Interior and closure operators on bounded residuated lattice ordered monoids

summary:$GMV$-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior $GMV$-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on $DRl$-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on $GMV$-algebras

Noncommutative Riemann integration and and Novikov-Shubin invariants for open manifolds

Given a C*-algebra A with a semicontinuous semifinite trace tau acting on the Hilbert space H, we define the family R of bounded Riemann measurable elements w.r.t. tau as a suitable closure, a la Dedekind, of A, in analogy with one of the classical characterizations of Riemann measurable functions, and show that R is a C*-algebra, and tau extends to a semicontinuous semifinite trace on R. Then, unbounded Riemann measurable operators are defined as the closed operators on H which are affiliated to A'' and can be approximated in measure by operators in R, in analogy with unbounded Riemann integration. Unbounded Riemann measurable operators form a tau-a.e. bimodule on R, denoted by R^, and such bimodule contains the functional calculi of selfadjoint elements of R under unbounded Riemann measurable functions. Besides, tau extends to a bimodule trace on R^. Type II_1 singular traces for C*-algebras can be defined on the bimodule of unbounded Riemann-measurable operators. Noncommutative Riemann integration, and singular traces for C*-algebras, are then used to define Novikov-Shubin numbers for amenable open manifolds, show their invariance under quasi-isometries, and prove that they are (noncommutative) asymptotic dimensions.Comment: 34 pages, LaTeX, a new section has been added, concerning an application to Novikov-Shubin invariants, the title changed accordingl

Elementary operators and their lengths

Elementary operators on an algebra, which are finite sums of operators $x \mapsto axb$, provide a way to study properties of the algebra. In particular, for C*-algberas we consider results that are related to the length $\ell$ of the operator, defined as the minimal number of summands required. We will review some results concerning complete positivity or complete boundedness. Although all elementary operators on a C*-algebra $A$ are completely bounded, that is induce uniformly bounded operators on the algebras $M_n(A)$, the supremum is always attained for $n =\ell$, or for smaller $n$ in case $A$ has special structure. For positivity, there are also results couched in analagous terms, but with different bounds. In recent work with I.~Gogi\\u27c, we have shown that for prime C*-algebras $A$ the elementary operators of length (at most) $1$ are norm closed, but that for the rather tractable class of homogeneous C*-algebras more subtle considerations are required for closure. For instance $A = C_0(X, M_n)$ fails to have this closure property if $X$ is an open set in $\mathbb{R}^d$ with $d \geq 3$, $n \geq 2$ ($X \neq \emptyset$)

Stable isomorphism and strong Morita equivalence of operator algebras

We introduce a Morita type equivalence: two operator algebras $A$ and $B$ are called strongly $\Delta$-equivalent if they have completely isometric representations $\alpha$ and $\beta$ respectively and there exists a ternary ring of operators $M$ such that $\alpha (A)$ (resp. $\beta (B)$) is equal to the norm closure of the linear span of the set $M^*\beta (B)M,$ (resp. $M\alpha (A)M^*$). We study the properties of this equivalence. We prove that if two operator algebras $A$ and $B,$ possessing countable approximate identities, are strongly $\Delta$-equivalent, then the operator algebras A\otimes \cl K and B\otimes \cl K are isomorphic. Here \cl K is the set of compact operators on an infinite dimensional separable Hilbert space and $\otimes$ is the spatial tensor product. Conversely, if A\otimes \cl K and B\otimes \cl K are isomorphic and $A, B$ possess contractive approximate identities then $A$ and $B$ are strongly $\Delta$-equivalent.Comment: We present some shorter proofs using references from the literature. Also example 3.7 is new and provides a new proof of the fact our notion of strong Morita equivalence is stronger than "BMP strong Morita equivalence " introduced by Blecher, Muhly and Paulse

Belief revision in the propositional closure of a qualitative algebra

Belief revision is an operation that aims at modifying old be-liefs so that they become consistent with new ones. The issue of belief revision has been studied in various formalisms, in particular, in qualitative algebras (QAs) in which the result is a disjunction of belief bases that is not necessarily repre-sentable in a QA. This motivates the study of belief revision in formalisms extending QAs, namely, their propositional clo-sures: in such a closure, the result of belief revision belongs to the formalism. Moreover, this makes it possible to define a contraction operator thanks to the Harper identity. Belief revision in the propositional closure of QAs is studied, an al-gorithm for a family of revision operators is designed, and an open-source implementation is made freely available on the web