Dual 𝐴*-algebras of the first kind

Johnson, David L.; Lahr, Charles D.

doi:10.1090/S0002-9939-1979-0524307-1

Dual $A^{\ast }$-algebras of the first kind
HTML articles powered by AMS MathViewer

by David L. Johnson and Charles D. Lahr PDF

Proc. Amer. Math. Soc. 74 (1979), 311-314 Request permission

Abstract:

Let A be an ${A^\ast }$-algebra of the first kind. It is proved that A has property P2 of Máté if and only if ${A^2}$ is dense in A if and only if A possesses an (operator-bounded) approximate identity. Further, it is shown that an ${A^\ast }$-algebra of the first kind having property P2 is a dual algebra if and only if it is a modular annihilator algebra. As applications, these results are used to strengthen certain theorems about Hilbert algebras.

References

Bruce A. Barnes, Banach algebras which are ideals in a Banach algebra, Pacific J. Math. 38 (1971), 1–7; correction, ibid. 39 (1971), 828. MR 310640, DOI 10.2140/pjm.1971.38.1
Bruce A. Barnes, Examples of modular annihilator algebras, Rocky Mountain J. Math. 1 (1971), no. 4, 657–665. MR 288581, DOI 10.1216/RMJ-1971-1-4-657
Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). Deuxième édition. MR 0246136
Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773
David L. Johnson and Charles D. Lahr, Multipliers and derivations of Hilbert algebras, Math. Japon. 25 (1980), no. 1, 43–54. MR 571262

The trace class of an arbitrary Hilbert algebra

Approximate identities and multipliers

Michael R. W. Kervin, The trace-class of a full Hilbert algebra, Trans. Amer. Math. Soc. 178 (1973), 259–270. MR 318900, DOI 10.1090/S0002-9947-1973-0318900-8
L. Máté, The Arens product and multiplier operators, Studia Math. 28 (1966/67), 227–234. MR 215095, DOI 10.4064/sm-28-3-227-234
Edith A. McCharen, A characterization of dual $B^{\ast }$-algebras, Proc. Amer. Math. Soc. 37 (1973), 84. MR 306927, DOI 10.1090/S0002-9939-1973-0306927-7
Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5
B. J. Tomiuk, Modular annihilator $A^{\ast }$-algebras, Canad. Math. Bull. 15 (1972), 421–426. MR 317060, DOI 10.4153/CMB-1972-076-0
B. J. Tomiuk, Multipliers on dual $A^*$-algebras, Proc. Amer. Math. Soc. 62 (1977), no. 2, 259–265. MR 433215, DOI 10.1090/S0002-9939-1977-0433215-4
Bertram Yood, Ideals in topological rings, Canadian J. Math. 16 (1964), 28–45. MR 158279, DOI 10.4153/CJM-1964-004-2
Bertram Yood, Hilbert algebras as topological algebras, Ark. Mat. 12 (1974), 131–151. MR 380429, DOI 10.1007/BF02384750