Maps preserving zero products

J. Alaminos (Author), Matej Brešar (Author), J. Extremera (Author), A. R. Villena (Author)

Abstract

Linearna preslikava ▫$T$▫ iz Banachove algebre ▫$A$▫ v Banachovo algebro ▫$B$▫ ohranja ničelni produkt, če je ▫$T(a)T(b) = 0$▫, kadarkoli je ▫$ab = 0$▫. Glavna tema članka je vprašanje, kdaj je zvezna linearna surjektivna preslikava ▫$T: A \to B$▫, ki ohranja ničelni produkt, uteženi homomorfizem. Dokažemo, da to velja za velik razred algeber, ki vključuje grupne algebre. Naša metoda sloni na obravnavi bilinearnih preslikav ▫$\phi : A \times A \to X$▫ (kjer je ▫$X$▫ Banachov prostor) z lastnostjo, da iz ▫$ab=0$▫ sledi ▫$\phi(a,b) = 0$▫. Dokažemo, da taka preslikava zadošča ▫$\phi(a\mu, b) = \phi(a,\mu b)$▫ za vse ▫$a,b \in A$▫ in vse ▫$\mu$▫ iz zaprtja glede na krepko operatorsko topologijo podalgebre multiplikacijske algebre ▫${\mathcal M}(A)$▫ generirane z dvostranko potenčno omejenimi elementi. Ta metoda je uporabna tudi za karakterizacijo odvajanj s pomočjo ničelnega produkta.

Keywords

matematika;teorija operatorjev;grupna algebra;▫$C^\ast$▫-algebra;homomorfizem;uteženi homomorfizem;odvajanje;posplošeno odvajanje;mathematics;operator theory;group algebra;homomorphism;weighted homomorphism;derivation;generalized derivation;separating map;disjointness preserving map;zero product preserving map;doubly power-bounded element;

Data

Language:	English
Year of publishing:	2009
Typology:	1.01 - Original Scientific Article
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
UDC:	517.983
COBISS:	15201369
ISSN:	0039-3223
Views:	48
Downloads:	5
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Uhranjevalci ničelnega produkta
Secondary abstract:	A linear map ▫$T$▫ from a Banach algebra ▫$A$▫ into another ▫$B$▫ preserves zero products if ▫$T(a)T(b) = 0$▫ whenever ▫$a,b \in A$▫ are such that ▫$ab = 0$▫. This paper is mainly concerned with the question of whether every continuous linear surjective map ▫$T: A \to B$▫ that preserves zero products is a weighted homomorphism. We show that this is indeed the case for a large class of Banach algebras which includes group algebras. Our method involves continuous bilinear maps ▫$\phi : A \times A \to X$▫ (for some Banach space ▫$X$▫) with the property that ▫$\phi(a,b) = 0$▫ whenever ▫$a,b \in A$▫ are such that ▫$ab = 0$▫. We prove that such a map necessarily satises ▫$\phi(a\mu, b) = \phi(a, \mu b)$▫ for all ▫$a,b \in A$▫ and for all ▫$\mu$▫ from the closure with respect to the strong operator topology of the subalgebra of ▫${\mathcal M}(A)$▫ (the multiplier algebra of ▫$A$▫) generated by doubly power-bounded elements of ▫${\mathcal M}(A)$▫. This method is also shown to be useful for characterizing derivations through the zero products.
Secondary keywords:	matematika;teorija operatorjev;grupna algebra;▫$C^\ast$▫-algebra;homomorfizem;uteženi homomorfizem;odvajanje;posplošeno odvajanje;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 131-159
Volume:	ǂVol. ǂ193
Issue:	ǂno. ǂ2
Chronology:	2009
ID:	1474353