We prove that, under rather general conditions, the 1-cohomology of a von Neumann algebra $M$ with values in a Banach $M$ -bimodule satisfying a combination of smoothness and operatorial conditions vanishes. For instance, we show that, if $M$ acts normally on a Hilbert space ${\mathcal{H}}$ and ${\mathcal{B}}_{0}\subset {\mathcal{B}}({\mathcal{H}})$ is a norm closed $M$ -bimodule such that any $T\in {\mathcal{B}}_{0}$ is smooth (i.e., the left and right multiplications of $T$ by $x\in M$ are continuous from the unit ball of $M$ with the $s^{\ast }$ -topology to ${\mathcal{B}}_{0}$ with its norm), then any derivation of $M$ into ${\mathcal{B}}_{0}$ is inner. The compact operators are smooth over any $M\subset {\mathcal{B}}({\mathcal{H}})$ , but there is a large variety of non-compact smooth elements as well.

Let $P$ be a finitely generated cancellative abelian monoid. A $P$-graph ${\rm\Lambda}$ is a natural generalization of a $k$-graph. A pullback of ${\rm\Lambda}$ is constructed by pulling it back over a given monoid morphism to $P$, while a pushout of ${\rm\Lambda}$ is obtained by modding out its periodicity, which is deduced from a natural equivalence relation on ${\rm\Lambda}$. One of our main results in this paper shows that, for some $k$-graphs ${\rm\Lambda}$, ${\rm\Lambda}$ is isomorphic to the pullback of its pushout via a natural quotient map, and that its graph $\text{C}^{\ast }$-algebra can be embedded into the tensor product of the graph $\text{C}^{\ast }$-algebra of its pushout and $\text{C}^{\ast }(\text{Per}\,{\rm\Lambda})$. As a consequence, in this case, the cycline algebra generated by the standard generators corresponding to equivalent pairs is a maximal abelian subalgebra, and there is a faithful conditional expectation from the graph $\text{C}^{\ast }$-algebra onto it.

We define the quotient and complete NUOS-quotient map (NUOS stands for nonunital operator system) in the category of nonunital operator systems. We prove that the greatest reduced tensor product max0 is projective in some sense. Moreover, we define a pseudo unit in a nonunital operator system and give some necessary and sufficient conditions under which a nonunital operator system has an operator system structure.

We prove that every incidence graph of a finite projective plane allows a partitioning into incident point-line pairs. This is used to determine the order of the identity in the K0-group of so-called polygonal algebras associated with cocompact group actions on Ã2-buildings with three orbits. These C*-algebras are classified by the K0-group and the class of the identity in K0. To be more precise, we show that 2(q − 1) = 0, where q is the order of the links of the building. Furthermore, if q = 22l−1 with l ∈ ℤ, then the order of is q − 1.

An algebra A is said to be directly finite if each left-invertible element in the (conditional) unitization of A is right invertible. We show that the reduced group C*-algebra of a unimodular group is directly finite, extending known results for the discrete case. We also investigate the corresponding problem for algebras of p-pseudofunctions, showing that these algebras are directly finite if G is amenable and unimodular, or unimodular with the Kunze–Stein property. An exposition is also given of how existing results from the literature imply that L1(G) is not directly finite when G is the affine group of either the real or complex line.

It is well known that a finitely generated group ${\rm\Gamma}$ has Kazhdan’s property (T) if and only if the Laplacian element ${\rm\Delta}$ in $\mathbb{R}[{\rm\Gamma}]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in $\mathbb{R}[{\rm\Gamma}]$. Namely, ${\rm\Gamma}$ has property (T) if and only if there exist a constant ${\it\kappa}>0$ and a finite sequence ${\it\xi}_{1},\ldots ,{\it\xi}_{n}$ in $\mathbb{R}[{\rm\Gamma}]$ such that ${\rm\Delta}^{2}-{\it\kappa}{\rm\Delta}=\sum _{i}{\it\xi}_{i}^{\ast }{\it\xi}_{i}$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in $\mathbb{R}[{\rm\Gamma}]$, possibly with the assistance of computers.

We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.

We study the metric entropy of the metric space ${\mathcal{B}}_{n}$ of all $n$-dimensional Banach spaces (the so-called Banach–Mazur compactum) equipped with the Banach–Mazur (multiplicative) “distance” $d$. We are interested either in estimates independent of the dimension or in asymptotic estimates when the dimension tends to $\infty$. For instance, we prove that, if $N({\mathcal{B}}_{n},d,1+{\it\varepsilon})$ is the smallest number of “balls” of “radius” $1+{\it\varepsilon}$ that cover ${\mathcal{B}}_{n}$, then for any ${\it\varepsilon}>0$ we have

$$\begin{eqnarray}0<\liminf _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})\leqslant \limsup _{n\rightarrow \infty }n^{-1}\log \log N({\mathcal{B}}_{n},d,1+{\it\varepsilon})<\infty .\end{eqnarray}$$

$n$

$d_{N}$

$N\times N$

$N$

$N$

$N=1$

$n$

$N\times N$

It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear) $\mathrm{C}^*$ -algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in $\mathrm{C}^*$ -algebras and show that our method cannot produce a separable counterexample.

We show Exel’s tight representation of an inverse semigroup can be described in terms of joins and covers in the natural partial order. Using this, we show that the ${C}^{\ast } $-algebra of a finitely aligned category of paths, developed by Spielberg, is the tight ${C}^{\ast } $-algebra of a natural inverse semigroup. This includes as a special case finitely aligned higher-rank graphs: that is, for such a higher-rank graph $\Lambda $, the tight ${C}^{\ast } $-algebra of the inverse semigroup associated to $\Lambda $ is the same as the ${C}^{\ast } $-algebra of $\Lambda $.

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

Suppose that ${\Gamma }^{+ } $ is the positive cone of a totally ordered abelian group $\Gamma $, and $(A, {\Gamma }^{+ } , \alpha )$ is a system consisting of a ${C}^{\ast } $-algebra $A$, an action $\alpha $ of ${\Gamma }^{+ } $ by extendible endomorphisms of $A$. We prove that the partial-isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ is a full corner in the subalgebra of $\L ({\ell }^{2} ({\Gamma }^{+ } , A))$, and that if $\alpha $ is an action by automorphisms of $A$, then it is the isometric crossed product $({B}_{{\Gamma }^{+ } } \otimes A)\hspace{0.167em} {\mathop{\times }\nolimits }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $, which is therefore a full corner in the usual crossed product of system by a group of automorphisms. We use these realizations to identify the ideal of $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{piso} } \hspace{0.167em} {\Gamma }^{+ } $ such that the quotient is the isometric crossed product $A\hspace{0.167em} { \mathop{\times }\nolimits}_{\alpha }^{\mathrm{iso} } \hspace{0.167em} {\Gamma }^{+ } $.

The class of $\lambda $-synchronizing subshifts generalizes the class of irreducible sofic shifts. A $\lambda $-synchronizing subshift can be presented by a certain $\lambda $-graph system, called the $\lambda $-synchronizing $\lambda $-graph system. The $\lambda $-synchronizing $\lambda $-graph system of a $\lambda $-synchronizing subshift can be regarded as an analogue of the Fischer cover of an irreducible sofic shift. We will study algebraic structure of the ${C}^{\ast } $-algebra associated with a $\lambda $-synchronizing $\lambda $-graph system and prove that the stable isomorphism class of the ${C}^{\ast } $-algebra with its Cartan subalgebra is invariant under flow equivalence of $\lambda $-synchronizing subshifts.

We define the class of weakly approximately divisible unital ${C}^{\ast } $-algebras and show that this class is closed under direct sums, direct limits, any tensor product with any ${C}^{\ast } $-algebra, and quotients. A nuclear ${C}^{\ast } $-algebra is weakly approximately divisible if and only if it has no finite-dimensional representations. We also show that Pisier’s similarity degree of a weakly approximately divisible ${C}^{\ast } $-algebra is at most five.

In the third and latest paper in this series, we recover the imprimitivity theorems of Mansfield and Fell using our technique of Fell bundles over groupoids. Also, we apply the Rieffel surjection of the first paper in the series to relate our version of Mansfield’s theorem to that of an Huef and Raeburn, and to give an automatic amenability result for certain transformation Fell bundles.

For a countable discrete space $V$, every nondegenerate separable ${C}^{\ast } $-correspondence over ${c}_{0} (V)$ is isomorphic to one coming from a directed graph with vertex set $V$. In this paper we demonstrate why the analogous characterizations fail to hold for higher-rank graphs (where one considers product systems of ${C}^{\ast } $-correspondences) and for topological graphs (where $V$ is locally compact Hausdorff), and we discuss the obstructions that arise.

Results of Fowler and Sims show that every k-graph is completely determined by its k-coloured skeleton and collection of commuting squares. Here we give an explicit description of the k-graph associated with a given skeleton and collection of squares and show that two k-graphs are isomorphic if and only if there is an isomorphism of their skeletons which preserves commuting squares. We use this to prove directly that each k-graph Λ is isomorphic to the quotient of the path category of its skeleton by the equivalence relation determined by the commuting squares, and show that this extends to a homeomorphism of infinite-path spaces when the k-graph is row finite with no sources. We conclude with a short direct proof of the characterization, originally due to Robertson and Sims, of simplicity of the C*-algebra of a row-finite k-graph with no sources.

Paterson showed how to construct an étale groupoid from an inverse semigroup using ideas from functional analysis. This construction was later simplified by Lenz. We show that Lenz’s construction can itself be further simplified by using filters: the topological groupoid associated with an inverse semigroup is precisely a groupoid of filters. In addition, idempotent filters are closed inverse subsemigroups and so determine transitive representations by means of partial bijections. This connection between filters and representations by partial bijections is exploited to show how linear representations of inverse semigroups can be constructed from the groups occurring in the associated topological groupoid.

Let A be a unital C*-algebra with the canonical (H) C*-bundle over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.