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

Arzikulov, Farkhad Nematjonovich; Ayupov, Shavkat Abdullayevich

Algebr. Represent. Theory 16, No. 1, 289-301 (2013).

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., \(C^*(\mathcal{A}_{sa})= \mathcal{A}+i\mathcal{A}\).

MSC:

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

Keywords:

JC-algebra; real AW\(^*\)-algebra; AW\(^*\)-algebra; AJW-algebra

References:

[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