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 $V_i^\ast V_j=\overline{z}_{ij} V_j V_i^\ast$ 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.Analysis and Stochastic

Dynamical systems and commutants in crossed products

Article / Letter to editorMathematisch Instituu

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

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.Analysis and Stochastic

The closure of ideals of ℓ 1 (Σ) in its enveloping C*-algebra

If X X is a compact Hausdorff space and σ σ is a homeomorphism of X X , then an involutive Banach algebra ℓ 1 (Σ) ℓ1(Σ) of crossed product type is naturally associated with the topological dynamical system Σ=(X,σ) Σ=(X,σ) . We initiate the study of the relation between two-sided ideals of ℓ 1 (Σ) ℓ1(Σ) and  C ∗ (Σ) C∗(Σ) , the enveloping C ∗  C∗ -algebra C(X)⋊ σ Z C(X)⋊σZ of ℓ 1 (Σ) ℓ1(Σ) .  Among others, we prove that the closure of a proper two-sided ideal of ℓ 1 (Σ) ℓ1(Σ) in  C ∗ (Σ) C∗(Σ) is again a proper two-sided ideal of C ∗ (Σ) C∗(Σ) .Analysis and Stochastic

Compact groups of positive operators on Banach lattices

In this paper, we study groups of positive operators on Banach lattices. If a certain factorization property holds for the elements of such a group, the group has a homomorphic image in the isometric positive operators which has the same invariant ideals as the original group. If the group is compact in the strong operator topology, it equals a group of isometric positive operators conjugated by a single central lattice automorphism, provided an additional technical assumption is satisfied, for which we have only examples. We obtain a characterization of positive representations of a group with compact image in the strong operator topology, and use this for normalized symmetric Banach sequence spaces to prove an ordered version of the decomposition theorem for unitary representations of compact groups. Applications concerning spaces of continuous functions are also considered.Analysis and Stochastic

Simultaneous power factorization in modules over Banach algebras

Let A be a Banach algebra with a bounded left approximate identity {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ, let π" role="presentation">π be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that limλπ(eλ)s=s" role="presentation">limλπ(eλ)s=s uniformly on S. If S is bounded, or if {eλ}λ∈Λ" role="presentation">{eλ}λ∈Λ is commutative, then we show that there exist a∈A" role="presentation">a∈A and maps xn:S→X" role="presentation">xn:S→X for n≥1" role="presentation">n≥1 such that s=π(an)xn(s)" role="presentation">s=π(an)xn(s) for all n≥1" role="presentation">n≥1 and s∈S" role="presentation">s∈S. The properties of a∈A" role="presentation">a∈A and the maps xn" role="presentation">xn, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of C0(R)" role="presentation">C0(R) and unbounded S is included.Analysis and Stochastic

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.Analysis and Stochastic

An uncertainty principle for integral operators

The classical uncertainty principle for the Fourier transform has been extended to the spherical transform for Gelfand pairs by Wolf. We sharpen the principle and extend its validity to the context of integral operators with a bounded kernel for which there is a Plancherel theorem.Analysis and Stochastic