Recent work by the author
which was independently duplicated in part by Giles and Kummer has made it
possible to generalize the Gelfand representation theorem for abelian C −∗algebras
to the non-abelian case. Let A be a C-algebra with unit. If A is abelian, it can be
identified with the algebra of all continuous complex-valued functions on
its maximal ideal space (with the hull-kernel topology). A less precise way
of looking at this result would be to say that an abelian A is completely
recoverable from the set of maximal ideals and a certain structure thereon (in
this case, a topology). If we use the latter description as the basis for a
theory applicable to non-abelian A, we find immediately that two changes are
necessary. The set of maximal ideals is replaced by the set of maximal left
ideals, and secondly, the structure defined thereon will not be a topology,
though it will have many similar properties when viewed correctly. This paper
shows how the C∗-algebra is recovered from the maximal left ideals (with
structure).
