The proposed use of operator algebras has several advantages. First, operator algebras constitute themselves as a natural link between algebraic and analytical models.
Second, C
-algebras are well studied object. From the mathematical
point of view, there exists a quite extensive knowledge. At the
same time C
-algebras are also a still developing branch of
mathematics. To make use of this mathematical knowledge should help
to gain a better understanding of various aspects of computation;
it might prevent us from reinventing the wheel all over again.
Third, C
-algebras are used within several fields closely related
to neural networks and other models of dynamic systems. Some of these
concern various ``dynamic'' and ``physical'' aspects computation,
like Markov Chains [Cuntz and Krieger, 1980], Dynamical Systems
[Connes, 1994], and -- last but not least -- Quantum Mechanics
[Bratteli and Robinson, 1979]. And there are also several recent
approaches in logic and computer science which are
related to concepts based in C
-algebras (Girard's Geometry
of Interaction, Quantales, etc).
Common models in computer science usually are based on discrete mathematics. Analytical models like operator algebras might thus seem superficial. But incorporating elements and methods from mathematical disciplines like functional analysis together with logical frameworks will increase our knowledge about computational processes, e.g. by embedding discrete models (e.g. a proof tree) in a continuous model (e.g. operator algebra) -- after all, it is not obvious that discrete problems need discrete methods to solve them, neither are discrete problems easier to handle (e.g. diophantine equations are by no means simpler than those over the real numbers).