Given a unital and commutative algebra A associated to a spectral triple, we show how a differentiable structure is constructed on the spectrum of such an algebra whenever the spectral triple satisfies eight so-called “axioms”, in such a way that A ∼= C∞(M). This construction is the celebrated “reconstruction theorem” of Alain Connes [14], [21]. We discuss two spin manifolds, the circle and the...

Let A be a finite dimensional, unital, and associative algebra which is endowed with a non-degenerate and invariant inner product. We give an explicit description of an action of cyclic Sullivan chord diagrams on the normalized Hochschild cochain complex of A. As a corollary, the Hochschild cohomology of A becomes a Frobenius algebra which is endowed with a compatible BV operator. If A is also ...

In this paper we put forward the definition of particular subsets on a unital C∗algebra, that we call isocones, and which reduce in the commutative case to the set of continuous non-decreasing functions with real values for a partial order relation defined on the spectrum of the algebra, which satisfies a compatibility condition with the topology (complete separateness). We prove that this spac...

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...

Let A be a unital, commutative and finitely generated ring. We prove that if n ≥ 4, then the group G = ELn(A) has a fixed point property for affine isometric actions on B. Here B stands for any L space or any Banach space isomorphic to a Hilbert space. We also verify that the comparison map Ψ : H b (G,B) → H(G,B) from bounded to ordinary cohomology is injective, where G and B are as in above. F...

In this paper, we prove that for a unital commutative and finitely generated ring A, the group G = ELn(A) has a fixed point property for affine isometric actions on B if n ≥ 4. Here B stands for any L space or any Banach space isomorphic to a Hilbert space. We also verify that the comparison map Ψ : H b (G,B) → H (G,B) from bounded to usual cohomology is injective, where G and B are same as in ...

Given a finite (associative, unital) ring R, let K(R) denote the set of polynomials in R[x] that send each element of R to 0 under evaluation. We study K(R) and its elements. We conjecture that K(R) is a two-sided ideal of R[x] for any finite ring R, and prove the conjecture for several classes of finite rings (including commutative rings, semisimple rings, local rings, and all finite rings of ...

(2) Let n be an ordinal number, T be an admissible connected term order of n, L be an addassociative right complementable right zeroed commutative associative well unital distributive Abelian field-like non degenerated non empty double loop structure, and f , p, g be polynomials of n, L. Suppose f reduces to g, p, T . Then there exists a monomial m of n, L such that g = f −m∗ p and HT(m∗ p,T ) ...

The aim of this short note is to study some algebraic and topological questions associated with the “Bézout equation” b1a1 + · · · + bNaN = e, where bi , ai (1 ≤ i ≤ N) are elements of the commutative unital topological ring (DA(R), +, *, TD′A(Rd)), defined below, and e denotes the identity element (which will be the locally finite sum of Dirac distributions placed at a lattice formed by the pe...

It is shown that every locally idempotent (locallym-pseudoconvex) Hausdorff algebra A with pseudoconvex vonNeumannbornology is a regular (respectively, bornological) inductive limit of metrizable locallym-(kB-convex) subalgebras AB of A. In the case where A, in addition, is sequentially BA-complete (sequentially advertibly complete), then every subalgebraAB is a locally m-(kB-convex) Fréchet al...