Multiplicativity Factors for Function Norms

AbstractLet (T, Ω, m) be a measure space; let ρ be a function norm on M = M(T, Ω, m), the algebra of measurable functions on T; and let Lρ be the space {f ∈ M : ρ(f) < ∞} modulo the null functions. If Lρ, is an algebra, then we call a constant μ > 0 a multiplicativity factor for ρ if ρ(fg) ≤ μρ(f) ρ(g) for all f, g ∈ Lρ. Similarly, λ > 0 is a quadrativity factor if ρ(f2) ≤ λρ(f)2 for all f. The main purpose of this paper is to give conditions under which Lρ, is indeed an algebra, and to obtain in this case the best (least) multiplicativity and quadrativity factors for ρ. The first of our two principal results is that if ρ is σ-subadditive, then Lρ is an algebra if and only if Lρ is contained in L∞. Our second main result is that if (T, Ω, m) is free of infinite atoms, ρ is σ-subadditive and saturated, and Lρ, is an algebra, then the multiplicativity and quadrativity factors for ρ coincide, and the best such factor is determined by sup{||f||∞: f ∈ Lρ, ρ(f) ≤ 1}

Multiplicativity Factors for Orlicz Space Function Norms

AbstractLet ρφ be a function norm defined by a Young function φ with respect to a measure space (T, Ω, m), and let Lφ be the Orlicz space determined by ρφ. If Lφ is an algebra, then a constant μ > 0 is called a multiplicativity factor for ρφ, if ρφ,(fg) ≤ μρφ(f) ρφ(g) for all f, g ∈ Lφ. The main objective of this paper is to give conditions under which Lφ is indeed an algebra, and to obtain in this case the best (least) multiplicativity factor for ρφ. The first of our principal results is that Lφ is an algebra if and only if minf ≡ inf{m(A) > 0 : A ∈ Ω} > 0 or x∞(φ) ≡ sup{x ≥ 0 : φ(x) < ∞} < ∞ Our second main result states that if Lφ is an algebra and (T, Ω, m) is free of infinite atoms, then the best multiplicativity factor for ρφ is φ−1(1/minf if minf > 0, and x∞(φ) if minf = 0

Multivariate risks and depth-trimmed regions

We describe a general framework for measuring risks, where the risk measure takes values in an abstract cone. It is shown that this approach naturally includes the classical risk measures and set-valued risk measures and yields a natural definition of vector-valued risk measures. Several main constructions of risk measures are described in this abstract axiomatic framework. It is shown that the concept of depth-trimmed (or central) regions from the multivariate statistics is closely related to the definition of risk measures. In particular, the halfspace trimming corresponds to the Value-at-Risk, while the zonoid trimming yields the expected shortfall. In the abstract framework, it is shown how to establish a both-ways correspondence between risk measures and depth-trimmed regions. It is also demonstrated how the lattice structure of the space of risk values influences this relationship.Comment: 26 pages. Substantially revised version with a number of new results adde

A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic