polynomials in a hermitian element

1988 ◽  
Vol 30 (2) ◽  
pp. 171-176 ◽  
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Banach Algebra ◽  
Dual Space ◽  
Numerical Range ◽  
Numerical Radius ◽  
Hermitian Element ◽  
Unital Banach Algebra

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

Closure properties of the quasi-centre of a Banach algebra

1990 ◽  
Vol 108 (2) ◽  
pp. 355-364
Author(s):  
J. F. Rennison
Keyword(s):  
Banach Algebra ◽  
Complex Field ◽  
Closure Properties ◽  
Image Position ◽  
Unital Banach Algebra

Throughout the paper, A will denote a unital Banach algebra with centre Z over the complex field ℂ. An element a of A is called quasi-central if, for some K ≥ 1,The set of elements a satisfying (1) for a particular value of K will be denoted by Q(K) and Q = UK≥1Q(K) will be called the quasi-centre of A.

scholarly journals A non-smooth extension of Frechet differentiability of the norm with applications to numerical ranges

1986 ◽  
Vol 28 (2) ◽  
pp. 121-137 ◽  
Author(s):  
C. Aparicio ◽  
F. Ocaña ◽  
R. Payá ◽  
A. Rodríguez
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Banach Algebras ◽  
Continuous States ◽  
Fréchet Differentiability ◽  
Frechet Differentiability ◽  
Smooth Extension ◽  
Unital Banach Algebra

The following result in the theory of numerical ranges in Banach algebras is well known (see [3, Theorem 12.2]). The numerical range of an element F in the bidual of a unital Banach algebra A is the closure of the set of values at F of the w*-continuous states of . As a consequence of the results in this paper the following

scholarly journals Imbedding elements whose numerical range has a vertex at zero in holomorphic semigroups

1985 ◽  
Vol 28 (1) ◽  
pp. 91-95
Author(s):  
J. Martinez-Moreno ◽  
A. Rodriguez-Palacios
Keyword(s):  
Banach Algebra ◽  
Numerical Range ◽  
Compact Convex ◽  
Banach Algebras ◽  
Point Of View ◽  
Holomorphic Semigroup ◽  
Compact Convex Set ◽  
Basic Ideas ◽  
Unital Banach Algebra ◽  
First Time

If a is an element of a complex unital Banach algebra whose numerical range is confined to a closed angular region with vertex at zero and angle strictly less than π, we imbed a in a holomorphic semigroup with parameter in the open right half plane.There has been recently a great development in the theory of semigroups in Banach algebras (see [6]), with attention focused on the relation between the structure of a given Banach algebra and the existence of continuous or holomorphic non-trivial semigroups with certain properties with range in this algebra. The interest of this paper arises from the fact that we relate in it, we think for the first time, this new point of view in the theory of Banach algebras with the already classic one of numerical ranges [2,3]. The proofs of our results use, in addition to some basic ideas from numerical ranges in Banach algebras, the concept of extremal algebra Ea(K) of a compact convex set K in ℂ due to Bollobas [1] and concretely the realization of Ea(K) achieved by Crabb, Duncan and McGregor [4].

Jordan algebras spanned by Hermitian elements of a Banach algebra

1977 ◽  
Vol 81 (1) ◽  
pp. 3-13 ◽  
Author(s):  
F. F. Bonsall
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Banach Algebras ◽  
Jordan Algebras ◽  
The Self ◽  
Image Position ◽  
Adjoint Operators ◽  
Unital Banach Algebra ◽  
Algebra Of Operators

The Vidav–Palmer theorem [(11), (5), (2) (p. 65)] characterizes C*-algebras among Banach algebras in terms of the algebra and norm structure alone, without reference to an involution, in the following way. Let B denote a complex unital Banach algebra, and let Her (B) denote the set of Hermitian elements of B, that is the elements of B with real numerical ranges. In this notation, the Vidav–Palmer theorem tells us that ifthen B is isometrically isomorphic to a C*-algebra of operators on a Hilbert space, with the Hermitian elements corresponding to the self-adjoint operators in the C*-algebra.

The numerical range in the second dual of a Banach algebra

1981 ◽  
Vol 89 (2) ◽  
pp. 301-307
Author(s):  
R. R. Smith
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Numerical Range ◽  
Special Property ◽  
Empty Interior ◽  
Banach Theorem ◽  
Second Dual ◽  
Algebra Theory ◽  
Unital Banach Algebra ◽  
Selection Of

An elementary consequence of the Hahn-Banach theorem is that every Banach space Y is ω*-dense in its second dual Y**, so that every element y ∈ Y** is the w*-limit of a net {ya}α ∈ Λ from Y. There is, of course, a great deal of choice in the selection of such a net, and so one may impose extra conditions on the net related to some special property of the limit point, and then ask for existence. The object of this paper is to present such a result in the context of a unital Banach algebra and its second dual , and then to give two applications to Banach algebra theory. The theorem to be proved is this: if the numerical range W(a) of an element in has non-empty interior then a is the ω*-limit of a net {aa}α ∈ Λ from whose numerical ranges are contained in W(a), while if W(a) has empty interior then the numerical ranges W(aα) are contained in a shrinking set of neighbourhoods of W(a).

A characterization of Jordan and 5-Jordan homomorphisms between Banach algebras

2018 ◽  
Vol 11 (02) ◽  
pp. 1850021 ◽  
Author(s):  
A. Zivari-Kazempour
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Commutative Banach Algebra ◽  
Jordan Homomorphism ◽  
Jordan Homomorphisms ◽  
Unital Banach Algebra

We prove that each surjective Jordan homomorphism from a Banach algebra [Formula: see text] onto a semiprime commutative Banach algebra [Formula: see text] is a homomorphism, and each 5-Jordan homomorphism from a unital Banach algebra [Formula: see text] into a semisimple commutative Banach algebra [Formula: see text] is a 5-homomorphism.

scholarly journals Use of otoliths to estimate size at sexual maturity in fish

2000 ◽  
Vol 43 (4) ◽  
pp. 437-440 ◽  
Author(s):  
Carlos Sérgio Agostinho
Keyword(s):  
Sexual Maturity ◽  
Line Segments ◽  
Alternative Method ◽  
Straight Line ◽  
Size At Sexual Maturity

The viability of an alternative method for estimating the size at sexual maturity of females of Plagioscion squamosissimus (Perciformes, Sciaenidae) was analyzed. This methodology was used to evaluate the size at sexual maturity in crabs, but has not yet been used for this purpose in fishes. Separation of young and adult fishes by this method is accomplished by iterative adjustment of straight-line segments to the data for length of the otolith and length of the fish. The agreement with the estimate previously obtained by another technique and the possibility of calculating the variance indicates that in some cases, the method analyzed can be used successfully to estimate size at sexual maturity in fish. However, additional studies are necessary to detect possible biases in the method.

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