For a commutative ring k, the homotopy category of commutative Hk-algebras (strictly unital E∞ ring spectra under the Eilenberg-Mac Lane spectrum Hk) is equivalent to the homotopy category of E∞ differential graded k-algebras. The functor from topology to algebra is a CW approximation and cellular chain functor; the inverse equivalence is constructed by Brown’s representability theorem.

Abstract We provide an abstract characterization for the Cuntz semigroup of unital commutative AI-algebras, as well a semigroups form $\operatorname {\mathrm {Lsc}} (X,\overline {\mathbb {N}})$ some $T_1$ -space X . In our investigations, we also uncover new properties that all AI-algebras satisfies.

We give a full description of the Poisson structures on finitary incidence algebra $FI(P,R)$ an arbitrary poset $P$ over commutative unital ring $R$.

When replacing the non-negative real numbers with their addition by a commutative quantale V, under a metric lens one may then view small V-categories as sets that come with a V-valued distance function. The ensuing category V-Cat is well known to be a concrete topological category that is symmetric monoidal closed. In this paper we show which concrete symmetric monoidalclosed topological categ...

We show that if T is an isometry (as metric spaces) from an open subgroup of the group of the invertible elements in a unital semisimple commutative Banach algebra onto an open subgroup of the group of the invertible elements in a unital Banach algebra, then T (1)T is an isometrical group isomorphism. In particular, T (1)T is extended to an isometrical real algebra isomorphism from A onto B.

The structure of arbitrary associative commutative unital artinian algebras is well-known: they are finite products of associative commutative unital local algebras [6, pg.351, Cor. 23.12]. In the semi-simple case, we have the Artin-Wedderburn Theorem which states that any semi-simple artinian algebra (which is assumed to be associative and unital but not necessarily commutative) is a direct pr...

let  be a banach algebra and  a derivation. in this paper, it is proved, under certain conditions, that , where  is the jacobson radical of . moreover, we prove that if  is unital and  is a continuous derivation, then , where  denotes the set of all primitive ideals such that  is commutative,  denotes the set of all maximal (modular) ideals such that  is commutative, and  is the set of all non-...

For a commutative associate unital ring R the notion of an Rconvolution type is introduced and applied to R-algebras. Some examples are presented and relations to known concepts are discussed. Mathematics Subject Classification (2010): 16S99, 13F99, 13F20, 13F25

We show that the algebra A of a commutative unital spectral triple (A,H,D) satisfying several additional conditions, slightly stronger than those proposed by Connes, is the algebra of smooth functions on a compact spin manifold.