AW\(^*\)-algebras which are enveloping \(C^*\)-algebras of JC-algebras. (English) Zbl 1271.46054

Summary: The authors consider enveloping \(C^*\)-algebras of AJW-algebras. Conditions are given for when the enveloping \(C^*\)-algebra of an AJW-algebra is an AW\(^*\)-algebra, and corresponding theorems are proved. In particular, we prove that if \(\mathcal{A}\) is a real AW\(^*\)-algebra, \(\mathcal{A}_{sa}\) is the JC-algebra of all self-adjoint elements of \(\mathcal{A}\), \(\mathcal{A}+i\mathcal{A}\) is an AW\(^*\)-algebra and \(\mathcal{A}\cap i\mathcal{A} = \{0\}\), then the enveloping \(C^*\)-algebra \(C^*(\mathcal{A}_{sa})\) of the JC-algebra \(\mathcal{A}_{sa}\) is an AW\(^*\)-algebra. Moreover, if \(\mathcal{A}+i\mathcal{A}\) does not have nonzero direct summands of type I\(_{2}\), then \(C^*(\mathcal{A}_{sa})\) coincides with the algebra \(\mathcal{A}+i\mathcal{A}\), i.e., \(C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}\).


46L70 Nonassociative selfadjoint operator algebras
17C65 Jordan structures on Banach spaces and algebras
47L30 Abstract operator algebras on Hilbert spaces
46L35 Classifications of \(C^*\)-algebras
Full Text: DOI


[1] Stacey, P.J.: Locally orientable JW-algebras of complex type. Quart. J. Math. 33(2), 247–251 (1982) · Zbl 0476.46053
[2] Arzikulov, F.N.: On two problems concerning enveloping von Neumann algebras. J. Algebr. Represent Theor. (2011, to appear) · Zbl 1247.46062
[3] Li, B-R.: Real operator algebras. Institute of Mathematics, p. 223. Academia Sinica Beijing 100080, China (2002)
[4] Arzikulov, F.N.: On abstract JW-algebras. Sib. Math. J. 39(1), 20–27 (1998) · Zbl 0917.46062
[5] Chilin V.I.: Partial ordered Bear involutive algebras. Itogi Nauki Teh., Ser. Sovrem. Probl. Mat. VINITI. 27 99–128 (1985, Russian)
[6] Albeverio, S., Ayupov, S.A., Abduvaitov, A.H.: On Real AW*-algebras. Preprint N37, SFB 611, p. 18. Bonn (2002) · Zbl 1095.46031
[7] Arzikulov, F.N.: On an analog of the Peirce decomposition. Sib. Math. J. 40(3), 413–419 (1999) · Zbl 0938.46054
[8] Kusrayev, A.G.: Boolean-valued analysis of normed Jordan algebras. Investigations in functional analysis and its applications. M. pp. 50–124. Nauka (2005, Russian)
[9] Albeverio, S., Ayupov, S.A., Abduvaitov A.H.: On coincidence of types of a real AW*-algebra and its comlexification. News of Russian Academy of Sciences. Mathematical series 68(5), 3–12 (2004, Russian) · Zbl 1074.46042
[10] Ayupov, S.A., Arzikulov, F.N.: Maximal real von Neumann algebras on a Hilbert space. UzbekUzb. Mat. Ž. (3), 7–12 (2006, Russian) · Zbl 1296.46057
[11] Kusrayev, A.G.: On functional realization of AW*-algebras of type I//. Sib. Math. J. 32(3), 78–88 (1991)
[12] Arzikulov, F.N.: On enveloping C*-algebras of JB-algebras. Vladikavk. Mat. Ž. 8(3), 3–15 (2006) · Zbl 1324.46071
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.