SK1 of graded division algebras

The reduced Whitehead group $\SK$ of a graded division algebra graded by a torsion-free abelian group is studied. It is observed that the computations here are much more straightforward than in the non-graded setting. Bridges to the ungraded case are then established by the following two theorems: It is proved that $\SK$ of a tame valued division algebra over a henselian field coincides with $\SK$ of its associated graded division algebra. Furthermore, it is shown that $\SK$ of a graded division algebra is isomorphic to $\SK$ of its quotient division algebra. The first theorem gives the established formulas for the reduced Whitehead group of certain valued division algebras in a unified manner, whereas the latter theorem covers the stability of reduced Whitehead groups, and also describes $\SK$ for generic abelian crossed products.

Unitary SK1 of unitary and graded division algebras

The reduced unitary Whitehead group $\SK$ of a graded division algebra equipped with a unitary involution (i.e., an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their nongraded counterparts. The bridge to the non-graded case is established by proving that the unitary $\SK$ of a tame valued division algebra wih a unitary involution over a henselian field coincides with the unitary $\SK$ of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary $\SK$ for much wider classes of division algebras over henselian fields.

Normalisers, nest algebras and tensor products

We show that if $\cl A$ is the tensor product of finitely many continuous nest algebras, $\cl B$ is a CDCSL algebra and $\cl A$ and $\cl B$ have the same normaliser semi-group then either $\cl A = \cl B$ or $\cl A^* = \cl B$.

Convergence of bimodules over maximal abelian selfadjoint algebras

We prove a continuity result for the map sending a masa-bimodule to its support. We characterise the convergence of a net of weakly closed convex hulls of bilattices in terms of the convergence of the corresponding supports, and establish a lower-semicontinuity result for the map sending a support to the corresponding masa-bimodule.

Angles in C*-algebras

In this work we characterise the C*-algebras $\mathcal{A}$ generated by projections with the property that every pair of projections in $\mathcal{A}$ has positive angle, as certain extensions of abelian algebras by algebras of compact operators. We show that this property is equivalent to a lattice theoretic property of projections and also to the property that the set of finite dimensional *-subalgebras of $\mathcal{A}$ is directed.

Multipliers of multidimensional Fourier algebras

Let $G$ be a locally compact $\sigma$-compact group. Motivated by an earlier notion for discrete groups due to Effros and Ruan, we introduce the multidimensional Fourier algebra $A^n(G)$ of $G$. We characterise the completely bounded multidimensional multipliers associated with $A^n(G)$ in several equivalent ways. In particular, we establish a completely isometric embedding of the space of all $n$-dimensional completely bounded multipliers into the space of all Schur multipliers on $G^{n+1}$ with respect to the (left) Haar measure. We show that in the case $G$ is amenable the space of completely bounded multidimensional multipliers coincides with the multidimensional Fourier-Stieltjes algebra of $G$ introduced by Ylinen. We extend some well-known results for abelian groups to the multidimensional setting.

On the simplicial cohomology of Banach operator algebras

We investigate the simplicial cohomology of certain Banach operator algebras. The two main examples considered are the Banach algebra of all bounded operators on a Banach space and its ideal of approximable operators. Sufficient conditions will be given forcing Banach algebras of this kind to be simplicially trivial.

Spectral isometries on non-simple C*-algebras

We prove that unital surjective spectral isometries on certain non-simple unital C*-algebras are Jordan isomorphisms. Along the way, we establish several general facts in the setting of semisimple Banach algebras.

C∗-Segal algebras with order unit

We introduce the notion of a (noncommutative) C *-Segal algebra as a Banach algebra (A, {norm of matrix}{dot operator}{norm of matrix} A) which is a dense ideal in a C *-algebra (C, {norm of matrix}{dot operator}{norm of matrix} C), where {norm of matrix}{dot operator}{norm of matrix} A is strictly stronger than {norm of matrix}{dot operator}{norm of matrix} C onA. Several basic properties are investigated and, with the aid of the theory of multiplier modules, the structure of C *-Segal algebras with order unit is determined.

Semiring induced valuation algebras: Exact and approximate local computation algorithms

Local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure, called valuation algebra.There are many models of this algebraic structure ranging from probability theory to numerical analysis, relational databases and various classical and non-classical logics. It turns out that many interesting models of valuation algebras may be derived from semiring valued mappings. In this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring. In particular, c-semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation. Also important are semirings whose multiplicative semigroup is embedded in a union of groups. They induce valuation algebras with a partially defined division. For these valuation algebras, the well-known architectures for Bayesian networks apply. We also extend the general computational framework to allow derivation of bounds and approximations, for when exact computation is not feasible.

Chaotic Banach algebras

We construct an infinite dimensional non-unital Banach algebra $A$ and $a\in A$ such that the sets $\{za^n:z\in\C,\ n\in\N\}$ and $\{({\bf 1}+a)^na:n\in\N\}$ are both dense in $A$, where $\bf 1$ is the unity in the unitalization $A^{\#}=A\oplus \spann\{{\bf 1}\}$ of $A$. As a byproduct, we get a hypercyclic operator $T$ on a Banach space such that $T\oplus T$ is non-cyclic and $\sigma(T)=\{1\}$.

Minimal Hilbert series for quadratic algebras and the Anick conjecture

We study the question on whether the famous Golod–Shafarevich estimate, which gives a lower bound for the Hilbert series of a (noncommutative) algebra, is attained. This question was considered by Anick in his 1983 paper ‘Generic algebras and CW-complexes’, Princeton Univ. Press, where he proved that the estimate is attained for the number of quadratic relations $d\leq n^2/4$
and $d\geq n^2/2$, and conjectured that it is the case for any number of quadratic relations. The particular point where the number of relations is equal to $n(n-1)/2$ was addressed by Vershik. He conjectured that a generic algebra with this number of relations is finite dimensional. We announce here the result that over any infinite field, the Anick conjecture holds for $d \geq 4(n2+n)/9$ and an arbitrary number of generators. We also discuss the result that confirms the Vershik conjecture over any field of characteristic 0, and a series of related
asymptotic results.