In 1996, Harris and Kadison posed the following problem: show that a linear bijection between C∗-algebras that preserves the identity and the set of invertible elements is a Jordan isomorphism. In this paper, we show that if A and B are semisimple Banach algebras andΦ : A→ B is a linear map onto B that preserves the spectrum of elements, thenΦ is a Jordan isomorphism if either A or B is a C∗-al...

Abstract. A compact closed bicategory is a symmetric monoidal bicategory where every object is equipped with a weak dual. The unit and counit satisfy the usual “zig-zag” identities of a compact closed category only up to natural isomorphism, and the isomorphism is subject to a coherence law. We give several examples of compact closed bicategories, then review previous work. In particular, Day a...

The Friedlander{Milnor Conjecture 1] asserts that if G is a reductive algebraic group over an algebraically closed eld k, then the comparison map H et (BG k ; Z=p) ?! H (BG; Z=p) is an isomorphism for all primes p not equal to the characteristic of k. Gabber's rigidity theorem 2] implies that this map is indeed an isomorphism for the stable general linear group GL (this is due to Suslin 6] for ...

In the present paper we discuss an independent on the Grothendieck-Sato isomorphism approach to the Riemann-RochHirzebruch formula for an arbitrary differential operator. Instead of the Grothendieck-Sato isomorphism, we use the Topological Quantum Mechanics (more or less equivalent to the well-known constructions with the Massey operations from [KS], [P], [Me]). The statement that the Massey op...

A theory is called κ-categorical, or categorical in power κ, if it has one model up to isomorphism of cardinality κ. Morley’s Categoricity Theorem states that if a theory of first order logic is categorical in some uncountable power κ, then it is categorical in every uncountable power. We provide an elementary exposition of this theorem, by showing that a theory is categorical in some uncountab...

We prove the following theorem characterizing Du Bois singularities. Suppose that Y is smooth and that X is a reduced closed subscheme. Let π : e Y → Y be a log resolution of X in Y that is an isomorphism outside of X. If E is the reduced pre-image of X in e Y , then X has Du Bois singularities if and only if the natural map ØX → Rπ∗ØE is a quasi-isomorphism. We also deduce Kollár’s conjecture ...

Every isomorphism invariant Borel subset of the space of structures on the natural numbers in a countable relational language is definable in Lω1ω by a theorem of Lopez-Escobar. We derive variants of this result for stabilizer subgroups of the symmetric group Sym(N) for families of relations and non-isomorphism invariant generalized quantifiers on the natural numbers such as “for all even numbe...

Recently, Cochran and Harvey defined torsion-free derived series of groups and proved an injectivity theorem on the associated torsion-free quotients. We show that there is a universal construction which extends such an injectivity theorem to an isomorphism theorem. Our result relates injectivity theorems to a certain homology localization of groups. In order to give a concrete combinatorial de...

The Friedlander{Milnor Conjecture 1] asserts that if G is a reductive algebraic group over an algebraically closed eld k, then the comparison map H et (BG k ; Z=p) ?! H (BG; Z=p) is an isomorphism for all primes p not equal to the characteristic of k. Gabber's rigidity theorem 2] implies that this map is indeed an isomorphism for the stable general linear group GL (this is due to Suslin 6], als...

Let X and Y be compact metric spaces and let a map /: X—> Y be onto. The Vietoris Mapping Theorem as proved by Vietoris [8] states that if for all Ofkrfkn-1 and all yEY, flr(/_1(y)) =0 (augmented Vietoris homology mod two) then the induced homomorphism /*: Hr(X)-+Hr(Y) is an isomorphism onto for rfkn — l and onto for r=n. Begle [l; 2] has generalized this theorem to nonmetric spaces and more ge...