
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 for locally compact Hausdorff having a countable basis? Making the ansatz , we expect that every unital -homomorphism extend canonically to a unital -homomorphism . Thus, we expect to extend the continuous functional calculus for a unitary operator on 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…
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Taxonomy
TopicsAdvanced Algebra and Logic
