The cyclical compactness in Banach \(C_\infty (Q)\)-modules. (English. Russian original) Zbl 1514.46035

Chilin, V. I.; Karimov, J. A.

J. Math. Sci., New York 265, No. 1, 129-145 (2022); translation from Sovrem. Mat., Fundam. Napravl. 65, No. 1, 137-155 (2019).

Summary: In this paper we study the class of laterally complete commutative unital regular algebras \(A\) over arbitrary fields. We introduce a notion of passport \(\Gamma (X)\) for a faithful regular laterally complete \(A\) modules \(X\), which consist of uniquely defined partition of unity in the Boolean algebra of all idempotents in \(A\) and of the set of pairwise different cardinal numbers. We prove that \(A\)-modules \(X\) and \(Y\) are isomorphic if and only if \(\Gamma (X) = \Gamma (Y)\). Further we study Banach \(A\)-modules in the case \(A = C_\infty (Q)\) or \(A = C_\infty (Q)+ i \cdot C_\infty (Q)\). We establish the equivalence of all norms in a finite-dimensional (respectively, \( \sigma \)-finite-dimensional) \(A\)-module and prove an \(A\)-version of Riesz theorem, which gives the criterion of a finite-dimensionality (respectively, \( \sigma \)-finite-dimensionality) of a Banach \(A\)-module.

MSC:

46H25	Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
16D99	Modules, bimodules and ideals in associative algebras
46L10	General theory of von Neumann algebras
16E50	von Neumann regular rings and generalizations (associative algebraic aspects)

Keywords:

laterally complete commutative unital regular algebras; \(A\)-module; Banach \(A\)-module

References:

[1]	Berberian, SK, The regular ring of a finite AW*-algebra, Ann. Math., 65, 2, 224-240 (1957) · Zbl 0085.09904
[2]	Chilin, VI, Partially ordered Baer involutive algebras, Totals Sci. Tech. Ser. Contemp. Probl. Math. New Progr., 27, 99-128 (1985)
[3]	Chilin, VI; Karimov, JA, Disjunct complete C_∞(Q)-modules, Vladikavkaz Math. Zh., 16, 2, 69-78 (2014) · Zbl 1448.46040
[4]	V. I. Chilin and J. A. Karimov, “Strictly homogeneous laterally complete modules,” J. Phys. Conf. Ser., 697 (2016).
[5]	A. N. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Am. Math. Soc., Providence (1961). · Zbl 0111.03403
[6]	Ganiev, IG; Kudaybergenov, KK, Finite dimensional modules over the ring of measurable functions, Uzbek Math. Zh., No., 4, 3-9 (2004)
[7]	Kaplansky, J., Projections in Banach algebras, Ann. Math, 53, 235-249 (1951) · Zbl 0042.12402
[8]	Kaplansky, J., Algebras of type I, Ann. Math, 56, 450-472 (1952) · Zbl 0047.35701
[9]	Kaplansky, J., Modules over operator algebras, Amer. J. Math, 75, 4, 839-858 (1953) · Zbl 0051.09101
[10]	J. A. Karimov, “Kaplansky-Hilbert modules over the algebra of measurable functions,” Uzbek math. Zh., No. 4, 74-81 (2010).
[11]	Karimov, JA, Equivalence of norms in finite dimensional C_∞(Q)-modules, Bull. Nat. Univ. Uzbek., No., 2, 1, 100-108 (2017)
[12]	Kusraev, AG, Vector Duality and Its Applications (1985), Novosibirsk: Nauka, Novosibirsk · Zbl 0616.49010
[13]	Kusraev, AG, Dominated Operators (2000), Dordrecht: Kluwer, Dordrecht
[14]	F. Maeda, Kontinuierliche Geometrien, Springer, Berlin-Heidelberg (1958).
[15]	M. A. Muratov and V. I. Chilin, Algebra of Measurable and Locally Measurbale Operators [in Russian], Inst. mat. NAN Ukr., Kiev (2007).
[16]	Saito, K., On the algebra of measurable operators for a general AW*-algebra, I, Tohoku Math. J., 21, 2, 249-270 (1969) · Zbl 0183.14302
[17]	Saito, K., On the algebra of measurable operators for a general AW*-algebra, II, Tohoku Math. J., 23, 3, 525-534 (1971) · Zbl 0231.46102
[18]	Segal, I., A noncommutative extension of abstract integration, Ann. Math., 57, 3, 401-457 (1953) · Zbl 0051.34201
[19]	Skornyakov, LA, Dedekind Complemented Structures and Regular Rings (1961), Moscow: Fizmatgiz, Moscow
[20]	Van Der Waerdenm, BL, Algebra (1991), New York: Springer, New York
[21]	Vulikh, BZ, Introduction to the Theory of Partially Ordered Spaces (1967), Groningen: Wolters-Noordhoff Sci. Publ, Groningen · Zbl 0186.44601
[22]	Yeadon, FJ, Convergence of measurable operators, Math. Proc. Cambridge Philos. Soc., 74, 2, 257-268 (1973) · Zbl 0272.46043