scholarly journals Interpolation and inequalities for functions of exponential type: the Arens irregularity of an extremal algebra

1993 ◽  
Vol 35 (3) ◽  
pp. 325-326
Author(s):  
M. J. Crabb ◽  
C. M. McGregor
Keyword(s):  
Exponential Type ◽  
Line Segment ◽  
Convex Set ◽  
Numerical Range ◽  
Compact Convex ◽  
Banach Algebras ◽  
Functions Of Exponential Type ◽  
Compact Convex Set ◽  
Unital Banach Algebra ◽  
Extremal Algebra

For any compact convex set K ⊂ ℂ there is a unital Banach algebra Ea(K) generated by an element h in which every polynomial in h attains its maximum norm over all Banach algebras subject to the numerical range V(h) being contained in K, [1]. In the case of K a line segment in ℝ, we show here that Ea(K) does not have Arens regular multiplication. We also show that ideas about Ea(K) give simple proofs of, and extend, two inequalities of C. Frappier [4].

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].

Tensor products of compact convex sets and Banach algebras

1978 ◽  
Vol 83 (3) ◽  
pp. 419-427 ◽  
Author(s):  
C. J. K. Batty
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Banach Algebras ◽  
Tensor Products ◽  
Unit Space ◽  
Compact Convex Set ◽  
Affine Functions ◽  
Left Annihilator ◽  
Order Unit Space

Alfsen and Andersen(2) defined the centre of the complete order-unit space A(K) associated with a compact convex set K to be the set of functions in A(K) which multiply with A(K) pointwise on the extreme boundary of K, thereby generalizing the concept of centres of C*-algebras. It is therefore possible to extend this definition to include the space A (K; B) of continuous affine functions of K into a Banach algebra B. Such spaces arise in the theory of weak tensor products E ⊗λB of B with a Banach space E, which may be embedded in A(K; B) where K is the unit ball of E* in the weak* topology. Andersen and Atkinson(4) considered multipliers in A(K; B) and showed that if B is unital, then the multipliers are precisely those functions which are continuous in the facial topology on the extreme boundary. It is shown here that this result extends to non-unital Banach algebras with trivial left annihilator.

scholarly journals Diameter-preserving linear bijections of function spaces

2001 ◽  
Vol 70 (3) ◽  
pp. 323-336 ◽  
Author(s):  
T. S. S. R. K. Rao ◽  
A. K. Roy
Keyword(s):  
Maximal Ideal ◽  
Convex Set ◽  
Compact Convex ◽  
Extreme Points ◽  
Continuous Functions ◽  
Ideal Space ◽  
Function Algebras ◽  
Compact Convex Set ◽  
Vector Valued Functions ◽  
Vector Valued

AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (02) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random setsX, densitiesDφ(X) of additive functionalsφare defined and formulas forare derived whenKis a compact convex set in. In particular, for the quermassintegrals and motioninvariantX, these formulas are in analogy with classical integral geometric formulas. The case whereXis the union set of a Poisson processYof convex particles is considered separately. Here, formulas involving the intensity measure ofYare obtained.

Densities for stationary random sets and point processes

1984 ◽  
Vol 16 (2) ◽  
pp. 324-346 ◽  
Author(s):  
Wolfgang Weil ◽  
John A. Wieacker
Keyword(s):  
Poisson Process ◽  
Point Processes ◽  
Convex Set ◽  
Compact Convex ◽  
Random Sets ◽  
Intensity Measure ◽  
Additive Functionals ◽  
Compact Convex Set ◽  
Classical Integral

For certain stationary random sets X, densities Dφ (X) of additive functionals φ are defined and formulas for are derived when K is a compact convex set in . In particular, for the quermassintegrals and motioninvariant X, these formulas are in analogy with classical integral geometric formulas. The case where X is the union set of a Poisson process Y of convex particles is considered separately. Here, formulas involving the intensity measure of Y are obtained.

Hyperplans poissoniens et compacts de Steiner

1974 ◽  
Vol 6 (03) ◽  
pp. 563-579 ◽  
Author(s):  
G. Matheron
Keyword(s):  
Stationary Process ◽  
Convex Set ◽  
Compact Convex ◽  
Minkowski Sum ◽  
Isotropic Case ◽  
Geometrical Properties ◽  
Steiner Formula ◽  
Compact Convex Set ◽  
Finite Sums ◽  
Linear Variety

A compact convex set in RN is Steiner if it is a finite Minkowski sum of line segments, or a limit of such finite sums, and then satisfies an extension of the Steiner formula. With each Poisson hyperplane stationary process A is uniquely associated a Steiner set M, and for any linear variety V, the Steiner set associated with is the projection of M on V. The density of the order k network Ak (i.e., the set of the intersections of k hyperplanes belonging to A) is linked with simple geometrical properties of M. In the isotropic case, the expression of the covariance measures associated with Ak is derived and compared with the analogous results obtained for (N — k)-dimensional Poisson flats.

Some Remarks on Translative Coverings of Convex Domains by Strips

1984 ◽  
Vol 27 (2) ◽  
pp. 233-237 ◽  
Author(s):  
H. Groemer
Keyword(s):  
Euclidean Plane ◽  
Convex Set ◽  
Compact Convex ◽  
Convex Domains ◽  
Minimal Width ◽  
Compact Convex Set

AbstractIn the euclidean plane let K be a compact convex set and Sl, S2,… strips of respective widths wl, w2,… Some conditions on Σ wi are given that imply that K can be covered by translates of the strips Si. These conditions involve the perimeter, the diameter, or the minimal width of K and yield improvements of previously known results.

Rates of convergence for random approximations of convex sets

1996 ◽  
Vol 28 (02) ◽  
pp. 384-393 ◽  
Author(s):  
Lutz Dümbgen ◽  
Günther Walther
Keyword(s):  
Convex Subset ◽  
Hausdorff Distance ◽  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Almost Sure Convergence ◽  
Rates Of Convergence ◽  
Random Sets ◽  
Compact Convex Set

The Hausdorff distance between a compact convex set K ⊂ ℝd and random sets is studied. Basic inequalities are derived for the case of being a convex subset of K. If applied to special sequences of such random sets, these inequalities yield rates of almost sure convergence. With the help of duality considerations these results are extended to the case of being the intersection of a random family of halfspaces containing K.

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