An Isomorphism Theorem for Bornological Groups

Transcription

1 International Mathematical Form, Vol. 12, 2017, no. 6, HIKARI Ltd, An Isomorphism Theorem for Bornological rops Dinamérico P. Pombo Jr. Institto de Matemática e Estatística Universidade Federal Flminense Ra Mário Santos Braga, s/n o Niterói, RJ Brasil Copyright c 2017 Dinamérico P. Pombo Jr. This article is distribted nder the Creative Commons Attribtion License, which permits nrestricted se, distribtion, and reprodction in any medim, provided the original work is properly cited. Abstract A necessary and sfficient condition for the validity of a general form of the isomorphism theorem in the context of bornological grops is established. Mathematics Sbject Classification: 46A17 Keywords: Bornological grops, isomorphism theorem 1. Introdction The notion of a bornological grop has been defined and stdied in [6]: a bornological grop is a pair (, B) consisting of a grop and a bornology B on in sch a way that the mapping (x, y) (, B B) xy 1 (, B) is bonded, where B B is the prodct bornology on. In that paper a version of the isomorphism theorem for bornological grops has been obtained, as well as some conseqences of it. The prpose of this note is to establish a more general version of the isomorphism theorem in the context of bornological grops.

2 272 Dinamérico P. Pombo Jr. 2. The reslt Theorem 1. Let (, B) and (H, C) be two bornological grops, : (, B) (H, C) a srjective bonded grop homomorphism, N a normal sbgrop of H and N = 1 (N ) (which is a normal sbgrop of ). Let : /N and : H H/N be the canonical srjections and consider /N and H/N endowed with the corresponding qotient bornologies B = (B) and C = (C) (which are grop bornologies). In order that there exists a (necessarily niqe) bornological grop isomorphism : (/N, B ) (H/N, C ) making the diagram H /N H/N commtative, it is necessary and sfficient that for each C C there exists a B B sch that 1 (C) BN. Proof. First, let s prove the necessity. Indeed, let : (/N, B ) (H/N, C ) be a bornological grop isomorphism as in the statement of the theorem. Pt w =, which is a bonded grop homomorphism from (, B) onto (H/N, C ) sch that Ker(w) = N and = w. Therefore, by the necessity of Theorem 3.16 of [6], for each C C there is a B B sch that w 1 ( (C)) B Ker(w) = BN. Conseqently, 1 (C) BN. In fact, if x 1 (C), w(x) = ((x)) (C) and hence there are b B and n N so that x = bn. Now, let s prove the sfficiency. Indeed, let w be as above and let C C be arbitrary. By hypothesis, there is a B B so that 1 (C) BN. We claim that w 1( (C) ) BN. In fact, if x w 1 ( (C)), (x)n (C) = (( 1 (C))) ((BN)) = ((B)) (N ) = ((B)), and so there is a b B so that (x)n = (b)n. Ths (xb 1 ) N, and xb 1 N, that is, x BN. Therefore, by the sfficiency of Theorem 3.16 of [6], there exists a (necessarily niqe) bornological grop isomorphism : (/N, B ) (H/N, C ) making the diagram w H/N /N commtative. This completes the proof of the theorem.

3 An isomorphism theorem for bornological grops 273 We have obtained Theorem 1 from Theorem 3.16 of [6], bt we cold have given a direct proof of Theorem 1, following the lines of that of Theorem 3.16 of [6]. On the other hand, Theorem 3.16 of [6] is an immediate conseqence of Theorem 1: Corollary 2. Let (, B), (H, C) and be as in Theorem 1. Let : /Ker() be the canonical srjection and let B be the qotient bornology on /Ker(). In order that there exists a (necessarily niqe) bornological grop isomorphism : (/Ker(), B ) (H, C) making the diagram /Ker() H commtative, it is necessary and sfficient that for each C C there exists a B B sch that 1 (C) B Ker(). Proof. Follows immediately from Theorem 1, by taking N as the sbgrop of H redced to its identity element. Remark 3. Theorem 1 implies a well-known isomorphism theorem for grops ([3], p.17, (v)), which reads: Let and H be two grops, : H a grop homomorphism, N a normal sbgrop of H and N = 1 (N ). Then there exists a niqe injective grop homomorphism : /N H/N making the diagram commtative. /N H H/N Indeed, by a straightforward argment, we may assme that is srjective. Consider (resp. H) endowed with the grop bornology B (resp. C) consisting of all sbsets of (resp. H). Then : (, B) (H, C) is a bonded grop homomorphism. Moreover, for all C C, 1 (C) B and 1 (C) 1 (C)N. Ths or claim is an immediate conseqence of Theorem 1, being a grop isomorphism from /N onto H/N in this case. Remark 4. Theorem 1 implies the corresponding reslts for bornological modles [5], linearly bornologized modles [4], bornological vector spaces [7], convex bornological vector spaces over R or C [2], and convex bornological vector spaces over a complete non-archimedean non-trivially valed field [1].

4 274 Dinamérico P. Pombo Jr. For instance, we shall prove the version of Theorem 1 for bornological R- modles, R being a topological ring with a non-zero identity element. Let s first observe that if (E, B) is a bornological R-modle, then ( (E, +), B ) is an abelian bornological grop; moreover, if (F, C) is a bornological R-modle and : (E, B) (F, C) is a bonded R-linear mapping, then : ( (E, +), B ) ( (F, +), C ) is a bonded grop homomorphism. We shall show the validity of the following reslt: Let (E, B) and (F, C) be two bornological R-modles, : (E, B) (F, C) a srjective bonded R-linear mapping, M a sbmodle of F and M = 1 (M ). Let : E E/M and : F F/M be the canonical srjections, and consider E/M and F/M endowed with the corresponding qotient bornologies B and C (which are R-modle bornologies). In order that there exists a (necessarily niqe) bornological R-modle isomorphism : (E/M, B ) (F/M, C ) making the diagram E F E/M F/M commtative, it is necessary and sfficient that for each C C there exists a B B sch that 1 (C) B + M. The existence of a bornological R-modle isomorphism as above and the necessity of Theorem 1 imply that for each C C there is a B B sch that 1 (C) B + M. Conversely, if the latter condition holds, the sfficiency of Theorem 1 garantees the existence of a (necessarily niqe) bornological grop isomorphism : ( (E/M, +), B ) ( (F/M, +), C ) sch that =. Finally, is a bornological R-modle isomorphism, since is an R-linear mapping in the present case. References [1] M. Akkar, Espaces vectoriels bornologiqes K-convexes, Indag. Math., 73 (1970), [2] H. Hogbe-Nlend, Bornologies and Fnctional Analysis, Notas de Matemática 62, North-Holland, [3] S. Lang, Algebra, radate Texts in Mathematics, Vol. 211, Revised third edition, Springer-Verlag, [4] D.P. Pombo Jr., Linear topologies and linear bornologies on modles, J. Indian Math. Soc., 59 (1993), , errata ibid., 60 (1994), 191.

5 An isomorphism theorem for bornological grops 275 [5] D.P. Pombo Jr., Modle bornologies, Comment Math.-Prace Mat., 36 (1996), [6] D.P. Pombo Jr., Bornological grops, Indian J. Math., 54 (2012), [7] C. Hozel (Ed.), Séminaire Banach, Lectre Notes in Mathematics, Vol. 277, Springer-Verlag, Received: Janary 3, 2017; Pblished: Febrary 3, 2017