Isometric isomorphisms between Banach algebras related to locally compact groups

Authors:
F. Ghahramani, A. T. Lau and V. Losert

Journal:
Trans. Amer. Math. Soc. **321** (1990), 273-283

MSC:
Primary 43A20; Secondary 43A10, 46J99

DOI:
https://doi.org/10.1090/S0002-9947-1990-1005079-2

MathSciNet review:
1005079

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Abstract: Let , be locally compact groups. We prove in this paper that if is an isometric isomorphism from the Banach algebra (the continuous dual of the Banach space of left uniformly continuous functions on , equipped with Arens multiplication) onto , then maps onto and onto . We also prove that any isometric isomorphism from (second conjugate algebra of ) onto maps onto .

**[1]**R. Arens,*The adjoint of a bilinear operation*, Proc. Amer. Math. Soc.**2**(1951), 939-948. MR**0045941 (13:659f)****[2]**M. M. Day,*Amenable semigroups*, Illinois J. Math.**1**(1957), 509-544. MR**0092128 (19:1067c)****[3]**J. Duncan and S. A. R. Hosseiniun,*The second dual of a Banach algebra*, Proc. Roy. Soc. Edinburgh Sect. A**84**(1979), 309-325. MR**559675 (81f:46057)****[4]**F. Ghahramani and A. T. Lau,*Isometric isomorphisms between the second conjugate algebras of the group algebras*, Bull. London Math. Soc.**20**(1988), 342-344. MR**940290 (89e:43008)****[5]**E. Granirer and M. Leinert,*On some topologies which coincide on the unit sphere of the Fourier algebra**and of the measure algebra*, Rocky Mountain J. Math.**11**(1981), 459-472. MR**722579 (85f:43009)****[6]**M. Grosser,*as an ideal in its second dual space*, Proc. Amer. Math. Soc.**73**(1979), 3633-3634. MR**518521 (82i:43005)****[7]**M. Grosser and V. Losert,*The norm-strict bidual of a Banach algebra and the dual of*, Manuscripta Math.**45**(1984), 127-146. MR**724731 (86b:46073)****[8]**E. Hewitt and K. A. Ross,*Abstract harmonic analysis*, vols. I & II, Springer-Verlag, Berlin, 1963 and 1970. MR**551496 (81k:43001)****[9]**N. Isik, J. S. Pym and A. Ülger,*The second dual of the group algebra of a compact group*, J. London Math. Soc.**35**(1987), 135-148. MR**871771 (88f:43012)****[10]**B. E. Johnson,*Isometric isomorphisms of measure algebras*, Proc. Amer. Math. Soc.**15**(1964), 186-188. MR**0160846 (28:4056)****[11]**A. T. Lau,*Operators which commute with convolutions on subspaces of*, Colloq. Math.**39**(1978), 351-359. MR**522378 (80h:43007)****[12]**-,*Continuity of Arens multiplication on the dual space of bounded uniformly continuous functions on locally compact groups and topological semigroups*, Math. Proc. Cambridge Philos. Soc.**99**(1986), 273-283. MR**817669 (87i:43001)****[13]**A. T. Lau and V. Losert,*On the second conjugate algebra of**of a locally compact group*, J. London Math. Soc.**37**(1988), 464-470. MR**939122 (89e:43007)****[14]**A. T. Lau and K. McKennon,*Isomorphisms of locally compact groups and Banach algebras*, Proc. Amer. Math. Soc.**79**(1980), 55-58. MR**560583 (81b:43004)****[15]**K. McKennon,*Multipliers, positive functionals, positive-definite functions, and Fourier-Stieltjes transforms*, Mem. Amer. Math. Soc. No. 111 (1971). MR**0440299 (55:13174)****[16]**S. A. McKilligan,*On the representation of the multiplier algebras of some Banach algebras*, J. London Math. Soc.**6**(1973), 399-402. MR**0318881 (47:7427)****[17]**P. Milnes and J. S. Pym,*Counterexample in the theory of continuous functions on topological groups*, Pacific J. Math.**66**(1976), 205-209. MR**0440279 (55:13157)****[18]**T. Mitchell,*Topological semigroups and fixed points*, Illinois J. Math.**14**(1970), 630-641. MR**0270356 (42:5245)****[19]**L. Pigno,*A multiplier theorem*, Pacific J. Math.**34**(1970), 755-757. MR**0284768 (44:1992)****[20]**S. Sakai,*-algebras and**-algebras*, Springer-Verlag, Berlin and New York, 1971. MR**0442701 (56:1082)****[21]**J. G. Wendel,*Left centralizers and isomorphism of group algebras*, Pacific J. Math.**2**(1952), 251-261. MR**0049911 (14:246c)****[22]**J. C. S. Wong,*Topologically stationary locally compact groups and amenability*, Trans. Amer. Math. Soc.**144**(1969), 351-363. MR**0249536 (40:2781)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1990-1005079-2

Keywords:
Locally compact groups,
uniformly continuous functions,
group algebra,
Arens multiplication,
measure algebra,
isometric isomorphisms,
right identities

Article copyright:
© Copyright 1990
American Mathematical Society