Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)

It has been known since the work of Duskin and Pelletier four decades ago that KH^op, the category opposite to compact Hausdorff spaces and continuous maps, is monadic over the category of sets. It follows that KH^op is equivalent to a possibly infinitary variety of algebras V in the sense of Slominski and Linton. Isbell showed in 1982 that the Lawvere-Linton algebraic theory of V can be generated using a finite number of finitary operations, together with a single operation of countably infinite arity. In 1983, Banaschewski and Rosicky independently proved a conjecture of Bankston, establishing a strong negative result on the axiomatisability of KH^op. In particular, V is not a finitary variety--Isbell's result is best possible. The problem of axiomatising V by equations has remained open. Using the theory of Chang's MV-algebras as a key tool, along with Isbell's fundamental insight on the semantic nature of the infinitary operation, we provide a finite axiomatisation of V.Comment: 26 pages. Presentation improve

Adelic versions of the Weierstrass approximation theorem

Let $\underline{E}=\prod_{p\in\mathbb{P}}E_p$ be a compact subset of $\widehat{\mathbb{Z}}=\prod_{p\in\mathbb{P}}\mathbb{Z}_p$ and denote by $\mathcal C(\underline{E},\widehat{\mathbb{Z}})$ the ring of continuous functions from $\underline{E}$ into $\widehat{\mathbb{Z}}$. We obtain two kinds of adelic versions of the Weierstrass approximation theorem. Firstly, we prove that the ring ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}}):=\{f(x)\in\mathbb{Q}[x]\mid \forall p\in\mathbb{P},\;\;f(E_p)\subseteq \mathbb{Z}_p\}$ is dense in the direct product $\prod_{p\in\mathbb{P}}\mathcal C(E_p,\mathbb{Z}_p)\,$ for the uniform convergence topology. Secondly, under the hypothesis that, for each $n\geq 0$, $\#(E_p\pmod{p})>n$ for all but finitely many $p$, we prove the existence of regular bases of the $\mathbb{Z}$-module ${\rm Int}_{\mathbb{Q}}(\underline{E},\widehat{\mathbb{Z}})$, and show that, for such a basis $\{f_n\}_{n\geq 0}$, every function $\underline{\varphi}$ in $\prod_{p\in\mathbb{P}}\mathcal{C}(E_p,\mathbb{Z}_p)$ may be uniquely written as a series $\sum_{n\geq 0}\underline{c}_n f_n$ where $\underline{c}_n\in\widehat{\mathbb{Z}}$ and $\lim_{n\to \infty}\underline{c}_n\to 0$.Comment: minor corrections the statement of Theorem 3.5, which covers the case of a general compact subset of the profinite completion of Z. to appear in Journal of Pure and Applied Algebra, comments are welcome

Hecke algebra isomorphisms and adelic points on algebraic groups

Let $G$ denote a linear algebraic group over $\mathbf{Q}$ and $K$ and $L$ two number fields. Assume that there is a group isomorphism of points on $G$ over the finite adeles of $K$ and $L$, respectively. We establish conditions on the group $G$, related to the structure of its Borel groups, under which $K$ and $L$ have isomorphic adele rings. Under these conditions, if $K$ or $L$ is a Galois extension of $\mathbf{Q}$ and $G(\mathbf{A}_{K,f})$ and $G(\mathbf{A}_{L,f})$ are isomorphic, then $K$ and $L$ are isomorphic as fields. We use this result to show that if for two number fields $K$ and $L$ that are Galois over $\mathbf{Q}$, the finite Hecke algebras for $\mathrm{GL}(n)$ (for fixed $n > 1$) are isomorphic by an isometry for the $L^1$-norm, then the fields $K$ and $L$ are isomorphic. This can be viewed as an analogue in the theory of automorphic representations of the theorem of Neukirch that the absolute Galois group of a number field determines the field if it is Galois over $\mathbf{Q}$.Comment: 19 pages - completely rewritte

Elementary approach to homogeneous C*-algebras

A C*-algebra is n-homogeneous (where n is finite) if every its nonzero irreducible representation acts on an n-dimensional Hilbert space. An elementary proof of Fell's characterization of n-homogeneous C*-algebras (by means of their spectra) is presented. A spectral theorem and a functional calculus for finite systems of elements which generate n-homogeneous C*-algebras are proposed.Comment: 22 page