A Survey of Operator Theory
The subject of operator algebras originated in a paper of
John von Neumann that appeared in 1929.
Motivated by quantum mechanics and group representation theory,
John von Neumann introduced in the early 30's certain algebras
of bounded operators on a Hilbert space, the so-called von Neumann
algebras.
Applications to the foundations of quantum mechanics were
a motivation for von Neumann's interest.
In his bicommutant theorem he showed that these algebras can be
characterized
either in purely topological or in purely algebraic terms,
a fact that has numerous beautiful and deep consequences.
Today the subject is one of the most dynamic areas of research
in modern mathematics.
The theory of operator algebras has many fruitful interrelations
and fruitful connections with other areas of mathematics and physics,
in particular with low dimensional topology, statistical mechanics, and
quantum field theory. In addition, Jones' theory of subfactors has far
reaching applications in knot theory, quantum groups and topology.
Modern research in C*-algebras via K-theory, KK-theory, and E-theory
connect this area to the mathematical development
related to the Atiyah-Singer index theorem.
For instance, Alain Connes' noncommutative geometry provides a framework
for the standard model in modern particle physics and allows one
to predict properties of elementary particles
(e.g. the mass of the Higgs particle).
Vaughan Jones' theory of subfactors led to
the discovery of the Jones polynomial, an invariant for knots and links,
and had far reaching applications to low dimensional topology
(e.g. new invariants for links and three manifolds and
Atiyah-Witten topological quantum field theories).
Dan Voiculescu's free probability theory introduced probabilistic methods
to the analysis of von Neumann algebras associated to free groups
and his new concept of free entropy can be viewed as a measure of freeness
in this context. K-theory, Kasporov's KK-theory
and Connes' cyclic (co)homology are used to study and classify
C*-algebras and have applications to geometry such as the Novikov
conjecture and various generalizations of the Atiyah-Singer index theorem.
A subfactor can be viewed as a mathematical object encoding symmetry
of a mathematical or physical problem much like a group does.
However, a subfactor is an infinite
dimensional, highly noncommutative object and the symmetry it represents
is more general than group symmetry.
Operator algebra methods can be used to decode this symmetry and one
obtains
finite dimensional data in this process,
which can then be described combinatorially and computed numerically.
For instance, certain weighted bipartite graphs appear
as basic structural ingredients and commuting squares
(certain inclusions of four finite dimensional algebras)
play a key role in this analysis.
There are numerous fruitful connections of the theory of subfactors
to statistical mechanics, algebraic quantum field theory,
low dimensional topology, and other areas of mathematics and physics.
Intuitively, a C*-algebra can be thought of as a quantization of space -
a topological space or a manifold. The best evidence for this is in the
study of the irrational rotation algebras. These are algebras generated by
2 unitaries U and V (position and momentum unitaries) satisfying the
Heisenberg commutation relation VU = zUV, where z=exp(2pi*i*theta) and
theta is the angle of rotation (Planck's constant). A deep result obtained
by the work of Rieffel, Voiculescu, and Pimsner in the early 1980's, is
that if we perturb Planck's constant however slightly, we get a very
different (non-isomorphic) algebra.
One contribution of C*-algebra theory to mathematical physics is
that the fabric of spacetime may not be locally Euclidean,
as is assumed in current physical theory (including differential
geometry). The locally Euclidean hypothesis is then replaced
by looking 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.
In fact, some prominant physicists
(including Fields medalist Ed Witten of Institute for Advanced Study
at Princeton) have taken this approach
and believe that it could form the mathematical base needed
to develop a unified field theory - so that quantum phenomena
and gravitation (general relativity) would be unified into one theory.
Another link to mathematical physics is
in the study of the Quantum Hall Effect where the Hall conductivity is
shown to be equal to Connes' noncommutative Chern character, which is the
Chern character of a projection (in a C*-algebra) associated to the Fermi
energy. 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.
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