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1.
Let A be an algebra without unit. If ∥ ∥ is a complete regular norm on A it is known that among the regular extensions of ∥ ∥ to the unitization of A there exists a minimal (operator extension) and maximal (ℓ1-extension) which are known to be equivalent. We shall show that the best upper bound for the ratio of these two extensions is exactly 3. This improves the results represented by A. K. Gaur and Z. V. Kovářík and later by T. W. Palmer. The second author was partially supported by the grant No. 201/03/0041 of GAČR.  相似文献   

2.
Let K be a complete ultrametric algebraically closed field and let A be the K-Banach algebra of bounded analytic functions in the disk D: |x| < 1. Let Mult(A, ∥ · ∥) be the set of continuous multiplicative semi-norms of A, let Mult m (A, ∥ · ∥) be the subset of the ϕMult(A, ∥ · ∥) whose kernel is a maximal ideal and let Mult a (A, ∥ · ∥) be the subset of the ϕMult m (A, ∥ · ∥) whose kernel is of the form (x − a)A, aD ( if ϕMult m (A, ∥ · ∥) \ Mult a (A, ∥ · ∥), the kernel of ϕ is then of infinite codimension). We examine whether Mult a (A, ∥ · ∥) is dense inside Mult m (A, ∥ · ∥) with respect to the topology of simple convergence. This a first step to the conjecture of density of Mult a (A, ∥ · ∥) in the whole set Mult(A, ∥ · ∥): this is the corresponding problem to the well-known complex corona problem. We notice that if ϕMult m (A, ∥ · ∥) is defined by an ultrafilter on D, then ϕ lies in the closure of Mult a (A, ∥ · ∥). Particularly, we show that this is case when a maximal ideal is the kernel of a unique ϕMultm(A, ∥ · ∥). Particularly, when K is strongly valued all maximal ideals enjoy this property. And we can prove this is also true when K is spherically complete, thanks to the ultrametric holomorphic functional calculus. More generally, we show that if ψMult(A, ∥ · ∥) does not define the Gauss norm on polynomials (∥ · ∥), then it is defined by a circular filter, like on rational functions and analytic elements. As a consequence, if ψ ∈ Multm(A, ∥ · ∥) \ Multa(A, ∥ · ∥) or if φ does not lie in the closure of Mult a (A, ∥ · ∥), then its restriction to polynomials is the Gauss norm. The first situation does happen. The second is unlikely. The text was submitted by the authors in English.  相似文献   

3.
Von Neumann-Jordan Constants of Absolute Normalized Norms on C^n   总被引:1,自引:0,他引:1  
In this note, we give some estimations of the Von Neumann-Jordan constant C N J (∥·∥ψ) of Banach space (ℂ n , ∥·∥ψ), where ∥·∥ψ is the absolute normalized norm on ℂ n given by function ψ. In the case where ψ and φ are comparable, n=2 and C N J (∥·∥ψ)=1, we obtain a formula of computing C N J (∥·∥ψ). Our results generalize some results due to Saito and others. Received May 11, 2002, Accepted November 20, 2002 This work is partly supported by NNSF of China (No. 19771056)  相似文献   

4.
For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖ G onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖ G ) is isomorphic toG × {−1, + 1}.  相似文献   

5.
When the base connected cochain DG algebra is cohomologically bounded, it is proved that the difference between the amplitude of a compact DG module and that of the DG algebra is just the projective dimension of that module. This yields the unboundedness of the cohomology of non-trivial regular DG algebras. When A is a regular DG algebra such that H(A) is a Koszul graded algebra, H(A) is proved to have the finite global dimension. And we give an example to illustrate that the global dimension of H(A) may be infinite, if the condition that H(A) is Koszul is weakened to the condition that A is a Koszul DG algebra. For a general regular DG algebra A, we give some equivalent conditions for the Gorensteiness. For a finite connected DG algebra A, we prove that Dc(A) and Dc(A op) admit Auslander-Reiten triangles if and only if A and A op are Gorenstein DG algebras. When A is a non-trivial regular DG algebra such that H(A) is locally finite, Dc(A) does not admit Auslander-Reiten triangles. We turn to study the existence of Auslander-Reiten triangles in Dlfb(A) and Dlfb (A op) instead, when A is a regular DG algebra. This work was supported by the National Natural Science Foundation of China (Grant No. 10731070) and the Doctorate Foundation of Ministry of Education of China (Grant No. 20060246003)  相似文献   

6.
UniversalC*-algebrasC*(A) exist for certain topological *-algebras called algebras with aC*-enveloping algebra. A Frechet *-algebraA has aC*-enveloping algebra if and only if every operator representation ofA mapsA into bounded operators. This is proved by showing that every unbounded operator representation π, continuous in the uniform topology, of a topological *-algebraA, which is an inverse limit of Banach *-algebras, is a direct sum of bounded operator representations, thereby factoring through the enveloping pro-C*-algebraE(A) ofA. Given aC*-dynamical system (G,A,α), any topological *-algebraB containingC c (G,A) as a dense *-subalgebra and contained in the crossed productC*-algebraC*(G,A,α) satisfiesE(B) =C*(G,A,α). IfG = ℝ, ifB is an α-invariant dense Frechet *-subalgebra ofA such thatE(B) =A, and if the action α onB ism-tempered, smooth and by continuous *-automorphisms: then the smooth Schwartz crossed productS(ℝ,B,α) satisfiesE(S(ℝ,B,α)) =C*(ℝ,A,α). WhenG is a Lie group, theC -elementsC (A), the analytic elementsC ω(A) as well as the entire analytic elementsC є(A) carry natural topologies making them algebras with aC*-enveloping algebra. Given a non-unitalC*-algebraA, an inductive system of idealsI α is constructed satisfyingA =C*-ind limI α; and the locally convex inductive limit ind limI α is anm-convex algebra with theC*-enveloping algebraA and containing the Pedersen idealK a ofA. Given generatorsG with weakly Banach admissible relationsR, we construct universal topological *-algebraA(G, R) and show that it has aC*-enveloping algebra if and only if (G, R) isC*-admissible.  相似文献   

7.
Given a C*-normed algebra A which is either a Banach *-algebra or a Frechet *-algebra, we study the algebras Ω A and Ωε A obtained by taking respectively the projective limit and the inductive limit of Banach *-algebras obtained by completing the universal graded differential algebra Ω*A of abstract non-commutative differential forms over A. Various quantized integrals on Ω A induced by a K-cycle on A are considered. The GNS-representation of Ω A defined by a d-dimensional non-commutative volume integral on a d +-summable K-cycle on A is realized as the representation induced by the left action of A on Ω*A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).  相似文献   

8.
Let A be a complex Banach algebra. It is well known that the second dual A** of A can be equipped with a multiplication that extends the original multiplication on A and makes A** a Banach algebra. We show that Rad(A) = (A * · A) and Rad(A **) = (A * · A) for some classes of Banach algebras A with scattered structure space. Some applications of these results are given.  相似文献   

9.
Résumé.  Soit A une algèbre réelle sans diviseurs de zéro. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a 2∥ ≤ ∥a2 pour tout . Alors A est de dimension finie dans chacun des quatre cas suivants :
1.  A est commutative contenant un élément non nul a tel que ∥ax∥ = ∥a∥ ∥x∥ pour tout ,
2.  A est commutative algébrique et ∥a 2∥ = ∥a2 pour tout ,
3.  A est alternative contenant un élément unité e tel que ∥e∥ = 1,
4.  A est alternative contenant un élément central non nul a tel que ∥ax∥ = ∥a∥ ∥x∥ pour tout .
A est isomorphe à ou dans les deux premiers cas et isomorphe à ou dans les deux derniers cas.
Let A be a real algebra without divisor of zero. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a 2∥ ≤ ∥a2 for all . Then A is finite dimensional in the four following cases :
1.  A is a commutative containing a nonzero element a such that ∥ax∥ = ∥a∥∥x∥ for all ,
2.  A is a commutative algebraic and ∥a 2∥ = ∥a2 for all ,
3.  A is an alternative containing a unit element e such that ∥e∥ = 1,
4.  A is an alternative containing a nonzero central element a such that ∥ax∥ = ∥ a∥∥x∥ for all .
A is isomorphic to or in the two first cases and isomorphic to or in the two last cases.
  相似文献   

10.
Résumé. Soit A une algèbre réelle. On suppose que l’espace vectoriel A est muni d’une norme ∥.∥ préhilbertienne vérifiant ∥a 2∥ = ∥a2 pour tout . Si A est flexible, sans diviseurs de zéro et de dimension ≤ 4, alors A est isomorphe à ou , ce qui généralise un théorème d’El-Mallah [1]. Si A est flexible, sans diviseurs de zéro, contenant un idempotent central et vérifiant la propriété d’Osborn, alors A est de dimension finie et isomorphe à , ou . Enfin nous montrons qu’une algèbre normée préhilbertienne unitaire d’unité e telle que ∥e∥ = 1 est flexible et vérifie ∥a 2∥ = ∥ a2.
Let A be a real algebra. Assuming that a vector space A is endowed with a pre-Hilbert norm ∥.∥ satisfying ∥a 2∥ = ∥a2 for all . If A is flexible, without divisor of zero and of a dimension ≤ 4, then A is isomorphic to or , which generalize El-Mallah’s theorem [1]. If A is flexible, without divisor of zero, containing a central idempotent and satisfying Osborn’s properties, then A is finite dimensional and isomorphic to , or . Finally we prove that a normed pre-Hilbert algebra with unit e such that ∥e∥ = 1 is flexible and satisfies ∥a 2∥ = ∥a2.
  相似文献   

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