10 research outputs found
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]
Constructive aspects of Riemann's permutation theorem for series
The notions of permutable and weak-permutable convergence of a series
of real numbers are introduced. Classically, these
two notions are equivalent, and, by Riemann's two main theorems on the
convergence of series, a convergent series is permutably convergent if and only
if it is absolutely convergent. Working within Bishop-style constructive
mathematics, we prove that Ishihara's principle \BDN implies that every
permutably convergent series is absolutely convergent. Since there are models
of constructive mathematics in which the Riemann permutation theorem for series
holds but \BDN does not, the best we can hope for as a partial converse to our
first theorem is that the absolute convergence of series with a permutability
property classically equivalent to that of Riemann implies \BDN. We show that
this is the case when the property is weak-permutable convergence
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
Bohrification
New foundations for quantum logic and quantum spaces are constructed by
merging algebraic quantum theory and topos theory. Interpreting Bohr's
"doctrine of classical concepts" mathematically, given a quantum theory
described by a noncommutative C*-algebra A, we construct a topos T(A), which
contains the "Bohrification" B of A as an internal commutative C*-algebra. Then
B has a spectrum, a locale internal to T(A), the external description S(A) of
which we interpret as the "Bohrified" phase space of the physical system. As in
classical physics, the open subsets of S(A) correspond to (atomic)
propositions, so that the "Bohrified" quantum logic of A is given by the
Heyting algebra structure of S(A). The key difference between this logic and
its classical counterpart is that the former does not satisfy the law of the
excluded middle, and hence is intuitionistic. When A contains sufficiently many
projections (e.g. when A is a von Neumann algebra, or, more generally, a
Rickart C*-algebra), the intuitionistic quantum logic S(A) of A may also be
compared with the traditional quantum logic, i.e. the orthomodular lattice of
projections in A. This time, the main difference is that the former is
distributive (even when A is noncommutative), while the latter is not.
This chapter is a streamlined synthesis of 0709.4364, 0902.3201, 0905.2275.Comment: 44 pages; a chapter of the first author's PhD thesis, to appear in
"Deep Beauty" (ed. H. Halvorson
Recommended from our members
Mathematical Logic: Proof Theory, Constructive Mathematics
The workshop âMathematical Logic: Proof Theory, Constructive Mathematicsâ was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexity
Fibred contextual quantum physics
Inspired by the recast of the quantum mechanics in a toposical framework, we develop a contextual quantum mechanics via the geometric mathematics to propose a quantum contextuality adaptable in every topos.
The contextuality adopted corresponds to the belief that the quantum world must only be seen from the classical viewpoints Ă la Bohr consequently putting forth the notion of a context, while retaining a realist understanding. Mathematically, the cardinal object is a spectral Stone bundle ÎŁ â B (between stably-compact locales) permitting a treatment of the kinematics, fibre by fibre and fully point-free. In leading naturally to a new notion of points, the geometricity permits to understand those of the base space B as the contexts C â the commutative C*âalgebras of a incommutative C*âalgebras â and those of the spectral locale ÎŁ as the couples (C, Ï), with Ï a state of the system from the perspective of such a C. The contexts are furnished with a natural order, the aggregation order which is installed as the specialization on B and ÎŁ thanks to (one part of) the Priestley's duality adapted geometrically as well as to the effectuality of the lax descent of the Stone bundles along the perfect maps
Constructive Results on Operator Algebras
Contains fulltext :
32356.pdf (publisher's version ) (Open Access