On local automorphisms of group algebras

of compact groups

Authors:
Lajos Molnár and Borut Zalar

Journal:
Proc. Amer. Math. Soc. **128** (2000), 93-99

MSC (1991):
Primary 43A15, 43A22, 46H99

DOI:
https://doi.org/10.1090/S0002-9939-99-05108-4

Published electronically:
June 30, 1999

MathSciNet review:
1637412

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that with few exceptions every local isometric automorphism of the group algebra of a compact metric group is an isometric automorphism.

**[BaMo]**C.J.K. Batty and L. Molnár,*On topological reflexivity of the groups of *-automorphisms and surjective isometries of*, Arch. Math.**67**(1996), 415-421. MR**97f:47034****[Bre]**M. Bre\v{s}ar,*Characterizations of derivations on some normed algebras with involution*, J. Algebra**152**(1992), 454-462. MR**94e:46098****[BrSe1]**M. Bre\v{s}ar and P. \v{S}emrl,*Mappings which preserve idempotents, local automorphisms, and local derivations*, Canad. J. Math.**45**(1993), 483-496. MR**94k:47054****[BrSe2]**M. Bre\v{s}ar and P. \v{S}emrl,*On local automorphisms and mappings that preserve idempotents*, Studia Math.**113**(1995), 101-108. MR**96i:47058****[Cri]**R.L. Crist,*Local derivations on operator algebras*, J. Funct. Anal.**135**(1996), 76-92. MR**96m:46128****[HeRo]**E. Hewitt and K.A. Ross,*Abstract Harmonic Analysis I., II.*, Springer Verlag, 1963, 1970. MR**28:158**; MR**41:7378****[Kad]**R.V. Kadison,*Local derivations*, J. Algebra**130**(1990), 494-509. MR**91f:46092****[Lar]**D.R. Larson,*Reflexivity, algebraic reflexivity and linear interpolation*, Amer. J. Math.**110**(1988), 283-299. MR**89d:47096****[LaSo]**D.R. Larson and A.R. Sourour,*Local derivations and local automorphisms of*, in Proc. Sympos. Pure Math. 51, Part 2, Providence, Rhode Island, 1990, 187-194. MR**91k:47106****[Mol1]**L. Molnár,*Algebraic difference between -classes of an -algebra*, Proc. Amer. Math. Soc.**124**(1996), 169-175. MR**96d:46072****[Mol2]**L. Molnár,*The set of automorphisms of is topologically reflexive in*, Studia Math.**122**(1997), 183-193. MR**98e:47068****[Mol3]**L. Molnár,*Reflexivity of the automorphism and isometry groups of -algebras in BDF theory*, Arch. Math., to appear.**[Mol4]**L. Molnár and M. Gy\H{o}ry,*Reflexivity of the automorphism and isometry groups of the suspension of*, J. Funct. Anal.**159**(1998), 568-586. CMP**99:04****[RuDy]**B. Russo and H.A. Dye,*A note on unitary operators in -algebras*, Duke Math. J.**33**(1966), 413-416. MR**33:1750****[Sco]**W.R. Scott,*Half-homomorphisms of groups*, Proc. Amer. Math. Soc.**8**(1957), 1141-1144. MR**20:2388****[Shu]**V.S. Shul'man,*Operators preserving ideals in -algebras*, Studia Math.**109**(1994), 67-72. MR**95b:46097****[Str]**R.S. Strichartz,*Isomorphisms of group algebras*, Proc. Amer. Math. Soc.**17**(1966), 858-862. MR**33:1751****[ZhXi]**J. Zhu and C. Xiong,*Bilocal derivations of standard operator algebras*, Proc. Amer. Math. Soc.**125**(1997), 1367-1370. MR**97g:46091**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
43A15,
43A22,
46H99

Retrieve articles in all journals with MSC (1991): 43A15, 43A22, 46H99

Additional Information

**Lajos Molnár**

Affiliation:
Institute of Mathematics, Lajos Kossuth University, 4010 Debrecen, P.O. Box 12, Hungary

Email:
molnarl@math.klte.hu

**Borut Zalar**

Affiliation:
Department of Basic Sciences, Faculty of Civil Engineering, University of Maribor, Smetanova 17, 2000 Maribor, Slovenia

Email:
borut.zalar@uni-mb.si

DOI:
https://doi.org/10.1090/S0002-9939-99-05108-4

Keywords:
Compact group,
group algebra,
isometric automorphism,
local isometric automorphism

Received by editor(s):
March 4, 1998

Published electronically:
June 30, 1999

Additional Notes:
This research was supported by the Joint Hungarian-Slovene research project supported by OMFB in Hungary and the Ministry of Science and Technology in Slovenia, Reg. No. SLO-2/96. The first author was supported in part by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. T–016846 F–019322, and by a grant from the Ministry of Education, Hungary, Reg. No. FKFP 0304/1997. The second author was supported in part by a grant from the Ministry of Science and Technology, Slovenia

Communicated by:
David R. Larson

Article copyright:
© Copyright 1999
American Mathematical Society