Dual spaces of operator systems Ng, Chi-Keung
Journal of mathematical analysis and applications,
04/2022, Volume:
508, Issue:
2
Journal Article
Peer reviewed
Open access
The aim of this article is to give an infinite dimensional analogue of a result of Choi and Effros concerning dual spaces of finite dimensional unital operator systems. An (not necessarily unital) ...operator system is a self-adjoint subspace of L(H), equipped with the induced matrix norm and the induced matrix cone. We say that an operator system T is dualizable if one can find an equivalent dual matrix norm on the dual space T⁎ such that under this dual matrix norm and the canonical dual matrix cone, T⁎ becomes a dual operator system. We show that an operator system T is dualizable if and only if the ordered normed space M∞(T)sa satisfies a form of bounded decomposition property. In this case,‖f‖d:=sup{‖fi,j(xk,l)‖:x∈Mn(T)+;‖x‖≤1;n∈N}(f∈Mm(T⁎);m∈N), is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on T⁎ that turns it into a dual operator system, denoted by Td. It can be shown that Td is again dualizable. For every completely positive completely bounded map ϕ:S→T between dualizable operator systems, there is a unique weak-⁎-continuous completely positive completely bounded map ϕd:Td→Sd which is compatible with the dual map ϕ⁎. From this, we obtain a full and faithful functor from the category of dualizable operator systems to that of dualizable dual operator systems. Moreover, we will verify that if S is either a C⁎-algebra or a unital operator system, then S is dualizable and the canonical weak-⁎-homeomorphism from the unital operator system S⁎⁎ to the operator system (Sd)d is a completely isometric complete order isomorphism. Furthermore, via the duality functor above, the category of C⁎-algebras and that of unital operator systems (both equipped with completely positive complete contractions as their morphisms) can be regarded as full subcategories of the category of dual operator systems (with weak-⁎-continuous completely positive complete contractions as its morphisms). Consequently, a nice duality framework for operator systems is obtained, which includes all C⁎-algebras and all unital operator systems.
We investigate effects of fermionic T‐duality on type II superstring in presence of Ramond‐Ramond (RR) field that has infinitesimal linear dependence on bosonic coordinate xμ. Other fields are ...assumed to be constant. Procedure that we employ for obtaining fermionic T‐dual theory is Buscher procedure, where we will consider two distinct cases. One, where action has not been T‐dualized along bosonic coordinates and other where it has. By analyzing these two cases, their actions and T‐dual transformation laws, we obtain some insight into how background fields transform and what are necessary ingredients for emergence of fermionic non‐commutativity.
The authors investigate effects of fermionic T‐duality on type II superstring in presence of Ramond‐Ramond (RR) field that has infinitesimal linear dependence on bosonic coordinate xμ. Other fields are assumed to be constant. The employed procedure for obtaining fermionic T‐dual theory is Buscher procedure with consideration of two distinct cases. One, where action has been T‐dualized along bosonic coordinates and the other where it has not. By analyzing these two cases, their actions and T‐dual transformation laws, one obtains some insight into how background fields transform and what are necessary ingredients for emergence of fermionic non‐commutativity.
Abstract Mirror symmetry has proven to be a powerful tool to study several properties of higher dimensional superconformal field theories upon compactification to three dimensions. We propose a ...quiver description for the mirror theories of the circle reduction of twisted A 2N theories of class S in four dimensions. Although these quivers bear a resemblance to the star-shaped quivers previously studied in the literature, they contain unitary, symplectic and special orthogonal gauge groups, along with hypermultiplets in the fundamental representation. The vacuum moduli spaces of these quiver theories are studied in detail. The Coulomb branch Hilbert series of the mirror theory can be matched with that of the Higgs branch of the corresponding four dimensional theory, providing a non-trivial check of our proposal. Moreover various deformations by mass and Fayet-Iliopoulos terms of such quiver theories are investigated. The fact that several of them flow to expected theories also gives another strong support for the proposal. Utilising the mirror quiver description, we discover a new supersymmetry enhancement renormalisation group flow.
Abstract In this note, we study the Swampland Distance Conjecture in TCS G 2 manifold compactifications of M-theory. In particular, we are interested in testing a refined version — the Emergent ...String Conjecture, in settings with 4d N = 1 supersymmetry. We find that a weakly coupled, tensionless fundamental heterotic string does emerge at the infinite distance limit characterized by shrinking the K3-fiber in a TCS G 2 manifold. Such a fundamental tensionless string leads to the parametrically leading infinite tower of asymptotically massless states, which is in line with the Emergent String Conjecture. The tensionless string, however, receives quantum corrections. We check that these quantum corrections do modify the volume of the shrinking K3-fiber via string duality and hence make the string regain a non-vanishing tension at the quantum level, leading to a decompactification. Geometrically, the quantum corrections modify the metric of the classical moduli space and are expected to obstruct the infinite distance limit. We also comment on another possible type of infinite distance limit in TCS G 2 compactifications, which might lead to a weakly coupled fundamental type II string theory.
Duality in algebra and topology Dwyer, W.G.; Greenlees, J.P.C.; Iyengar, S.
Advances in mathematics (New York. 1965),
03/2006, Volume:
200, Issue:
2
Journal Article
Peer reviewed
Open access
We apply ideas from commutative algebra, and Morita theory to algebraic topology using ring spectra. This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré ...duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson–Carlson duality for cohomology rings of finite groups, (4) Poincaré duality for groups and (5) Gross–Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon.
This paper considers an uncertain convex optimization problem, posed in a locally convex decision space with an arbitrary number of uncertain constraints. To this problem, where the uncertainty only ...affects the constraints, we associate a robust (pessimistic) counterpart and several dual problems. The paper provides corresponding dual variational principles for the robust counterpart in terms of the closed convexity of different associated cones.
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