scholarly journals On Kadison's condition for extreme points of the unit ball in a B*-algebra

1969 ◽  
Vol 16 (3) ◽  
pp. 245-250 ◽  
Author(s):  
Bertram Yood
Keyword(s):  
Unit Ball ◽  
Banach Algebra ◽  
Extreme Points ◽  
Complex Banach Algebra ◽  
Image Position

Let B be a complex Banach algebra with an identity 1 and an involution x→x*. Kadison (1) has shown that, if B is a B*-algebra, [the set of extreme points of its unit ball coincides with the set of elements x of B for which

Some Extreme Rays of the Positive Pluriharmonic Functions

1979 ◽  
Vol 31 (1) ◽  
pp. 9-16 ◽  
Author(s):  
Frank Forelli
Keyword(s):  
Unit Ball ◽  
Extreme Point ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Compact Open Topology ◽  
Open Unit Ball ◽  
Pluriharmonic Functions ◽  
Image Position ◽  
Extreme Rays

1.1. We will denote by B the open unit ball in Cn, and we will denote by H(B) the class of all holomorphic functions on B. LetThus N(B) is convex (and compact in the compact open topology). We think that the structure of N(B) is of interest and importance. Thus we proved in [1] that if(1.1)if(1.2)and if n≧ 2, then g is an extreme point of N(B). We will denote by E(B) the class of all extreme points of N(B). If n = 1 and if (1.2) holds, then as is well known g ∈ E(B) if and only if(1.3)

scholarly journals Banach algebra elements whose spectra intersect R+

1974 ◽  
Vol 19 (1) ◽  
pp. 59-69 ◽  
Author(s):  
F. F. Bonsall ◽  
A. C. Thompson
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Real Numbers ◽  
Complex Banach Algebra ◽  
Image Position ◽  
Invertible Elements

Let A denote a complex Banach algebra with unit, Inv(A) the set of invertible elements of A, Sp(a) and r(a) the spectrum and spectral radius respectively of an element a of A. Let Γ denote the set of elements of A whose spectra contain non-negative real numbers, i.e.

scholarly journals Spectral radius formulae

1979 ◽  
Vol 22 (3) ◽  
pp. 271-275 ◽  
Author(s):  
G. J. Murphy ◽  
T. T. West
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Quotient Algebra ◽  
Complex Banach Algebra ◽  
Image Position

If A is a complex Banach algebra (not necessarily unital) and x∈A, σ(x) will denote the spectrum and spectral radius of x in A. If I is a closed two-sided ideal in A let x + I denote the coset in the quotient algebra A/I containing x. Then

scholarly journals Extreme points in spaces between Dirichlet and Vanishing Mean Oscillation

2003 ◽  
Vol 67 (3) ◽  
pp. 365-375
Author(s):  
K. J. Wirths ◽  
J. Xiao
Keyword(s):  
Unit Ball ◽  
Unit Circle ◽  
Dirichlet Space ◽  
Extreme Points ◽  
Closed Unit Ball ◽  
Mean Oscillation ◽  
Image Position ◽  
Complex Valued

For p ∈ (0, ∞) define Qp0(∂Δ) as the space of all Lebesgue measurable complex-valued functions f; on the unit circle ∂Δ for which ∫∂Δf;(z)|dz|/(2π) = 0 andas the open subarc I of ∂Δ varies. Note that each Qp,0(∂Δ) lies between the Dirichlet space and Sarason's vanishing mean oscillation space. This paper determines the extreme points of the closed unit ball of Qp,0(∂Δ) equipped with an appropriate norm.

scholarly journals A formula for the resolvent of a Reynolds operator

1968 ◽  
Vol 8 (3) ◽  
pp. 447-456 ◽  
Author(s):  
J. B. Miller
Keyword(s):  
Linear Operator ◽  
Banach Algebra ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Complex Banach Algebra ◽  
Reynolds Operator ◽  
Image Position

Let be a complex Banach algebra, possibly non-commutative, with identity e. By a Reynolds operator we mean here a bounded linear operator T: → satisfying the Reynolds identity for all x, y ∈ . We prove that under certain conditions the resolvent of T, R(p, T) = (pI−T)−1, has the form where s = −log(e−Te) and exp y = e+y+y2/2!+….

Idempotents in Complex Banach Algebras

1987 ◽  
Vol 39 (3) ◽  
pp. 625-630
Author(s):  
G. N. Hile ◽  
W. E. Pfaffenberger
Keyword(s):  
Banach Algebra ◽  
Complex Number ◽  
Banach Algebras ◽  
Rectifiable Curve ◽  
Resolvent Set ◽  
Complex Banach Algebra ◽  
Image Position

The concept of the spectrum of A relative to Q, where A and Q commute and are elements in a complex Banach algebra with identity I, was developed in [1]. A complex number z is in the Q-resolvent set of A if and only if is invertible in otherwise, z is in the Q-spectrum of A, or spectrum of A relative to Q. One result from [1] was the following.THEOREM. Suppose no points in the ordinary spectrum of Q have unit magnitude. Let C be a simple closed rectifiable curve which lies in the Q-resolvent of A, and let*where P is defined asxs•

Kolmogorov, Linear and Pseudo-Dimensional Widths of Classes of s-Monotone Functions in 𝕃p, 0 < p < 1

2008 ◽  
Vol 51 (2) ◽  
pp. 236-248
Author(s):  
Victor N. Konovalov ◽  
Kirill A. Kopotun
Keyword(s):  
Unit Ball ◽  
Finite Interval ◽  
Divided Differences ◽  
Monotone Functions ◽  
Image Position

AbstractLet Bp be the unit ball in 𝕃p, 0 < p < 1, and let , s ∈ ℕ, be the set of all s-monotone functions on a finite interval I, i.e., consists of all functions x : I ⟼ ℝ such that the divided differences [x; t0, … , ts] of order s are nonnegative for all choices of (s + 1) distinct points t0, … , ts ∈ I. For the classes Bp := ∩ Bp, we obtain exact orders of Kolmogorov, linear and pseudo-dimensional widths in the spaces , 0 < q < p < 1:

scholarly journals Convex Bodies of Minimal Volume, Surface Area and Mean Width with Respect to Thin Shells

2008 ◽  
Vol 60 (1) ◽  
pp. 3-32 ◽  
Author(s):  
Károly Böröczky ◽  
Károly J. Böröczky ◽  
Carsten Schütt ◽  
Gergely Wintsche
Keyword(s):  
Unit Ball ◽  
Surface Area ◽  
Convex Bodies ◽  
Extreme Points ◽  
Thin Shells ◽  
Minimal Volume ◽  
Asymptotic Formulae ◽  
Mean Width

AbstractGiven r > 1, we consider convex bodies in En which contain a fixed unit ball, and whose extreme points are of distance at least r from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As r tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

scholarly journals A note on liftings of hermitian elements and unitaries

1980 ◽  
Vol 21 (1) ◽  
pp. 183-185
Author(s):  
C. K. Fong
Keyword(s):  
Banach Algebra ◽  
Present Note ◽  
Partial Answer ◽  
General Question ◽  
Quotient Mapping ◽  
Unitary Element ◽  
Finite Dimensional ◽  
Complex Banach Algebra ◽  
Special Cases

Let A be a complex Banach algebra with unit 1 satisfying ∥1∥ = 1. An element u in A is said to be unitary if it is invertible and ∥u∥ = ∥u−1∥ = 1. An element h in A is said to be hermitian if ∥exp(ith)∥ = 1 for all real t; that is, exp(ith) is unitary for all real t. Suppose that J is a closed two-sided ideal and π: A → A/J is the quotient mapping. It is easy to see that if x in A is hermitian (resp. unitary), then so is π(x) in A/J. We consider the following general question which is the converse of the above statement: given a hermitian (resp. unitary) element y in A/J, can we find a hermitian (resp. unitary) element x in A such that π(x)=y? (The author has learned that this question, in a more restrictive form, was raised by F. F. Bonsall and that some special cases were investigated; see [1], [2].) In the present note, we give a partial answer to this question under the assumption that A is finite dimensional.

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