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It is known that there exist topological division algebras which are not topologically isomorphic to C (see, for example, [42], pp.
Among the topics are the functoriality of Rieffel's generalized fixed-point algebras for proper actions, division algebras and supersymmetry, Riemann-Roch and index formulae in twisted K-theory, noncommutative Yang-Mills theory for quantum Heisenberg manifolds, distances between matrix algebras that converge to coadjoint orbits, and geometric and topological structures related to M-branes.
This time it was decided to expand the scope by including some further topics related to interpolation, such as inequalities, invariant theory, symmetric spaces, operator algebras, multilinear algebra and division algebras, operator monotonicity and convexity, functional spaces and applications and connections of these topics to nonlinear partial differential equations, geometry, mathematical physics, and economics.
Finally, the article by Erik Darpo presents interesting new aspects and results pertaining to the problem of classification of finite-dimensional real division algebras with deep connections to geometry and invariants.
Despite the simple definition, division algebras are highly non-trivial objects.
Over the real number field, R itself and the complex numbers C are immediate examples of finite-dimensional division algebras.
The four algebras R, C, H, and O are often referred to as the classical real division algebras and they share several important properties.
Moreover, according to a theorem by Zorn from 1931 [33], R, C, H, and O classify all finite-dimensional real division algebras that are alternative.
A new era in the theory of real division algebras was launched by Hopf [22] in 1940, when he proved that a finite-dimensional commutative real division algebra has either dimension one or two, and furthermore, that the dimension of any real division algebra is either a power of two or infinite.
The aim of the present article is to give an overview of some of the developments in the theory of finite-dimensional real division algebras that have occurred in the last 50 years.
The classification problem for one-dimensional real division algebras is trivial, every such algebra being isomorphic to R.
Thus, in Section 3 we treat quadratic division algebras over R, describing their classification in dimension four.

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