Operator algebras

Abstract

Linear maps acting on Hilbert spaces play an important role in quantum mechanics. Such a linear map is called an operator. Well-known examples are the Pauli matrix σz, which measures the spin in the z direction, and the position and momentum operators P and Q of a single particle on the line. The first acts on the Hilbert space ℂ2, while the latter are defined on dense subspaces of L2(ℝ). An operator algebra is an algebra of such operators, usually with additional conditions, such as being closed in a certain topology. Here we introduce some of the basic concepts in the theory of operator algebras. Thematerial here is standard, and by nowthere is a huge body of textbooks on the subject, most of which cover a substantially bigger part of the field than these notes. Particularly recommended are the two volumes by Bratteli and Robinson [2, 3], which contain many applications to physics. Many of the topics covered here are studied there in extenso. The books by Kadison& Ringrose provide a very thorough introduction to operator algebras, and contain many exercises [6, 7]. The book by Takesaki [14] (and the subsequent volumes II and III [15, 16]) is a classic, but is more technical. Volume I of Reed and Simon’s Methods of Modern Mathematical Physics [10] or Pedersen’s Analysis Now [9] cover the necessary tools of functional analysis (and much more), but do not cover most of the material on operator algebras we present here.