×

Commutativity of \(C^*\)-algebras and associativity of \(JB^*\)-algebras. (English) Zbl 0749.46032

Summary: A n.c. \(JB^*\)-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky for \(C^*\)-algebras. Different characterizations of associativity are given.

MSC:

46K70 Nonassociative topological algebras with an involution
46L05 General theory of \(C^*\)-algebras
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1007/BF01214510 · Zbl 0385.32002 · doi:10.1007/BF01214510
[2] DOI: 10.1017/S0013091500004429 · Zbl 0531.46038 · doi:10.1017/S0013091500004429
[3] DOI: 10.1017/S001309150001628X · Zbl 0414.46034 · doi:10.1017/S001309150001628X
[4] DOI: 10.1007/BF01181693 · Zbl 0514.46047 · doi:10.1007/BF01181693
[5] DOI: 10.1007/BF01165932 · Zbl 0512.46055 · doi:10.1007/BF01165932
[6] DOI: 10.1017/S0305004100055092 · Zbl 0392.46038 · doi:10.1017/S0305004100055092
[7] DOI: 10.1016/0022-247X(71)90193-4 · Zbl 0211.44401 · doi:10.1016/0022-247X(71)90193-4
[8] Bonsall, Complete Normed Algebras (1973) · doi:10.1007/978-3-642-65669-9
[9] DOI: 10.1307/mmj/1029001946 · Zbl 0384.46040 · doi:10.1307/mmj/1029001946
[10] Upmeier, Symmetric Banach Manifolds and Jordan (1985)
[11] Aupetit, Propriétés Spectrales des Algèbres de Banach (1979) · Zbl 0409.46054 · doi:10.1007/BFb0064204
[12] DOI: 10.1090/S0002-9947-1970-0290117-2 · doi:10.1090/S0002-9947-1970-0290117-2
[13] DOI: 10.1007/BF01214975 · Zbl 0513.46044 · doi:10.1007/BF01214975
[14] DOI: 10.1016/0022-1236(79)90010-7 · Zbl 0421.46043 · doi:10.1016/0022-1236(79)90010-7
[15] DOI: 10.1016/0001-8708(78)90044-0 · Zbl 0397.46065 · doi:10.1016/0001-8708(78)90044-0
[16] DOI: 10.2307/2372173 · Zbl 0042.35001 · doi:10.2307/2372173
[17] DOI: 10.2307/1970896 · doi:10.2307/1970896
[18] Schafer, An Introduction to Non-associative Algebras (1966) · Zbl 0145.25601
[19] DOI: 10.1112/plms/s3-47.2.258 · Zbl 0521.47036 · doi:10.1112/plms/s3-47.2.258
[20] DOI: 10.1112/jlms/s2-22.2.318 · Zbl 0483.46050 · doi:10.1112/jlms/s2-22.2.318
[21] DOI: 10.1112/plms/s3-48.3.428 · Zbl 0509.46052 · doi:10.1112/plms/s3-48.3.428
[22] DOI: 10.1007/BF01239948 · Zbl 0483.46049 · doi:10.1007/BF01239948
[23] Ogasawara, J. Sci. Hiroshima Univ. Ser. A 18 pp 307– (1955)
[24] Sz-Nagy, Harmonic Analysis of Operators in Hilbert Space (1970) · Zbl 0201.45003
[25] Martinez, Illinois J. Math. 29 pp 609– (1985)
[26] DOI: 10.1017/S0305004100056504 · Zbl 0425.46037 · doi:10.1017/S0305004100056504
[27] DOI: 10.1017/S0017089500006704 · Zbl 0617.46061 · doi:10.1017/S0017089500006704
[28] Kaplansky, Rings of Operators (1968)
[29] DOI: 10.1093/qmath/32.4.435 · Zbl 0446.46043 · doi:10.1093/qmath/32.4.435
[30] Iochum, Cônes Autopolaires et Algèbres de Jordan (1984) · Zbl 0556.46040 · doi:10.1007/BFb0071358
[31] Huruya, Proc. Amer. Math. Soc. 63 pp 289– (1977)
[32] Holbrook, Acta Sci. Math. (Szeged) 29 pp 299– (1968)
[33] Hanche-Olsen, Jordan operators algebras (1984)
[34] Edwards, Bull. Sci. Math. 104 pp 393– (1980)
[35] Duncan, Proc. Roy. Soc. Edinburgh Sect. A 75 pp 119– (1975) · Zbl 0331.46050 · doi:10.1017/S0308210500017832
[36] Doran, Characterizations of C*-algebras: The Gelfand-Naimark Theorem (1986) · Zbl 0597.46056
[37] DOI: 10.1017/S0017089500002378 · Zbl 0301.46047 · doi:10.1017/S0017089500002378
[38] DOI: 10.1017/S0013091500003965 · Zbl 0441.46042 · doi:10.1017/S0013091500003965
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.