2,465 research outputs found
Constructive Theory of Banach algebras
We present a way to organize a constructive development of the theory of
Banach algebras, inspired by works of Cohen, de Bruijn and Bishop. We
illustrate this by giving elementary proofs of Wiener's result on the inverse
of Fourier series and Wiener's Tauberian Theorem, in a sequel to this paper we
show how this can be used in a localic, or point-free, description of the
spectrum of a Banach algebra
Localic Metric spaces and the localic Gelfand duality
In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that
the constructive Gelfand duality can be extended into a duality between compact
regular locales and unital abelian localic C*-algebras. In order to do so we
develop a constructive theory of localic metric spaces and localic Banach
spaces, we study the notion of localic completion of such objects and the
behaviour of these constructions with respect to pull-back along geometric
morphisms.Comment: 57 page
Constructive Gelfand duality for C*-algebras
We present a constructive proof of Gelfand duality for C*-algebras by
reducing the problem to Gelfand duality for real C*-algebras.Comment: 6page
Locating Ax, where A is a subspace of B(H)
Let A be a linear space of operators on a Hilbert space H, x a vector in H,
and Ax the subspace of H comprising all vectors of the form Tx with T in A. We
discuss, within a Bishop-style constructive framework, conditions under which
the projection [Ax] of H on the closure of Ax exists. We derive a general
result that leads directly to both the open mapping theorem and our main
theorem on the existence of [Ax]
Bohrification of operator algebras and quantum logic
Following Birkhoff and von Neumann, quantum logic has traditionally been
based on the lattice of closed linear subspaces of some Hilbert space, or, more
generally, on the lattice of projections in a von Neumann algebra A.
Unfortunately, the logical interpretation of these lattices is impaired by
their nondistributivity and by various other problems. We show that a possible
resolution of these difficulties, suggested by the ideas of Bohr, emerges if
instead of single projections one considers elementary propositions to be
families of projections indexed by a partially ordered set C(A) of appropriate
commutative subalgebras of A. In fact, to achieve both maximal generality and
ease of use within topos theory, we assume that A is a so-called Rickart
C*-algebra and that C(A) consists of all unital commutative Rickart
C*-subalgebras of A. Such families of projections form a Heyting algebra in a
natural way, so that the associated propositional logic is intuitionistic:
distributivity is recovered at the expense of the law of the excluded middle.
Subsequently, generalizing an earlier computation for n-by-n matrices, we
prove that the Heyting algebra thus associated to A arises as a basis for the
internal Gelfand spectrum (in the sense of Banaschewski-Mulvey) of the
"Bohrification" of A, which is a commutative Rickart C*-algebra in the topos of
functors from C(A) to the category of sets. We explain the relationship of this
construction to partial Boolean algebras and Bruns-Lakser completions. Finally,
we establish a connection between probability measure on the lattice of
projections on a Hilbert space H and probability valuations on the internal
Gelfand spectrum of A for A = B(H).Comment: 31 page
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