Bicommutants and Arens Regularity

Abstract

Abstract. Let C and R be unital rings, and Z an injective cogenerator for right C-modules. For an R, C-bimodule U let U * = Hom C (U, Z), R ′ = End R (U ) and Biend R (U ) = End R ′ (U ), the biendomorphism ring of U . Under suitable requirements on U we show that B := Biend R (U ) can be identified with a subring ofB := Biend R (U * ), study conditions for the reverse inclusion and density of B inB. In the case C is contained in the center of R we describe Biend R (R * ) in terms of the Arens products in R * * and study Arens regularity of R in the context of duality of modules. We characterize Arens regular algebras over fields. IntroductionThe notion of a centralizer R ′ of a subring R in a ring L (that is, the set of all elements of L commuting with all elements of R) is fundamental in several areas of mathematics. The double centralizer ring R ′′ of R has continuously attracted attention in algebra (see e.g. [5], [7] and references there for recent results in this direction) and in functional analysis ([6] is a recent example). For an algebra R of operators on a vector space U over a field F the centralizer R ′ = End R (U ) of R in L = End F (U ) is usually called the commutant and the bicommutant is just the ring R ′′ = Biend R (U ) = End R ′ (U ) of R-biendomorphisms of U Regarding U as a left R-module, the dual space U * is a right R-module by ρa := ρ • a (a ∈ R, ρ ∈ U * ). Clearly the adjoint f * of each endomorphism f ∈ End R (U ) acts as an endomorphism of U * , but in general End R (U * ) also contains many elements which are not adjoint to any linear map f on U (if U is infinite dimensional). Therefore for a map g ∈ B := Biend R (U ) the adjoint g * is not necessarily an element ofB := Biend R (U * ). For a right R-module, such as U * , it is convenient to let biendomorphisms to act from the right, so that U * is a right module overB, and to take the ring multiplication inB to be the reverse of the composition of maps, so that R can be regarded as a subring ofB. If in End F (U * ) we reverse the composition of maps, then the involution * is a ring monomorphism End F (U ) → End F (U * ) and may be regarded as an inclusion. In this way the inclusion (1.1) Biend R (U ) ⊆ Biend R (U * ) makes sense, although it does not hold in general. Perhaps surprisingly, it turns out that (1.1) and even the reverse inclusion hold under the conditions which are 2010 Mathematics Subject Classification. Primary: 16S50; Secondary: 16D40, 16D50, 16D90, 16S85.Key words and phrases. Injective cogenerator, flat module, maximal ring of quotients, Morita duality, Arens products.Acknowledgment. The author is grateful to Matej Brešar for his comments on the paper and to PeterŠemrl for the discussion concerning Theorem 7.3. not very restrictive. Very special cases of the inclusion Biend R (U * ) ⊆ Biend R (U ) are known from functional analysis [13] (where susch an inclusion means weak*-continuity of certain maps) and in [15] the author (motivated by a problem concerning derivations) considered the case when R is generated by a...