Tensor Product of Dimension Effect Algebras
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Tensor Product of Dimension Effect Algebras Anna Jenˇcova´ 1 · Sylvia Pulmannova´ 1 Received: 29 May 2019 / Accepted: 3 November 2020 / © Springer Nature B.V. 2020
Abstract Dimension effect algebras were introduced in Jenˇcov´a and Pulmannov´a (Rep. Math. Phys. 62, 205–218, 2008), and it was proved that they are unit intervals in dimension groups. We prove that the effect algebra tensor product of dimension effect algebras is a dimension effect algebra, which is the unit interval in the unital abelian po-groups tensor product of the corresponding dimension groups. Keywords Dimension group · Dimension effect algebra · Tensor product · Direct limit
1 Introduction In [13], the notion of a dimension effect algebra was introduced as a counterpart of the notion of a dimension group. Recall that a dimension group (or a Riesz group) is a directed, unperforated interpolation group. By [5], dimension groups can be also characterized as direct limits of directed systems of simplicial groups. In analogy with the latter characterization, dimension effect algebras were defined as direct limits of directed systems of finite effect algebras with the Riesz decomposition property. It is well known that the latter class of effect algebras corresponds to the class of finite MV-algebras [2], and in analogy with simplicial groups, we call them simplicial effect algebras. It turns out that dimension effect algebras are exactly the unit intervals in unital dimension groups, and simplicial effect algebras are exactly the unit intervals in unital simplicial groups. In [13], an intrinsic characterization of dimension effect algebras was found, and also a categorical equivalence between countable dimension effect algebras and unital AF C*-algebras was shown [13, Theorem 5.2]. In this paper we continue the study of dimension effect algebras. In particular, we study the tensor product of dimension effect algebras in the category of effect algebras. We recall The authors were supported by grant VEGA No.2/0142/20 and by the grant of the Slovak Research and Development Agency grant No. APVV-16-0073. Sylvia Pulmannov´a
[email protected] Anna Jenˇcov´a [email protected] 1
ˇ anikova 49, SK-814 73 Mathematical Institute, Slovak Academy of Sciences, Stef´ Bratislava, Slovakia
Order
that the tensor product in the category of effect algebras (defined as a universal bimorphism) exists, and its construction was described in [3]. We first prove that the tensor product of simplicial effect algebras is again a simplicial effect algebra and is (up to isomorphism) the unit interval in the tensor product of the corresponding unital simplicial groups (Theorem 4.3). Then we extend this result to any dimension effect algebras, using the fact that every dimension effect algebra is a direct limit of a directed system of simplicial effect algebras. Namely, we prove that the tensor product of dimension effect algebras is a dimension effect algebra (Theorem 4.5), and is (up to isomorphism) the unit interval in the tensor product of the corresponding dimensio
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