Richard V. Kadison

A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras

In [2]2 we classified the isometric mappings of one C*-algebra (uniformly closed, self-adjoint operator algebra) onto another. It was remarked in that paper that the results obtained were a non-commutative extension of results of Banach [1] and Stone [7]. While this was true in spirit, we were well aware that it was not accurate to the letter. Banach and Stone deal with the algebra of real continuous functions on a compact-Hausdorff space, and our results concerning C*-algebras are actually the non-commutative analogue of results concerning the complex function algebra. The strict non-commutative analogue of the real function algebra is the Jordan algebra of self-adjoint elements in a C*-algebra (Jordan C*-algebra). The complex and real theorems follow very easily from one another in the commutative case, so that one might justifiably consider the C*-algebra theorem an extension of both of the function algebra theorems. Despite such trifling considerations, two questions still remain: what are the isometries of one C*-algebra onto another, and what are the isometries of one Jordan C*-algebra onto another? At the time [21 was written, the C*-algebra seemed the more natural object to consider. In view of the results obtained, answering the Jordan C*-algebra questions appeared to be an unnecessary decoration to the theory. We felt that the Jordan C*-algebra results could be obtained from the C*-algebra results in the same way that the real function algebra theorem follows from the complex function algebra theorem (viz., by showing that the complexified linear map is everywhere isometric). Subsequent investigations have changed our attitude in this matter. An important application of these considerations requires a Jordan C*-algebra theorem for one thing, and our attempts to derive this theorem directly from the C*-algebra theorem failed for another. The result in question is contained in Theorem 2 of ?2 and states (in normalized form) that an isometry between two Jordan C*-algebras which carries the identity into the identity is a C* (Jordan) -isomorphism. This theorem was eventually proved with the aid of a Generalized Schwarz Inequality (cf. Theorem 1 of ?2). In effect, an alternative ending has been given to the proof of [Theorem 7; 2]. This ending is by no means simpler or shorter than the one given in [2] (though it is, perhaps, less contrived), but it is flexible enough to allow us to draw the desired Jordan C*-algebra conclusion. The critical application of these results is contained in Corollary 3. A discussion accompanies Corollaries 3 and 4, but a few additional remarks are in order.

Transformations of states in operator theory and dynamics

Two BASIC constituents of a physical system are its family 2I of observable attributes and the family S of states in which the system can be found. In classical (particle) mechanics, the observables are algebraic combinations of the (canonical) coordinates and (conjugate) momenta. Each state is described by an assignment of numbers to these observables-the values certain to be found by measuring these observables in the given state. The totality of numbers associated with a given observable is its spectrum. In this view of classical statics, the observables 21 are represented as functions on the space S of states-they form an algebra (necessarily commutative) relative to pointwise operations. The dynamics (or law of motion) of this system describes the way the states evolve in time (i.e. specifies trajectories through states in S). The experiments involving atomic and sub-atomic phenomena made it clear that this Newtonian view of mechanics would not suffice for their basic theory. Speculation on the meaning of these experimental results eventually led to the conclusion that the only physically meaningful description of a state was in terms of an assignment of probability measures to the spectra of the observables (a measurement of the observable with the system in a given state will produce a value in a given portion of the spectrum with a specific probability). Moreover, it was necessary to assume, in this physical realm, that a state which assigns a “definite” value to one observable (position) assigns a dispersed measure to the spectrum of some other observable (momentum)--the amount of dispersion involving the experimentally reappearing Planck’s constant (The Uncertainty Principle). Further analysis shows that this entails the non-commutativity of the algebra of observables.

Derivations and automorphisms of operator algebras

AbstractThe theorem that each derivation of aC*-algebra

The Pythagorean Theorem: I. The finite case

\mathfrak{A}

Remarks on the Type of Von Neumann Algebras of Local Observables in Quantum Field Theory

extends to an inner derivation of the weak-operator closure ϕ(

DERIVATIONS OF OPERATOR GROUP ALGEBRAS.

\mathfrak{A}