Let A, B be two regular commutative unital Banach algebras such that B is integral over A. In 2003, Dawson and Feinstein showed that the topological stable rank tsr(B) = 1 whenever tsr(A) = 1. In this note, we investigate whether we will have tsr(A) = tsr(B) in general. For instance, when A is a commutative unital C∗-algebra, we show that tsr(A) ≤ tsr(B), and the equality holds at least when th...

Here, it is observe that ΛBV (p)(σ,B), the class of functions of bounded p−Λ− variation from a non-empty compact subset σ of R into a commutative unital Banach algebra B, is a commutative unital Banach algebra. Moreover, (Λ1, ...,ΛN )∗BV (Πi=1σi,B), the class of N−variables functions of bounded p−(Λ1, ...,ΛN )∗−variation from Πi=1σi into B, is a Banach space, where σi are non-empty compact subs...

Abstract. For any finite unital commutative idempotent semigroup S, a unital semilattice, we show how to compute the amenability constant of its semigroup algebra l(S), which is always of the form 4n+1. We then show that these give lower bounds to amenability constants of certain Banach algebras graded over semilattices. Our theory applies to certain natural subalgebras of Fourier-Stieltjes alg...

Let A be a unital amenable A∗-algebra, or a unital commutative hermitian Banach ∗-algebra. We show that if ρ : A → B(H) is a bounded homomorphism, then ρ is completely bounded and ‖ρ‖cb ≤ ‖ρ‖2. Mathematics Subject Classification: 46L07; 46k05

1.1. Definitions. 1) A (non-commutative) probability space consists of a pair (A, φ), where • A is a unital algebra • φ : A → C is a unital linear functional, i.e. in particular φ(1) = 1 2) Unital subalgebras A1, . . . ,An ⊂ A are called free, if we have φ(a1 . . . ak) = 0 whenever ai ∈ Aj(i) (i = 1, . . . , k) j(1) 6= j(2) 6= · · · 6 = j(k) φ(ai) = 0 (i = 1, . . . , k) 3) Random variables x1, ...

Recent investigations of lax algebras—in generalization of Barr’s relational algebras—make an essential use of lax extensions of monad functors on Set to the category Rel(V) of sets and V-relations (where V is a unital quantale). For a given monad there may be many such lax extensions, and different constructions appear in the literature. The aim of this article is to shed a unifying light on t...

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of étale groupoid is subsumed in a natural way by that of quantale. In particular, to each étale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic étale groupoids and their quantales, whic...

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define may not have unit. We develop the elementary theory of these structures and show, in particular, how to nuclei whose quotients quantales. This yields phase semantics representation theorem via Important examples arise from Raney’s notion tight Galois connection: endomap...

For two algebras $A$ and $B$, a linear map $T:A longrightarrow B$ is called separating, if $xcdot y=0$ implies $Txcdot Ty=0$ for all $x,yin A$. The general form and the automatic continuity of separating maps between various Banach algebras have been studied extensively. In this paper, we first extend the notion of separating map for module case and then we give a description of a linear se...

Each divisible and unital quantale is associated with two quantaloids. Categories enriched over these two quantaloids can be regarded respectively as crisp sets endowed with fuzzy preorders and fuzzy sets endowed with fuzzy preorders. This paper deals with the relationship between these two kinds of enriched categories. This is a special case of the change-base issue in enriched category theory...