Isometric isomorphisms between Banach algebras related to locally compact groups

Ghahramani, F.; Lau, A. T.; Losert, V.

doi:10.1090/S0002-9947-1990-1005079-2

Isometric isomorphisms between Banach algebras related to locally compact groups
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by F. Ghahramani, A. T. Lau and V. Losert PDF

Trans. Amer. Math. Soc. 321 (1990), 273-283 Request permission

Abstract:

Let ${G_1}$, ${G_2}$ be locally compact groups. We prove in this paper that if $T$ is an isometric isomorphism from the Banach algebra ${\text {LUC}}{({G_1})^*}$ (the continuous dual of the Banach space of left uniformly continuous functions on ${G_1}$, equipped with Arens multiplication) onto ${\text {LUC}}{({G_2})^*}$, then $T$ maps $M({G_1})$ onto $M({G_2})$ and ${L^1}({G_1})$ onto ${L^1}({G_2})$. We also prove that any isometric isomorphism from ${L^1}{({G_1})^{**}}$ (second conjugate algebra of ${L^1}({G_1})$) onto ${L^1}{({G_2})^{**}}$ maps ${L^1}({G_1})$ onto ${L^1}({G_2})$.

References

Richard Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848. MR 45941, DOI 10.1090/S0002-9939-1951-0045941-1
Mahlon M. Day, Amenable semigroups, Illinois J. Math. 1 (1957), 509–544. MR 92128
J. Duncan and S. A. R. Hosseiniun, The second dual of a Banach algebra, Proc. Roy. Soc. Edinburgh Sect. A 84 (1979), no. 3-4, 309–325. MR 559675, DOI 10.1017/S0308210500017170
F. Ghahramani and A. T. Lau, Isometric isomorphisms between the second conjugate algebras of group algebras, Bull. London Math. Soc. 20 (1988), no. 4, 342–344. MR 940290, DOI 10.1112/blms/20.4.342
E. E. Granirer and M. Leinert, On some topologies which coincide on the unit sphere of the Fourier-Stieltjes algebra $B(G)$ and of the measure algebra $M(G)$, Rocky Mountain J. Math. 11 (1981), no. 3, 459–472. MR 722579, DOI 10.1216/RMJ-1981-11-3-459
Michael Grosser, $L^{1}(G)$ as an ideal in its second dual space, Proc. Amer. Math. Soc. 73 (1979), no. 3, 363–364. MR 518521, DOI 10.1090/S0002-9939-1979-0518521-9
Michael Grosser and Viktor Losert, The norm-strict bidual of a Banach algebra and the dual of $C_{u}(G)$, Manuscripta Math. 45 (1984), no. 2, 127–146. MR 724731, DOI 10.1007/BF01169770
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. I, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR 551496
Nilgün Işık, John Pym, and Ali Ülger, The second dual of the group algebra of a compact group, J. London Math. Soc. (2) 35 (1987), no. 1, 135–148. MR 871771, DOI 10.1112/jlms/s2-35.1.135
B. E. Johnson, Isometric isomorphisms of measure algebras, Proc. Amer. Math. Soc. 15 (1964), 186–188. MR 160846, DOI 10.1090/S0002-9939-1964-0160846-2
Anthony To Ming Lau, Operators which commute with convolutions on subspaces of $L_{\infty }(G)$, Colloq. Math. 39 (1978), no. 2, 351–359. MR 522378, DOI 10.4064/cm-39-2-351-359
Anthony To Ming Lau, 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), no. 2, 273–283. MR 817669, DOI 10.1017/S0305004100064197
Anthony To Ming Lau and Viktor Losert, On the second conjugate algebra of $L_1(G)$ of a locally compact group, J. London Math. Soc. (2) 37 (1988), no. 3, 464–470. MR 939122, DOI 10.1112/jlms/s2-37.3.464
Anthony To Ming Lau and Kelly McKennon, Isomorphisms of locally compact groups and Banach algebras, Proc. Amer. Math. Soc. 79 (1980), no. 1, 55–58. MR 560583, DOI 10.1090/S0002-9939-1980-0560583-5
Kelly McKennon, Multipliers, positive functionals, positive-definite functions, and Fourier-Stieltjes transforms, Memoirs of the American Mathematical Society, No. 111, American Mathematical Society, Providence, R.I., 1971. MR 0440299
Sheila A. McKilligan, On the representation of the multiplier algebras of some Banach algebras, J. London Math. Soc. (2) 6 (1973), 399–402. MR 318881, DOI 10.1112/jlms/s2-6.3.399
P. Milnes and J. S. Pym, Counterexample in the theory of continuous functions on topological groups, Pacific J. Math. 66 (1976), no. 1, 205–209. MR 440279
Theodore Mitchell, Topological semigroups and fixed points, Illinois J. Math. 14 (1970), 630–641. MR 270356
Louis Pigno, A multiplier theorem, Pacific J. Math. 34 (1970), 755–757. MR 284768
Shôichirô Sakai, $C^*$-algebras and $W^*$-algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60, Springer-Verlag, New York-Heidelberg, 1971. MR 0442701
J. G. Wendel, Left centralizers and isomorphisms of group algebras, Pacific J. Math. 2 (1952), 251–261. MR 49911
James C. S. Wong, Topologically stationary locally compact groups and amenability, Trans. Amer. Math. Soc. 144 (1969), 351–363. MR 249536, DOI 10.1090/S0002-9947-1969-0249536-4