zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Approximately multiplicative maps between Banach algebras. (English) Zbl 0652.46031
A pair (${\cal A},{\cal B})$ of Banach algebras is said to have the property AMNM (almost multiplicative maps are near multiplicative maps), if on bounded subsets of L(${\cal A},{\cal B})$ (the Banach space of bounded linear operators from ${\cal A}$ into ${\cal B})$ for any $\epsilon >0$ there exists a $\delta <0$ such that for any $T\in L({\cal A},{\cal B})$ the inequality $\Vert T(ab)-T(a)T(b)\Vert \le \delta \Vert a\Vert \Vert b\Vert (a,b\in {\cal A})$ implies $\Vert T-T'\Vert \le \epsilon$ for some multiplicative map T’$\in L({\cal A},{\cal B})$. This paper is devoted to the question, which pairs of Banach algebras are AMNM pairs. As a central result this property is proven, when ${\cal A}$ is an amenable algebra (these are studied by the author in [Cohomology in Banach algebras, Mem. Am. Math. soc. 127 (1972; Zbl 0256.18014)]) and ${\cal B}$ is the dual of a ${\cal B}$-bimodule. This leads to results for the combination of group algebras with commutative algebras. Further positive answers are obtained for the case where ${\cal B}$ is the algebra of all continuous functions on a compact Hausdorff space. Finally it is shown that the property AMNM holds, if ${\cal A}$ and ${\cal B}$ both equal to the algebra of all bounded linear operators on a separable Hilbert space. A corresponHeisenberg group. This class is substantially larger than in the one-dimensional case, but the additional condition of invariance under affine automorphisms distinguishes two nontrivial algebras on $H\sp n$ analogous to the Phragmén-Lindelöf algebra (this is due to the nontriviality of the center of the group $H\sp n)$.
Reviewer: J.B.Prolla

46H05General theory of topological algebras
46H25Normed modules and Banach modules, topological modules
Full Text: DOI