C*-algebras

The purpose of this paper is to walk the reader through a mathematical development of physics, motivating everything along the way, sometimes with physical arguments, sometimes with mathematical ones, starting with Newtonian mechanics and ending with a modern axiomatization of quantum mechanics. From a purely mathematical standpoint, to achieve this goal, we introduce what the authors calls an \emph{$F^*$-algebra} and proves what are, to be best knowledge of the author, original results about such objects, the proofs of which were motivated by ideas that can be found in many functional analysis textbooks.

UPDATE:  There is at least one error I've caught so far.  Theorem 4.1 is wrong.  Essentially the same argument that shows there can be no such $C^*$-algebra should show there is no such $F^*$-algebra.  As for where the error is, at least one I believe is in the claim that "each $p_n$ is clearly a seminorm".  This is why you don't say things are "clear" unless you've already checked the details.

UPDATED:  I have since found that what I called $F^*$-algebras had been studied in the literature under the different name of \emph{$\sigma$-$C^*$-algebras}---see, for example, \textit{Inverse Limits of $C^*$-algebras} by N. Christopher Phillips (http://www.theta.ro/jot/archive/1988-019-001/1988-019-001-010.pdf).

A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli’s notion of equivalence, we obtain in particular a functorial formulation of Bratteli’s classification of AF algebras (and at the same time, of Glimm’s classification of UHF algebras). It is shown that the three approaches to classification of AF algebras, namely, through Bratteli diagrams, K-theory, and a certain natural abstract classifying category, are essentially the same from a categorical point of view.

Abstract: From N-tensor powers of the Toeplitz algebra, we construct a multipullback C*-algebra that is a noncommutative deformation of the complex projective space CP (N). Using Birkhoff's Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multipullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.

Abstract: From N-tensor powers of the Toeplitz algebra, we construct a multipullback C*-algebra that is a noncommutative deformation of the complex projective space CP (N). Using Birkhoff's Representation Theorem, we prove that the lattice of kernels of the canonical projections on components of the multipullback C*-algebra is free. This shows that our deformation preserves the freeness of the lattice of subsets generated by the affine covering of the complex projective space.