Some Automatic Continuity Theorems for Operator Algebras and Centralizers of Pedersen's Ideal

We prove automatic continuity theorems for "decomposable" or "local" linear transformations between certain natural subspaces of operator algebras. The transformations involved are not algebra homomorphisms but often are module homomorphisms. We show that all left (respectively quasi-) centralizers of the Pedersen ideal of a C*-algebra A are locally bounded if and only if A has no infinite dimensional elementary direct summand. It has previously been shown by Lazar and Taylor and Phillips that double centralizers of Pedersen's ideal are always locally bounded.Comment: A slightly revised version of this paper is being published by Integral Equations and Operator Theory (published online on October 11 2016

On contra λsi-continuous functions and their applications

In this work, we introduce and study the classes of contra ΛsI-continuous,contra quasi-ΛsI-continuous and contra ΛsI-irresolute functions in a topological spaceendowed with an ideal. We investigate the relationships among these functions andtheir respective characterizations. Also, we analyze the behavior of certain topologicalnotions under direct and inverse images of these new classes of functions

Contributions to the theory of distance functions and its application in general topology

As non-Hausdorff spaces are becoming more important in topology, there is a need to consider new notions in topology to supplement the usual structures. This work uses distance functions to find useful generalizations (in a non-Hausdorff context) of the classes of spaces that are important in the Hausdorff setting. We begin, in the first part, with a historical overview that traces the evolution of the notion of distance and its role in the development of general topology. In the second part of this work, we launch our study of distance functions. Using non-symmetric distance functions, called asymmetric, we generalize the class of symmetrizable spaces which itself includes Moore spaces and metrizable spaces. We also introduce generalizations of Gamma spaces, Nagata spaces and developable spaces. We conclude this work with a number of results about pseudo-metrizable and metrizable spaces. An underlying theme of this work is that distance functions can provide intuitively-appealing proofs for known theorems that usually have more complex derivations and are often presented with explicit use of the Hausdorff property

TT‾\mathrm{T}\overline{\mathrm{T}}-deformed 1d Bose gas

$\mathrm{T}\overline{\mathrm{T}}$ deformation was originally proposed as an irrelevant solvable deformation for 2d relativistic quantum field theories (QFTs). The same family of deformations can also be defined for integrable quantum spin chains which was first studied in the context of integrability in AdS/CFT. In this paper, we construct such deformations for yet another type of models, which describe a collection of particles moving in 1d and interacting in an integrable manner. The prototype of such models is the Lieb-Liniger model. This shows that such deformations can be defined for a very wide range of systems. We study the finite volume spectrum and thermodynamics of the $\mathrm{T}\overline{\mathrm{T}}$-deformed Lieb-Liniger model. We find that for one sign of the deformation parameter $(\lambda<0)$, the deformed spectrum becomes complex when the volume of the system is smaller than certain critical value, signifying the break down of UV physics. For the other sign $(\lambda>0)$, there exists an upper bound for the temperature, similar to the Hagedorn behavior of the $\mathrm{T}\overline{\mathrm{T}}$ deformed QFTs. Both behaviors can be attributed to the fact that $\mathrm{T}\overline{\mathrm{T}}$ deformation changes the size the particles. We show that for $\lambda>0$, the deformation increases the spaces between particles which effectively increases the volume of the system. For $\lambda<0$, $\mathrm{T}\overline{\mathrm{T}}$ deformation fattens point particles to finite size hard rods. This is similar to the observation that the action of $\mathrm{T}\overline{\mathrm{T}}$-deformed free boson is the Nambu-Goto action, which describes bosonic strings -- also an extended object with finite size.Comment: Clarifications added for the derivation of the deformed S-matrix; Typos corrected; References adde

Weak and strong structures and the T3.5 property for generalized topological spaces

We investigate weak and strong structures for generalized topological spaces, among others products, sums, subspaces, quotients, and the complete lattice of generalized topologies on a given set. Also we introduce T3.5 generalized topological spaces and give a necessary and sufficient condition for a generalized topological space to be a T3.5 space: they are exactly the subspaces of powers of a certain natural generalized topology on [0,1]. For spaces with at least two points here we can have even dense subspaces. Also, T3.5 generalized topological spaces are exactly the dense subspaces of compact T4 generalized topological spaces. We show that normality is productive for generalized topological spaces. For compact generalized topological spaces we prove the analogue of the Tychonoff product theorem. We prove that also Lindelöfness (and κ-compactness) is productive for generalized topological spaces. On any ordered set we introduce a generalized topology and determine the continuous maps between two such generalized topological spaces: for | X| , | Y| ≧ 2 they are the monotonous maps continuous between the respective order topologies. We investigate the relation of sums and subspaces of generalized topological spaces to ways of defining generalized topological spaces. © 2016, Akadémiai Kiadó, Budapest, Hungary