Elements of mathematics

Probably the most famous of Grothendieck’s contributions to Banach space theory is the result that he himself described as “the fundamental theorem in the metric theory of tensor products”. That is now commonly referred to as “Grothendieck’s theorem” (GT in short), or sometimes as “Grothendieck’s inequality”. This had a major impact first in Banach space theory (roughly after 1968), then, later on, in C ¤ -algebra theory, (roughly after 1978). More recently, in this millennium, a new version of GT has been successfully developed in the framework of “operator spaces” or non-commutative Banach spaces. In addition, GT independently surfaced in several quite unrelated fields: in connection with Bell’s inequality in quantum mechanics, in graph theory where the Grothendieck constant of a graph has been introduced and in computer science where the Grothendieck inequality is invoked to replace certain NP hard problems by others that can be treated by “semidefinite programming’ and hence solved in polynomial time. In this expository paper, we present a review of all these topics, starting from the original GT. We concentrate on the more recent developments and merely outline those of the first Banach space period since detailed accounts of that are already available, for instance the author’s 1986 CBMS notes.

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Grothendieck's Theorem, past and present

The basics of the theory of symmetric tensor products of normed spaces and some applications are presented.

Natural norms on symmetric tensor products of normed spaces

This thesis consists of two parts.
In Part I, projection algorithms for solving convex feasibility problems in Hilbert space are studied. Powerful techniques from Convex Analysis are employed within a very general framework that covers and extends many well-known results. Ostensibly different looking conditions sufficient for linear convergence are shown to be special instances of regularity--a concept new in this context. Numerous examples, including subgradient algorithms, are presented.
Several notions of monotonicity of operators on Banach spaces are analyzed in Part II. Utilizing Convex and Functional Analysis, it is shown that for a bounded linear positive semi-definite operator, all these "monotonicities" coincide with the monotonicity of the conjugate operator. Moreover, monotonicity of the conjugate operator is automatic in many classical Banach spaces but not in spaces containing a complemented copy of the space of absolutely convergent sequences.

https://summit.sfu.ca/system/files/iritems1/7015/b18025766.pdf

Projection algorithms and monotone operators

Weak compactness in the dual of a C*-algebra is determined commutatively.

We provide a variety of results for quasiconvex, law-invariant functionals defined on a general Orlicz space, which extend well-known results from the setting of bounded random variables. First, we show that Delbaen’s representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can always be achieved under the assumption of law-invariance. Second, we identify the class of Orlicz spaces where the characterization of the Fatou property in terms of norm-lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka’s representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovic and Svindland by replacing norm-lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.

Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

Unbounded order convergence has lately been systematically studied as a generalization of almost everywhere convergence to the abstract setting of vector and Banach lattices This paper presents a duality theory for unbounded order convergence We define the unbounded order dual (or uo-dual) $${X_{uo}^\sim }$$
 of a Banach lattice X and identify it as the order continuous part of the order continuous dual $${X_n^\sim }$$
  The result allows us to characterize the Banach lattices that have order continuous preduals and to show that an order continuous predual is unique when it exists Applications to the Fenchel–Moreau duality theory of convex functionals are given The applications are of interest in the theory of risk measures in Mathematical Finance

Duality for unbounded order convergence and applications

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and isc
 0-saturated, i.e., each closed infinite dimensional subspace contains an isomorph ofc
 0. In this paper, we show that the Orlicz sequence spaceh
 M is isomorphic to a polyhedral Banach space if lim
 t→0
 M(Kt)/M(t)=∞ for someK<∞. We also construct an Orlicz sequence spaceh
 M which isc
 0-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that beingc
 0-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.

Some isomorphically polyhedral Orlicz sequence spaces

A Banach space is polyhedral if the unit ball of each of its finite dimensional subspaces is a polyhedron. It is known that a polyhedral Banach space has a separable dual and is $c_0$-saturated, i.e., each closed infinite dimensional subspace contains an isomorph of $c_0$. In this paper, we show that the Orlicz sequence space $h_M$ is isomorphic to a polyhedral Banach space if $\lim_{t\to 0}M(Kt)/M(t) = \infty$ for some $K < \infty$. We also construct an Orlicz sequence space $h_M$ which is $c_0$-saturated, but which is not isomorphic to any polyhedral Banach space. This shows that being $c_0$-saturated and having a separable dual are not sufficient for a Banach space to be isomorphic to a polyhedral Banach space.

Denny H. Leung

Papers

Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

Duality for unbounded order convergence and applications

Some isomorphically polyhedral Orlicz sequence spaces

Some isomorphically polyhedral Orlicz sequence spaces

All 2-positive linear maps from M3(C) to M3(C) are decomposable