Maps preserving zero products

Alaminos, J.; Brešar, Matej; Extremera, J.; Villena, A. R.

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Title:	Maps preserving zero products
Authors:	ID Alaminos, J. (Author) ID Brešar, Matej (Author) ID Extremera, J. (Author) ID Villena, A. R. (Author)
Files:	http://dx.doi.org/10.4064/sm193-2-3
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	FNM - Faculty of Natural Sciences and Mathematics
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 ▫$phi(amu, b) = phi(a,mu b)$▫ 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, ▫$C^ast$▫-algebra, homomorphism, weighted homomorphism, derivation, generalized derivation, separating map, disjointness preserving map, zero product preserving map, doubly power-bounded element
Year of publishing:	2009
Number of pages:	str. 131-159
Numbering:	Vol. 193, no. 2
PID:	20.500.12556/DKUM-51794
UDC:	517.983
ISSN on article:	0039-3223
COBISS.SI-ID:	15201369
NUK URN:	URN:SI:UM:DK:CHMSVEF1
Publication date in DKUM:	10.07.2015
Views:	1301
Downloads:	99
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Categories:	Misc.
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Record is a part of a journal

Title:	Studia Mathematica
Shortened title:	Stud. Math.
Publisher:	Państwowe Wydawnictwo Naukowe
ISSN:	0039-3223
COBISS.SI-ID:	26464256

Secondary language

Language:	Unknown
Title:	Uhranjevalci ničelnega produkta
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 ▫$phi(amu, b) = phi(a, mu b)$▫ 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.

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