Search

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set $R$ of finitary algebraic relations yields a duality on a class of algebras $\mathcal{A} = \operatorname{\mathbb{I}\mathbb{S}\mathbb{P}}( \underline{M})$, those subsets $R'$ of $R$ which yield optimal dualities are characterised. Further, the manner in which the relations in $R$ are constructed from those in $R'$ is revealed in the important special case that $ \underline{M}$ generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on $\underline{M}$. Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties $\mathbf{B}_{n}$ of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results. [ABSTRACT FROM AUTHOR]

Let $\Lambda$ be a module-finite algebra over a commutative noetherian ring $R$ of Krull dimension 1. We determine when a collection of finitely generated modules over the localizations $\Lambda_{\mathbf{m}}$, at maximal ideals of $R$, is the family of all localizations $M_{\mathbf{m}}$ of a finitely generated $\Lambda$-module $M$. When $R$ is semilocal we also determine which finitely generated modules over the $J(R)$-adic completion of $\Lambda$ are completions of finitely generated $\Lambda$-modules. If $\Lambda$ is an $R$-order in a semisimple artinian ring, but not contained in a maximal such order, several of the basic tools of integral representation theory behave differently than in the classical situation. The theme of this paper is to develop ways of dealing with this, as in the case of localizations and completions mentioned above. In addition, we introduce a type of order called a ``splitting order'' of $\Lambda$ that can replace maximal orders in many situations in which maximal orders do not exist. [ABSTRACT FROM AUTHOR]

The space of congruence classes of full spherical minimal immersions $f:S^m\to S^n$ of a given source dimension $m$ and algebraic degree $p$ is a compact convex body $\mathcal{M}_m^p$ in a representation space $\mathcal{F}_m^p$ of the special orthogonal group $SO(m+1)$. In Ann. of Math. {\bf 93} (1971), 43--62 DoCarmo and Wallach gave a lower bound for $\mathcal{F}_m^p$ and conjectured that the estimate was sharp. Toth resolved this ``exact dimension conjecture'' positively so that all irreducible components of $\mathcal{F}_m^p$ became known. The purpose of the present paper is to characterize each irreducible component $V$ of $\mathcal{F}_m^p$ in terms of the spherical minimal immersions represented by the slice $V\cap \mathcal{M}_m^p$. Using this geometric insight, the recent examples of DeTurck and Ziller are located within $\mathcal{M}_m^p$. [ABSTRACT FROM AUTHOR]

We say that a unital $\mathrm{C}^{*}$-algebra $A$ has the approximate positive factorization property (APFP) if every element of $A$ is a norm limit of products of positive elements of $A$. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of $\mathrm{C}^{*}$-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial $K_{1}$, and stable rank 1. In the unital case, the $K_{0}$ group separates the tracial states. The APFP passes to matrix algebras, and if $I$ is an ideal in $A$ such that $I$ and $A/I$ have the APFP, then so does $A$. We also give some new examples of $\mathrm{C}^{*}$-algebras with the APFP, including type $\mathrm{II}_{1}$ factors and infinite-dimensional simple unital direct limits of homogeneous $\mathrm{C}^{*}$-algebras with slow dimension growth, real rank zero, and trivial $K_{1}$ group. Simple direct limits of homogeneous $\mathrm{C}^{*}$-algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a $\mathrm{C}^{*}$-algebra that may be of interest in the future. [ABSTRACT FROM AUTHOR]