A remark on the rank conjecture

We prove a result about the action of -operations on the homology of linear groups. We use this to give a sharper formulation of the rank conjecture as well as some shorter proofs of various known results. We formulate a conjecture about how the sharper formulation of the rank conjecture together with another conjecture could give rise to a different point of view on the isomorphism between and K_n^{(p)} (F)$ for an infinite field F, and we prove part of this new conjecture

Amenable crossed product Banach algebras associated with a class of C∗\mathrm{C}^\ast-dynamical systems

We prove that the crossed product Banach algebra $\ell^1(A,G,\alpha)$ that is associated with a $\mathrm{C}^\ast$-dynamical system $(A,G,\alpha)$ is amenable if $G$ is a discrete amenable group and $A$ is a commutative or finite dimensional $\mathrm{C}^\ast$-algebra. Perspectives for further developments are indicated.Comment: Improved discussion of the relation with the existing literature and of perspectives for further developments. 8 pages. To appear in Integral Equations and Operator Theor

The structure of doubly non-commuting isometries

Suppose that $n\geq 1$ and that, for all $i$ and $j$ with $1\leq i,j\leq n$ and $i\neq j$, $z_{ij}\in{\mathbb T}$ are given such that $z_{ji}=\overline{z}_{ij}$ for all $i\neq j$. If $V_1,\dotsc, V_n$ are isometries on a Hilbert space such that $V_i^\ast V_j^{\phantom{\ast}}\!=\overline{z}_{ij} V_j^{\phantom{\ast}}\!V_i^\ast$ for all $i\neq j$, then $(V_1,\dotsc,V_n)$ is called an $n$-tuple of doubly non-commuting isometries. The generators of non-commutative tori are well-known examples. In this paper, we establish a simultaneous Wold decomposition for $(V_1,\dotsc,V_n)$. This decomposition enables us to classify such $n$-tuples up to unitary equivalence. We show that the joint listing of a unitary equivalence class of a representation of each of the $2^n$ non-commutative tori that are naturally associated with the structure constants is a classifying invariant. A dilation theorem is also established, showing that an $n$-tuple of doubly non-commuting isometries can be extended to an $n$-tuple of doubly non-commuting unitary operators on an enveloping Hilbert space.Comment: A remark on the relation between the dilation theorem in this paper and other dilation theorems for multiple operators in the literature has been added. Otherwise, there are only a few minor editorial changes compared to the first version; some typos have also been corrected. Final version, to appear in Advances in Mathematic

Nonsimplicity of certain universal C^*-algebras

Given n≥2" role="presentation">n≥2, zij∈T" role="presentation">zij∈T such that zij=z¯ji" role="presentation">zij=z¯ji for 1≤i,j≤n" role="presentation">1≤i,j≤n and zii=1" role="presentation">zii=1 for 1≤i≤n" role="presentation">1≤i≤n, and integers p1,…,pn≥1" role="presentation">p1,…,pn≥1, we show that the universal C*" role="presentation">C∗-algebra generated by unitaries u1,…,un" role="presentation">u1,…,un such that uipiujpj=zijujpjuipi" role="presentation">upiiupjj=zijupjjupii for 1≤i,j≤n" role="presentation">1≤i,j≤n is not simple if at least one exponent pi" role="presentation">pi is at least two. We indicate how the method of proof by “working with various quotients” can be used to establish nonsimplicity of universal C*" role="presentation">C∗-algebras in other cases.Article / Letter to editorMathematisch Instituu

Introduction to the issue on heterogeneous data access and use for geospatial user communities - part II

The four papers in this second part of this special issue focus on heterogeneous data access and use for geospatial user communities. The first two papers relate to satellite remote sensing data and the second two are from the hydro-meteorological domain

Phase diagram and critical properties in the Polyakov--Nambu--Jona-Lasinio model

We investigate the phase diagram of the so-called Polyakov--Nambu--Jona-Lasinio model at finite temperature and nonzero chemical potential with three quark flavours. Chiral and deconfinement phase transitions are discussed, and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic expectations and lattice data. A special attention is payed to the critical end point: as the strength of the flavour-mixing interaction becomes weaker, the critical end point moves to low temperatures and can even disappear.Comment: Talk given at the 9th International Conference on Quark Confinement and the Hadron Spectrum - QCHS IX, Madrid, Spain, 30 August - September 201