On triangular algebras with noncommutative diagonals

We construct a triangular algebra whose diagonals form a noncommutative algebra and its lattice of invariant projections contains only two nontrivial projections. Moreover we prove that our triangular algebra is maximal.     

On a category of cluster algebras

We introduce a category of cluster algebras with fixed initial seeds. This category has countable coproducts, which can be constructed combinatorially, but no products. We characterise isomorphisms and monomorphisms in this category and provide combinatorial methods for constructing special classes of monomorphisms and epimorphisms. In the case of cluster algebras from surfaces, we describe interactions between this category and the geometry of the surfaces.     

Left-right noncommutative Poisson algebras

The notions of left-right noncommutative Poisson algebra (NP^lr-algebra) and left-right algebra with bracket AWB^lr are introduced. These algebras are special cases of NLP-algebras and algebras with bracket AWB, respectively, studied earlier. An NP^lr-algebra is a noncommutative analogue of the classical Poisson algebra. Properties of these new algebras are studied. In the categories AWB^lr and NP^lr-algebras the notions of actions, representations, centers, actors and crossed modules are described as special cases of the corresponding wellknown notions in categories of groups with operations. The cohomologies of NP^lr-algebras and AWB^lr (resp. of NP^r-algebras and AWB^r) are defined and the relations between them and the Hochschild, Quillen and Leibniz cohomologies are detected. The cases P is a free AWB^r, the Hochschild or/and Leibniz cohomological dimension of P is ≤ n are considered separately, exhibiting interesting possibilities of representations of the new cohomologies by the well-known ones and relations between the corresponding cohomological dimensions.     

Derivations of noncommutative Arens algebras

The present paper deals with derivations of noncommutative Arens algebras. We prove that every derivation of an Arens algebra associated with a von Neumann algebra and a faithful normal finite trace is inner. In particular, each derivation on such algebras is automatically continuous in the natural topology, and in the commutative case, even for semi-finite traces, all derivations are identically zero. At the same time, the existence of noninner derivations is proved for noncommutative Arens algebras with a semi-finite but nonfinite trace.     

On the category of hyper MV‐algebras

In this paper we study the category of hyper MV‐algebras and we prove that it has a terminal object and a coequalizer. We show that Jia's construction can be modified to provide a free hyper MV‐algebra by a set. We use this to show that in the category of hyper MV‐algebras the monomorphisms are exactly the one‐to‐one homomorphisms. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Derivations of noncommutative Banach algebras II

Supported in part by a grant from the Ministry of Science of Slovenia.     

On the existence of approximate identities in ideals of group algebras

A short proof is given of the theorem by Wik, who, in 1965, showed that the nowhere-dense strong Ditkin subsets ofT, the circle group, are finite.     

Twisted split category algebras as quasi-hereditary algebras

We show that if C is a finite split category, k is a field of characteristic 0, and α is a 2-cocycle of C with values in k  ^× , then the twisted category algebra k _α C is quasi-hereditary.     

On existence of finite universal Korovkin sets in Segal algebras

We prove that there exists a finite universal Korovkin set w.r.t positive operators for the centre of a Segal algebra on a compact groupG if and only ifG is metrizable. As a consequence it follows that a Segal algebra on a compact abelian group admits a finite universal Korovkin set w.r.t. positive operators iff the group is metrizable.     

Hadamard algebras possessing the unique noncommutative simple component

The famous conjecture on the orders of Hadamard matrices can be reformulated as follows: a commutative algebra is Hadamard if and only if its dimension is divisible by 4. In this paper we study the Hadamard algebras closed to commutative ones, namely, the algebras possessing the unique noncommutative simple component being the matrix algebra of order 2.