Abstract
We show that a unital n.c. (noncommutative) JB*-algebra has a faithful family of factor-representations of type I and determine the structure of n.c. JB*-factors: A n.c. JB*-factor is a commutative Jordan algebra, or flexible quadratic, or a quasi CC*-algebra.
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ALBERT, A.A.: Power-associative rings, A,M,S,64, 552–593 (1948)
ALBERT, A.A.: A theory of trace admissable algebras. Proc. Mat. Acad. Sei. USA35, 317–322 (1949)
ALFSEN, E.M., HANCHE-OLSEN, H., SHULTZ, F.W.: State Spaces of C*-Algebras, Acta Math.144, 267–305 (1980)
ALFSEN, E.M., SHULTZ, F.W., STORMER, E.: A Gelfand-Neumark Theorem for Jordan Algebras. Advances in Math.28, 11–56 (1978)
ALVERMANN, K.: Normierte nichtkommutative Jordan-Algebren vom Typ I. Dissertation T.U. Braunschweig (1982)
ARAKI, H., ELLIOTT, A.: On the Definition of C*-algebras. Publ. RIMS, Kyoto Univ.9, 93–112 (1972)
ARENS, R.: The adjoint of a bilinear operation. Proc. A.N.S.2, 839–848 (1951)
ARENS, R.: Operations induced in function classes. Monatshefte für Mathematik55, 1–19 (1951)
BONSALL, F.F., DUNEAN, J.: Numerical Ranges I. London Math. Soc. Lecture Note Series 2, Cambridge, University Press 1971
BONSALL, F.F., DUNCAN, J.: Numerical Ranges II. London Math. Soc. Lecture Note Series 10, Cambridge, University Press 1973
BONSALL, F.F., DUNCAN, J.: Complete Normed Algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 80. Heidelberg: Springer 1973
BRAUN, H., KOECHER, M.: Jordan-Algebren. Die Grundlehren der Mathematischen Wissenschaften Bd,128. Heidelberg: Springer 1965
BRAUN, R.B.: Ein Gelfand-Neumark Theorem für C* -alternative Algebren. Habilitationsschrift, Universität Tübingen (1981)
BRAUN, R.B,: Coordinatization of Alternative Algebras. Preprint, University of Tübingen (1982)
BRAUN, R.B.: C* -alternative Algebras. Preprint, University of Tübingen (1982)
BRAUN, R.B.: Structure and Representation of C* -alternative Algebras. Preprint, University of Tübingen (1982)
DUNFORD, N., SCHWARTZ, J.T.: Linear Operators I. New York: Interscience Publishers Inc. 1957
EDWARDS, C.M.: On Jordan W*-Algebras. Bull. Sei. Math. 2e série104, 393–403 (1980)
HANCHE-OLSEN, H.: A Note on the Bidual of a JB-Algebra. Math. Z.175. 29–31 (1980)
JACOBSON, N.: Structure and Representations of Jordan Algebras. Rhode Island: A.M.S. Colloquium Publications Vol. 39, 1968
JANSSEN, G.: Factor Representations of Typ I for Noncommutative JB- and JB*-Algebras. Preprint T.U. Braunschweig (1982)
KAPLANSKY, I.: Lecture on the 1976 St. Andrews Colloquium of the Edinburgh Math. Soc. (see also [35])
MARTINEZ-MORENO, J., MOJTAR-KAIDI, A., RODRIGUEZ-PALACIOS, A.: On a Nonassociative Vidav-Palmer Theorem. To appear in Quarterly J. of Math.
McCrimmon, K.: Noncommutative Jordan Rings. Transact. A.M.S.158. 1–33 (1971)
McCrimmon, K.: Structure and Representations of Noncommutative Jordan Algebras. Transact. A.M.S.121, 187–199 (1966)
OEHMKE, R.H.: On flexible algebras. Ann. of Math68, 211–230 (1958)
PAYA, R., PEREZ, J., RODRIGUEZ, A.: Noncommutative Jordan C*-Algebras. Manuscripta Math,37, 87–120 (1982)
RODRIGUEZ, A.: A Vidav-Palmer Theorem for Jordan C*-Algebras and Related Topics. J. London Math. Soc.22, 318–332 (1980)
SAKAI, S.: C-Algebras and W*-Algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd60. Heidelberg: Springer 1971
SCHAFER, R,D.: An Introduction to Nonassociative Algebras. New York: Academic Press 1960
SCHAFER, R.D.: Noncommutative Jordan Algebras of Characteristic Zero. Proc. A.M.S.6, 472–475 (1955)
SHULTZ, F.W.: On normed Jordan Algebras which are Banach Dual Spaces. J, Func, Anal,31. 360–376 (1979)
SINCLAIR, A.M.: The norm of a Hermitian element in a Banach algebra, Proc. A.M.S.28, 446–450 (1971)
SINCLAIR, A.M.: The Banach algebra generated by a hermitian operator. London Math. Soc,24, 681–691 (1972)
WRIGHT, J.D.M.: Jordan C*-algebras. Michigan J. Math,24, 291–302 (1977)
YOUNGSON, M.A.: A Vidav Theorem for Banach Jordan Algebras. Math. Proc. Camb. Phil. Soc.84, 263–272 (1978)
YOUNGSON, M.A.: Non Unital Banach Jordan Algebras and C*-Triple Systems, Proceedings Edinburgh Math. Soc.24, 19–29 (1981)
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Braun, R.B. Structure and representations of noncommutative C*-Jordan algebras. Manuscripta Math 41, 139–171 (1983). https://doi.org/10.1007/BF01165932
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DOI: https://doi.org/10.1007/BF01165932