Mathematics

Ando, Matsuzawa, Thom, and Törnquist have resolved a problem by Sorin Popa by constructing an example of a Polish group of unitary operators with the strong operator topology, whose left and right uniform structures coincide, but which does not embed into the unitary group of a finite von Neumann algebra. The question remained whether such a group can be connected. Here we observe that a connected (in fact, homeomorphic to the Hilbert space) example is obtained from the example of the above authors via the Hartman--Mycielski construction.

The first author and Latrémolière had introduced a quantum metric (in the sense of Rieffel) on the algebra of complex-valued continuous functions on the Cantor space. We show that this quantum metric is distinct from the quantum metric induced by a classical metric on the Cantor space. We accomplish this by showing that the seminorms induced by each quantum metric (Lip-norms) are distinct on a dense subalgebra of the algebra of complex-valued continuous functions on the Cantor space. In the process, we develop formulas for each Lip-norm on this dense subalgebra and show these Lip-norms agree on a Hamel basis of this subalgebra. Then, we use these formulas to find families of elements for which these Lip-norms disagree.

The Peterson-Thom conjecture asserts that any diffuse, amenable subalgebra of a free group factor is contained in a unique maximal amenable subalgebra. This conjecture is motivated by related results in Popa's deformation/rigidity theory and Peterson-Thom's results on L^{2}-Betti numbers. We present an approach to this conjecture in terms of so-called strong convergence of random matrices by formulating a conjecture which is a natural generalization of the Haagerup-Thorbjornsen theorem whose validity would imply the Peterson-Thom conjecture. This random matrix conjecture is related to recent work of Collins-Guionnet-Parraud.

We construct an example of a simple approximately homogeneous C*-algebra such that its Elliott invariant admits an automorphism which is not induced by an automorphism of the algebra.

The aim of this note is to give a simpler proof of a result of Avsec, which states that q -Gaussian algebras have the complete metric approximation property.

A Dirac operator is presented that will yield a 1+ summable regular even spectral triple for all noncommutative compact surfaces defined as subalgebras of the Toeplitz algebra. Connes' conditions for noncommutative spin geometries are analyzed and it is argued that the failure of some requirements is mainly due to a wrong choice of a noncommutative spin bundle.

For a countable group G we construct a small, idempotent complete, symmetric monoidal, stable ??-category KK G sep whose homotopy category recovers the triangulated equivariant Kasparov category of separable G - C ??-algebras, and exhibit its universal property. Likewise, we consider an associated presentably symmetric monoidal, stable ??-category KK G which receives a symmetric monoidal functor kk G from possibly non-separable G - C ??-algebras and discuss its universal property. In addition to the symmetric monoidal structures, we construct various change-of-group functors relating these KK-categories for varying G . We use this to define and establish key properties of a (spectrum valued) equivariant, locally finite K -homology theory on proper and locally compact G -topological spaces, allowing for coefficients in arbitrary G - C ??-algebras. Finally, we extend the functor kk G from G - C ??-algebras to G - C ??-categories. These constructions are key in a companion paper about a form of equivariant Paschke duality and assembly maps.

We survey the symmetry preserving properties for the dual propinquity, under natural non-degeneracy and equicontinuity conditions. These properties are best formulated using the notion of the covariant propinquity when the symmetries are encoded via the actions of proper monoids and groups. We explore the issue of convergence of Cauchy sequences for the covariant propinquity, which captures, via a compactness result, the fact that proper monoid actions can pass to the limit for the dual propinquity.

We investigate an invariant for continuous fields of the Cuntz algebra O n+1 introduced in our previous work, and find a way to obtain a continuous field of M n ( O ∞ ) from that of O n+1 using the construction of the invariant. By Brown's representability theorem, this gives a bijection from the set of the isomorphism classes of continuous fields of O n+1 to those of M n ( O ∞ ) . As a consequence, we obtain a new proof for M. Dadarlat's classification result of continuous fields of O n+1 arising from vector bundles, which corresponds to those of M n ( O ∞ ) stably isomorphic to the trivial field.

For a continuous field of the Cuntz algebra over a finite CW complex, we introduce a topological invariant, which is an element in Dadarlat-Pennig's generalized cohomology group, and prove that the invariant is trivial if and only if the field comes from a vector bundle via Pimsner's construction.