${\mathrm{C}}^{\ast }$-Algebras

Abstract

A ${\mathrm{C}}^{\ast }$-algebra is an involutive Banach algebra $A$ which satisfies the ${\mathrm{C}}^{\ast }$-condition $\Vert a^{\ast }a\Vert =\Vert a{\Vert }^2$ for all $a\in A$. The theory of ${\mathrm{C}}^{\ast }$-algebras goes back to work of Murray and von Neumann, who first studied a special variant now known as von Neumann algebras. The theory developed rapidly after some ground breaking work of Gelfand and Naimark in 1943 in which they showed that

• every commutative ${\mathrm{C}}^{\ast }$-algebra can be realized up to isomorphism as an algebra of continuous - functions which vanish at infinity on a locally compact Hausdorff space,
• every ${\mathrm{C}}^{\ast }$-algebra can be realized as a closed ${}^{\ast }$-subalgebra of the algebra of bounded operators $B(\mathcal{H})$ on some Hilbert space $\mathcal{H}$. In the 70's and 80's of the last century, the first of the above items lead to the point of view that non-commutative ${\mathrm{C}}^{\ast }$-algebras should be regarded as function spaces of “non-commutative” topological spaces. As a consequence, completely new areas in mathematics, like Non-commutative Geometry or Free Probability evolved and we now see that the theory of ${\mathrm{C}}^{\ast }$-algebras became a very active field with applications in and interactions with almost all areas of modern mathematics.

The aim of the workshop ${\mathrm{C}}^{\ast }$-algebras, organized by Claire Anantharaman–Delaroche, Siegfried Echterhoff, Uffe Haagerup, and Dan Voiculescu, is to bring together leading researchers from basically all areas related to the field. This gives a unique opportunity to maintain a broad view on the subject and to create new cooperations between researchers with different background. Among the 42 participants was a good number of young researchers, some of them already on the top of the field. There have been 27 lectures presented at the workshop with topics ranging from classification of ${\mathrm{C}}^{\ast }$-algebras, group actions on ${\mathrm{C}}^{\ast }$-algebras, orbit equivalence of dynamical systems, Jones' theory of subfactors, fundamental groups of II${}_1$-factors, $L^2$-invariants, duality for quantum groups and quantum groupoids, Index theorems, ${\mathrm{C}}^{\ast }$-algebras related to number theory, free probability, the relation between ${\mathrm{C}}^{\ast }$-algebras and Harmonic Analysis, and others.

Among the most exciting recent developments in the field we mention the breathtaking advances of Popa and Vaes on the fundamental groups of II${}_1$-factors in the theory of von Neumann algebras, with beautiful applications to the classification of equivalence relations and dynamical systems. Other milestones are the progress of Kirchberg and coworkers on the classification of actions on ${\mathcal{O}}_2$ and the beautiful results of Dadarlat on continuous bundles of ${\mathrm{C}}^{\ast }$-algebras. Important progress has been achieved also in the classification theory of simple and non-simple ${\mathrm{C}}^{\ast }$-algebras by Elliot, Rørdam, Toms, and Winter–with some further progress being made during the workshop. There were many other exciting contributions to this very successful workshop, which can be seen from the reports presented in this issue.

It is a pleasure for the organizers to thank all participants of the workshop for their beautiful lectures and fruitful discussions. We also want to use this opportunity to thank the Mathematisches Forschungs-institut Oberwolfach for providing a very stimulating environment and strong support for organizing this conference. Special thanks also go to the very competent and helpful staff of the institute.