George Elliott - -algebras--the first fifty years

	GEORGE ELLIOTT, Department of Mathematics, University of Toronto, Toronto, Ontario  M5S 3G3, Canada (also affiliated with the Fields Institute and the University of Copenhagen)
	-algebras--the first fifty years

It is now seventy years since the publication of von Neumann's bicommutant theorem, which might be said to be the beginning of operator algebra theory. During this time, not only have von Neumann algebras--the bicommutants, or double centralizers, of self-adjoint sets of bounded operators in Hilbert space--been studied--in some settings exhaustively--but also, especially in the last fifty years, general norm-closed self-adjoint subalgebras of such operators have been considered. In both categories, the so-called amenable objects are of special interest. Work of Connes and others has resulted in a surprisingly complete theory of amenable von Neumann algebras, including, in particular, a complete classification, in terms of simple invariants. (The conjugacy theory of subalgebras of such algebras, due to Jones and others, should also of course be mentioned.) The theory of amenable -algebras has developed more or less at the same time and in the same way. Far from being complete, however, it is perhaps now entering its most exciting phase of development.