The Fubini Product and Its Applications
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The Fubini Product and Its Applications Otgonbayar Uuye1
· Joachim Zacharias2
Received: 9 August 2019 / Revised: 5 January 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract The Fubini product of operator spaces provides a powerful tool for analysing properties of tensor products. In this paper, we apply the theory of Fubini products to the problem of computing invariant parts of dynamical systems. In particular, we study the invariant translation approximation property of discrete groups. Keywords Fubini product · Slice map property · Invariant translation approximation property Mathematics Subject Classification Primary 46B28; Secondary 46L07 · 47L25 · 20F65
1 Introduction The Fubini product of C ∗ -algebras was first defined and studied by Tomiyama [20–23] and Wassermann [24]. It has been used in the study of operator algebras and operator spaces for a long time. See, for instance, [1,3,10,11,13–15]. In this paper, we use old and new results about Fubini products for the study of the invariant translation approximation property. In Sect. 2, we begin with a brief survey of the theory of Fubini products and its applications. Most of these results appear scattered in numerous articles, most notably [6,9,10,12,15,21,24]. After briefly recalling some definitions concerning operator spaces in Sect. 2.1, we define the Fubini product
Communicated by Mohammad Sal Moslehian.
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Otgonbayar Uuye [email protected] Joachim Zacharias [email protected]
1
Institute of Mathematics, National University of Mongolia, Ikh Surguuliin Gudamj 1, Ulaanbaatar, Mongolia
2
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G128QW, Scotland, UK
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O. Uuye, J. Zacharias
and prove its fundamental properties regarding functoriality, intersections, kernels, relative commutants, invariant elements, combinations in Sects. 2.2–2.8. In Sect. 2.9, we review the relation between the operator approximation property and the slice map property. In Sect. 3, we apply the results in Sect. 2 to the study of groups with the invariant translation approximation property (ITAP) of Roe [17, Section 11.5.3]. In Sect. 3.1, we recall the definition of the uniform Roe algebra. In Sect. 3.2, we analyse the uniform Roe algebra of a product space. Finally, in Sect. 3.3, we study the ITAP of product groups. We show that for countable discrete groups G and H , if G has the approximation property (AP) of Haagerup–Kraus [6], the product G × H has the ITAP if and only if H has the ITAP. Finally, in Sect. 4, we study the crossed product version of the Fubini product.
2 The Fubini Product In this section, we recall the Fubini product and prove its fundamental properties. We study intersections, kernels, relative commutants, invariant elements, combinations in terms of the slice map property in Sects. 2.2–2.8. In Sect. 2.9, we review the relation between the operator approximation property and the slice map property. 2.1 Operator Spaces For the sake of complete
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