What is a C*-algebra?

It is a quantization of space.
It is a generalization of space.

It is a way of speaking about space without referring to "points". So you are allowing space to be a little "cloudy" or "fuzzy", very much like the idea of a wavefunction of an electron in quantum mechanics. The wavefunction does not tell you at precisely what "point" the electron is located (which would violate the Heisenberg Uncertaintly Principle), but gives you a "probability amplitude" of where it is.

Mathematically, a space is a topological space (as is studied in Topology). So, the study of C*-algebras is a generalization of Topology---sometimes referred to as Noncommutative Topology. It studies noncommutative topological spaces.

What is its history?

The field of Operator Algebras might not have come into existence without modern physics. For it gradually evolved during the development of quantum theory in the 1920's and in subsequent pioneering works of Murray and von Neumann---who studied the mathematical structure that was naturally emerging from Quantum Theory in the works of Planck, Bohr, Einstein, Schrodinger, Dirac, Heisenberg, Born, and others around the same time.

Operator Algebras emerged from the necessity, in Quantum Theory, that important observables do not commute with each other. For example, position and momentum observables, time and energy, or angular momenta or spin observables---all of which came to be best represented by noncommuting operators or matrices. (Hence the name "noncommutative geometry".) It is amazing that these abstract mathematical ideas became "necessitated" by numerous experiments in the real world. Physicists proved that!

Throughout its development, however, Operator Algebras has taken a life of its own and became an independent field of mathematical investigation. When you have newly emerging objects, you don't just leave them unstudied! So people began to probe the structure of these algebras, without worrying about whether they will have any 'practical' application. Extensive study of them has been made, and continues to be made. And now, after it has developed significantly over the last seventy years since its inception, Operator Algebras---mainly through the new geometry of Alain Connes---is now promising to furnish a new mathematical foundation on which it can be useful to its original ancestor (Physics), and perhaps help in unifying gravitation together with Quantum Physics and so eventually leading to Einstein's long sought unified field theory (where all the four known forces would be unified into a single force).

What is a good example? And what is its physical relevance?

The most concrete example of C*-algebras that illustrate the aforementioned "generalization" and "quantization" of space comes from the study of the rotation C*-algebras. These algebras are generated by two unitaries U and V ("position" and "momentum" unitaries of Quantum Mechanics) enjoying the Heisenberg commutation relation
        VU = e2(pi)ihUV,
where h is the angle of rotation (Planck's constant). (This is the unitary form of the famous relation of Heisenberg px-xp=ih.) A deep result of Rieffel, Pimsner, and Voiculescu in the early 1980's, is that if we perturb Planck's constant however slightly, we get a very different (non-isomorphic) algebra. Inside this algebra live some important Hamiltonians, such as that considered by Hofstader of electrons in a crystal lattice. Granting some scalings, this Hamiltonian can be written as
        H = U + V + U-1 + V-1
and is called the Harper operator. (When scales are introduced it is called the almost Mathieu operator.) The energy spectrum of H as a function of magnetic field strength has a beautiful structure, called Hofstader's "butterfly". (The mysterious fact is that if the magnetic field strength is an irrational number, then the energy spectrum is a totally disconnected space, called a Cantor set.) Further reference to such connections is found in A. Connes' classic book, Noncommutative Geometry (Academic Press, 1994), and especially to the references to J. Bellisard's work therein.

The most fascinating contribution of this new geometry is related to the fabric of spacetime. In physics one often thinks of space as being locally Euclidean. (This is assumed in differential geometry and in Einstein's General Relativity.) This locally Euclidean hypothesis is removed by looking instead at a (noncommutative) C*-algebra. The noncommutative aspect of space plays a central part in the formulation of Quantum Mechanics, as in the Heisenberg uncertainty principle, which basically says that the coordinates describing the position and momentum of a quantum particle (like the electron) do no commute with each other. In fact, it is encouraging (and telling) that some prominant theoretical physicists---including Fields medalist Ed Witten of the Institute for Advanced Study---have taken this approach and used it in their research investigations. It gives hope that this new Mathematics could form a more promising basis on which can be develop a unified field theory (where quantum phenomena and gravitation would be unified, as well as the unification of the four knows forces of nature).

Another link to mathematical physics arises in the study of the Quantum Hall Effect where, for example, the Hall conductivity of electrons in a crystal in a magnetic field is shown to be equal to Connes' noncommutative Chern character---the Chern character of a projection (in a C*-algebra) associated to the Fermi energy level. In 1980, von Klitzing showed that the Hall conductivity of a disordered crystal is quantized and in 1985 he won the Nobel prize in Physics (at the age of 42) for this discovery.

Connes managed to reformulate the Standard Model (which unifies the weak, strong, and electromagnetic interactions) in terms of Noncommutative Geometry. This is an extraordinary feat, and is done in the last chapter of his book (Ibid).



It is proved that the noncommutative Fourier transform auto- morphism σ of the irrational rotation algebra has the exact tracial Rokhlin property - a slightly stronger version of N. Christopher Phillips' tracial Rokhlin property. It essentially means that there are approximately central projections g such that g, σ(g), σ2(g), σ3(g) are mutually orthogonal and 1 - g - σ(g) - σ2(g) - σ3(g) is a `small' projection in the sense that it is Murray von Neumann equivalent to a subprojection of any prescribed projection. Consequently, the flip automorphism and the restriction of the Fourier transform to the flip-fixed subalgebra also have the exact tracial Rokhlin property.

Address: Department of Mathematics & Statistics, University of Northern British Columbia, Prince George, B.C. V2N 4Z9, Canada. E-mail address: walters@unbc.ca URL: http://hilbert.unbc.ca/walters