V*-algebras

TL;DR

This paper introduces V*-algebras, a new class of operator algebras defined via a continuum-weak topology, generalizing the functional calculus for unitaries in a noncommutative setting, within a set-theoretic framework assuming the existence of an inaccessible cardinal.

Contribution

It defines V*-algebras as C*-algebras closed in the continuum-weak topology and establishes their structure and relation to classical algebras like $ ext{ell}^ty$, extending the functional calculus in a noncommutative context.

Findings

V*-algebras generalize the notion of bounded operators with a new topology.

Every C*-algebra has an enveloping V*-algebra, $V^*(A)$.

For locally compact spaces, $V^*(C_0(X)) \u2208 ext{ell}^ty(X)$.

Abstract

What is the correct noncommutative generalization of the functor $C_0(X) \mapsto \ell^\infty(X)$ for locally compact Hausdorff $X$ having a countable basis? Making the ansatz $K(\ell^2) \mapsto B(\ell^2)$, we expect that every unital $*$-homomorphism $C(\mathbb T) \rightarrow B(\ell^2)$ extend canonically to a unital $*$-homomorphism $\ell^\infty(\mathbb T) \rightarrow B(\ell^2)$. Thus, we expect to extend the continuous functional calculus for a unitary operator on $\ell^2$ to all bounded complex-valued functions. Therefore, we work in a model of set theory where every set of real numbers is Lebesgue measurable; we must assume the consistency of an inaccessible cardinal in order to do so. The axiom of choice necessarily fails in such a model, but our model is carefully chosen to enable the verification of many familiar theorems via a scrutinization of their statements rather than their proofs. This technique significantly lowers the cost of doing interesting mathematics in this unfamiliar setting, and it is explained in detail. By analogy with the ultraweak topology, we define the continuum-weak topology on bounded operators to be the topology given by functionals of the form $x \mapsto \int_0^1 <\eta_t| x \xi_t>\, dt$. We then define a V*-algebra to be a C*-algebra of bounded operators that is closed in the continuum-weak topology. Every C*-algebra $A$ has an enveloping V*-algebra $V^*(A)$, and if $X$ is a locally compact Hausdorff space with a countable basis, then $V^*(C_0(X)) \cong \ell^\infty(X)$. More generally, if $A$ is any separable C*-algebra of type I, then $V^*(A)$ is canonically isomorphic to an $\ell^\infty$-direct sum of type I factors, with one summand for each irreducible representation of $A$. The self-adjoint part of any unital separable C*-algebra is isomorphic to the Banach space of strongly affine real-valued functions on its state space.