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The Boundaries of Von Neumann AlgebrasThe theory of von Neumann algebras was initially developed in the 1930s by Murray and von Neumann, providing a far reaching generalization of classical measure theory. Von Neumann algebras themselves were therefore initially studied as inherently analytic objects. This perspective was reinforced, and then almost immediately afterword dismantled, by Alain Connes starting in the 1970s when he showed that there is a unique separable injective II1 factor, and then also provided the first example of rigidity phenomena in von Neumann algebras through the use of Kazhdan's property (T). The juxtaposition of these two extremes led Popa's breakthrough in establishing deformation/rigidity theory, leading to numerous other breakthroughs and a deeper understanding of structural theory of von Neumann algebras. Von Neumann algebras therefore have a rich structure and it is only by studying the field from many different perspectives that we can gain a clearer picture of these mysterious objects. Each of the perspectives from ergodic theory, Voiculescu's free probability theory, abstract harmonic analysis, and other contribute to our understanding of the area. We describe a new perspective stemming from the dynamics of boundary actions. This has its origins in the work of Ozawa on applications of amenable actions, but it has only recently been realized that these boundaries are inherent to the von Neumann algebras themselves. These boundaries allow for the incorporation of geometric boundary techniques in the von Neumann algebraic setting. We survey some properties described using these boundaries as well as some rigidity results obtained using these techniques.Uploaded 06-10-26 Learn more about your ad choices. Visit megaphone.fm/adchoices
