C ∗-algebras are widely used in mathematical physics to represent the observables of physical systems, and are sometimes taken as the starting point for rigorous formulations of quantum mechanics and classical statistical mechanics. Nevertheless, in many cases the näıve choice of an algebra of observables does not admit a C∗-algebra structure, and some massaging is necessary. In this paper we i...

We clarify in which precise sense the theory of principal bundles and the theory of groupoids are equivalent; and how this equivalence of theories, in the differentiable case, reflects itself in the theory of connections. The method used is that of synthetic differential geometry. Introduction. In this note, we make explicit a sense in which the theory of principal fibre bundles is equivalent t...

We consider the vector-valued extension of the Fefferman–Stein– Strömberg sharp maximal inequality under growth condition. As an application we obtain a vector-valued extension of the boundedness of the commutator. Furthermore, we prove the boundedness of the commutator. 2000 Mathematics Subject Classification: Primary 42B35; Secondary 42B25.

We prove a Structure Identity Principle for theories defined on types of h-level 3 by defining a general notion of saturation for a large class of structures definable in the Univalent Foundations. Formalizing mathematics in the framework of the Univalent Foundations [Uni13] presents unique challenges and opportunities. One of the main opportunities is the ability to formalize higher-level math...

We define the phrase ‘category enriched in an fc-multicategory’ and explore some examples. An fc-multicategory is a very general kind of 2-dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fc-multicategory extends the (more or less well-known) theories of enrichment in a monoidal category, in a b...

In a recent paper we have shown that most non-expansive Lipschitz functions (in the sense of Baire’s category) have a maximal Clarke subdifferential. In the present paper, we show that in a separable Banach space the set of non-expansive Lipschitz functions with a maximal Clarke subdifferential is not only of generic, but also staunch. 1991 Mathematics Subject Classification: Primary 49J52.

In this note we provide a characterization, in terms of additional algebraic structure, of those strict intervals (certain cocategory objects) in a symmetric monoidal closed category E that are representable in the sense of inducing on E the structure of a finitely bicomplete 2-category. Several examples and connections with the homotopy theory of 2-categories are also discussed. Introduction A...

In this work we are concerned with maximality of monotone operators representable by certain convex functions in non-reflexive Banach spaces. We also prove that these maximal monotone operators satisfy a Brønsted-Rockafellar type property. 2000 Mathematics Subject Classification: 47H05, 49J52, 47N10.

In this article, we will show that the category of quaternion vector spaces, the category of (both one-sided and two sided) quaternion Hilbert spaces and the category of quaternion B∗-algebras are equivalent to the category of real vector spaces, the category of real Hilbert spaces and the category of real C∗-algebras respectively. We will also give a Riesz representation theorem for quaternion...

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset mona...