Full subalgebras of Jordan-Banach algebras and algebra norms on 𝐽𝐵*-algebras

Pérez, J.; Rico, L.; Rodríguez, A.

doi:10.1090/S0002-9939-1994-1195486-9

Full subalgebras of Jordan-Banach algebras and algebra norms on $\textrm {JB}^ *$-algebras
HTML articles powered by AMS MathViewer

by J. Pérez, L. Rico and A. Rodríguez PDF

Proc. Amer. Math. Soc. 121 (1994), 1133-1143 Request permission

Abstract:

We introduce normed Jordan Q-algebras, namely, normed Jordan algebras in which the set of quasi-invertible elements is open, and we prove that a normed Jordan algebra is a Q-algebra if and only if it is a full subalgebra of its completion. Homomorphisms from normed Jordan Q-algebras onto semisimple Jordan-Banach algebras with minimality of norm topology are continuous. As a consequence, the topology of the norm of a $J{B^ \ast }$-algebra is the smallest normable topology making the product continuous, and $J{B^ \ast }$-algebras have minimality of the norm. Some applications to (associative) ${C^ \ast }$-algebras are also given: (i) the associative normed algebras that are ranges of continuous (resp. contractive) Jordan homomorphisms from ${C^ \ast }$-algebras are bicontinuously (resp. isometrically) isomorphic to ${C^ \ast }$-algebras, and (ii) weakly compact Jordan homomorphisms from ${C^ \ast }$-algebras are of finite rank.

References

Bernard Aupetit, The uniqueness of the complete norm topology in Banach algebras and Banach Jordan algebras, J. Functional Analysis 47 (1982), no. 1, 1–6. MR 663830, DOI 10.1016/0022-1236(82)90097-0
V. K. Balachandran and P. S. Rema, Uniqueness of the norm topology in certain Banach Jordan algebras, Publ. Ramanujan Inst. 1 (1968/69), 283–289. MR 270153
A. Beddaa and M. Oudadess, On a question of A. Wilansky in normed algebras, Studia Math. 95 (1989), no. 2, 175–177. MR 1038503, DOI 10.4064/sm-95-2-175-177
Ali Bensebah, Weakness of the topology of a $\textrm {JB}^*$-algebra, Canad. Math. Bull. 35 (1992), no. 4, 449–454. MR 1191503, DOI 10.4153/CMB-1992-059-9
Mohamed Benslimane and Ángel Rodríguez-Palacios, Caractérisation spectrale des algèbres de Jordan Banach non commutatives complexes modulaires annihilatrices, J. Algebra 140 (1991), no. 2, 344–354 (French). MR 1120427, DOI 10.1016/0021-8693(91)90160-A
Frank F. Bonsall and John Duncan, Complete normed algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 80, Springer-Verlag, New York-Heidelberg, 1973. MR 0423029, DOI 10.1007/978-3-642-65669-9
Paul Civin and Bertram Yood, Lie and Jordan structures in Banach algebras, Pacific J. Math. 15 (1965), 775–797. MR 188813, DOI 10.2140/pjm.1965.15.775
Sandra Barkdull Cleveland, Homomorphisms of non-commutative $^{\ast }$-algebras, Pacific J. Math. 13 (1963), 1097–1109. MR 158274, DOI 10.2140/pjm.1963.13.1097
Antonio Fernández López, Modular annihilator Jordan algebras, Comm. Algebra 13 (1985), no. 12, 2597–2613. MR 811525, DOI 10.1080/00927878508823292
Antonio Fernández López and Ángel Rodríguez-Palacios, A Wedderburn theorem for nonassociative complete normed algebras, J. London Math. Soc. (2) 33 (1986), no. 2, 328–338. MR 838645, DOI 10.1112/jlms/s2-33.2.328
José E. Galé, Weakly compact homomorphisms in nonassociative algebras, Nonassociative algebraic models (Zaragoza, 1989) Nova Sci. Publ., Commack, NY, 1992, pp. 167–171. MR 1189619
J. E. Galé, T. J. Ransford, and M. C. White, Weakly compact homomorphisms, Trans. Amer. Math. Soc. 331 (1992), no. 2, 815–824. MR 1085942, DOI 10.1090/S0002-9947-1992-1085942-9
Leslie Hogben and Kevin McCrimmon, Maximal modular inner ideals and the Jacobson radical of a Jordan algebra, J. Algebra 68 (1981), no. 1, 155–169. MR 604300, DOI 10.1016/0021-8693(81)90291-X
Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099, DOI 10.1090/coll/039
Irving Kaplansky, Normed algebras, Duke Math. J. 16 (1949), 399–418. MR 31193

JV-algebras

87

Martin Mathieu, Weakly compact homomorphisms from $C^*$-algebras are of finite rank, Proc. Amer. Math. Soc. 107 (1989), no. 3, 761–762. MR 998737, DOI 10.1090/S0002-9939-1989-0998737-4
Kevin McCrimmon, Macdonald’s theorem with inverses, Pacific J. Math. 21 (1967), 315–325. MR 232815, DOI 10.2140/pjm.1967.21.315
Kevin McCrimmon, Noncommutative Jordan rings, Trans. Amer. Math. Soc. 158 (1971), 1–33. MR 310024, DOI 10.1090/S0002-9947-1971-0310024-7
Theodore W. Palmer, Spectral algebras, Rocky Mountain J. Math. 22 (1992), no. 1, 293–328. MR 1159960, DOI 10.1216/rmjm/1181072812

Non-Commutative Jordan

algebras

37

R. Payá-Albert, J. Pérez González, and Á. Rodríguez-Palacios, Type $\textrm {I}$ factor representations of noncommutative $JB^{\ast }$-algebras, Proc. London Math. Soc. (3) 48 (1984), no. 3, 428–444. MR 735223, DOI 10.1112/plms/s3-48.3.428
T. J. Ransford, A short proof of Johnson’s uniqueness-of-norm theorem, Bull. London Math. Soc. 21 (1989), no. 5, 487–488. MR 1005828, DOI 10.1112/blms/21.5.487

Non-associative normed algebras spanned by hermitian elements

47

Ángel Rodríguez-Palacios, The uniqueness of the complete norm topology in complete normed nonassociative algebras, J. Funct. Anal. 60 (1985), no. 1, 1–15. MR 780101, DOI 10.1016/0022-1236(85)90055-2
Ángel Rodríguez-Palacios, Jordan axioms for $C^*$-algebras, Manuscripta Math. 61 (1988), no. 3, 297–314. MR 949820, DOI 10.1007/BF01258441
Ángel Rodríguez-Palacios, Automatic continuity with application to $C^*$-algebras, Math. Proc. Cambridge Philos. Soc. 107 (1990), no. 2, 345–347. MR 1027786, DOI 10.1017/S0305004100068602
Shôichirô Sakai, Weakly compact operators on operator algebras, Pacific J. Math. 14 (1964), 659–664. MR 163185, DOI 10.2140/pjm.1964.14.659
C. Viola Devapakkiam, Jordan algebras with continuous inverse, Math. Japon. 16 (1971), 115–125. MR 297830

Letter to the editor

91

J. D. Maitland Wright, Jordan $C^*$-algebras, Michigan Math. J. 24 (1977), no. 3, 291–302. MR 487478
Bertram Yood, Homomorphisms on normed algebras, Pacific J. Math. 8 (1958), 373–381. MR 104164, DOI 10.2140/pjm.1958.8.373
Bertram Yood, Ideals in topological rings, Canadian J. Math. 16 (1964), 28–45. MR 158279, DOI 10.4153/CJM-1964-004-2