Relative discrete spectrum of W*-dynamical system
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Tusi Mathematical Research Group
ORIGINAL PAPER
Relative discrete spectrum of W*‑dynamical system Rocco Duvenhage1 · Malcolm King2,3 Received: 13 April 2020 / Accepted: 13 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract A definition of relative discrete spectrum of noncommutative W*-dynamical systems is given in terms of the basic construction of von Neumann algebras, motivated from three perspectives: First, as a complementary concept to relative weak mixing of W*-dynamical systems. Second, by comparison with the classical (i.e., commutative) case. And, third, by noncommutative examples. Keywords W*-dynamical systems · Relative discrete spectrum · Relative weak mixing · Relatively independent joinings Mathematics Subject Classification 46L55
1 Introduction In his study of ergodic actions of locally compact groups, Zimmer [20, 21] introduced relative discrete spectrum and proved what was to become known as the Furstenberg–Zimmer Structure Theorem. Proving the same structure theorem independently, Furstenberg [6] gave an ergodic theoretic proof of Szemeredi’s Theorem. In the noncommutative setting of W*-dynamical systems, Austin, Eisner, and Tao [1] proved a partial analogue of the Furstenberg–Zimmer Structure Theorem, providing conditions under which a certain case of relative weak mixing holds. In their approach, Communicated by Fedor Sukochev. * Rocco Duvenhage [email protected] Malcolm King [email protected] 1
Department of Physics, University of Pretoria, Pretoria 0002, South Africa
2
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
3
Present Address: Department of Decision Sciences, University of South Africa, Pretoria 0003, South Africa
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R. Duvenhage, M. King
which builds on the work by Popa [13], the basic construction of von Neumann algebras is an essential tool, although they do not define relative weak mixing in terms of the basic construction, and do not define relative discrete spectrum at all. Their use of the basic construction forms the basis for our approach to relative discrete spectrum in this paper, where we employ the basic construction for the von Neumann algebra of a W*-dynamical system and the subalgebra relative to which we want to define discrete spectrum of the W*-dynamical system. Of particular importance is [1]’s characterization of systems which are not relatively weakly mixing in terms of the existence of a non-trivial submodule, invariant under the dynamics, and finite with respect to the trace on the basic construction. In the noncommutative case, these kinds of submodules play an analogous role to the finite rank submodules which appear in the classical case. The paper has two main parts. The first, consisting of Sects. 2 and 3, treats our noncommutative definition of relative discrete spectrum. The definition is given in terms of the basic construction, and is motivated by the need to make relative discrete spectrum complementary to relative weak mixing as
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