The aim of this paper is to study the heat kernel and the jump kernel of the Dirichlet form associated to the ultrametric Cantor set $\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$ that is the infinite path space of the stationary $k$-Bratteli diagram ${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$, where $\unicode[STIX]{x1D6EC}$ is a finite strongly connected $k$-graph. The Dirichlet form which we are interested in is induced by an even spectral triple $(C_{\operatorname{Lip}}(\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}),\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}},{\mathcal{H}},D,\unicode[STIX]{x1D6E4})$ and is given by

$$\begin{eqnarray}Q_{s}(f,g)=\frac{1}{2}\int _{\unicode[STIX]{x1D6EF}}\operatorname{Tr}(|D|^{-s}[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(f)]^{\ast }[D,\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D719}}(g)])\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D719}),\end{eqnarray}$$

$\unicode[STIX]{x1D6EF}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}\times \unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$\unicode[STIX]{x1D6E5}_{s}$

$Q_{s}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$w_{\unicode[STIX]{x1D6FF}}$

${\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x1D6FF}\in (0,1)$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

$d^{(s)}$

$d_{w_{\unicode[STIX]{x1D6FF}}}$

$Q_{s}$

$Q_{s}$

${\mathcal{Q}}_{J_{s},\unicode[STIX]{x1D707}}$

$J_{s}$

$\unicode[STIX]{x1D707}$

$\unicode[STIX]{x2202}{\mathcal{B}}_{\unicode[STIX]{x1D6EC}}$

Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

Let $\mathbb{L}$ be a length function on a group $G$, and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$-algebra $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

In this paper, we review the parametrized strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of general torus bundles and their parametrized strict deformation quantization. Rieffel’s basic construction of an algebra deformation can be mimicked to deform a monoidal category, which deforms not only algebras but also modules. As a special case, we consider the parametrized strict deformation quantization of Hilbert C*-modules over C*-bundles with fibrewise torus action.

We show that the analytic rank, as defined by Murphy, of a unital graph C*-algebra is either 1 or 0, depending on whether or not the underlying graph possesses an initial loop. As a corollary, we show that the analytic rank of certain quantum spaces (including some quantum spheres, projective spaces and lens spaces) is 1.

We use the homological perturbation lemma to give an algebraic proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.

We study conjugacy invariants for completely positive maps that are inspired by the concept of curvature introduced for commuting d-tuples of contractions by Arveson.

Given a coaction $\alpha$ of a Hopf ${\rm C}^*$-algebra $A$ on a ${\rm C}^*$-algebra $B$ with an $\alpha$-invariant ${\rm C}^*$-subalgebra $C$, and a conditional expectation $E:B \rightarrow C$ commuting with $\alpha$, it is shown that if $(\pi, u)$ is a covariant representation of the system ($C, A, \alpha\mid_C$), then there is an associated covariant representation ($\tilde{\pi}, \tilde{u}$) of the system ($B, A, \alpha$), where $\tilde{\pi}$ is the representation induced from $\pi$ up to $B$ via $E$, and $\tilde{u}$ is a unitary corepresentation of $A$ naturally associated with $u$. Some applications are also discussed, including a lifting of ergodic coactions to von Neumann algebras, and a characterization of the amenability of multiplicative unitary operators via infinite tensor product covariant representations.

We examine Hilbert bimodules which possess a (generally unbounded) involution. Topics considered include a linking algebra representation, duality, locality, and the role of these bimodules in noncommutative differential geometry

We study estimates of the type

1

$${{\left\| \phi (D)\,-\,\phi ({{D}_{0}}) \right\|}_{E(\mathcal{M},\,\tau )}}\,\le \,C\,\cdot \,{{\left\| D\,-\,{{D}_{0}} \right\|}^{\alpha }},\,\,\,\,\,\,\,\alpha \,=\,\frac{1}{2},\,1$$

where $\phi (t)\,=\,t{{(1\,+\,{{t}^{2}})}^{-1/2}},\,{{D}_{0}}\,=\,D_{0}^{*}$ is an unbounded linear operator affiliated with a semifinite von Neumann algebra $M,D-{{D}_{0}}$ is a bounded self-adjoint linear operator from $\mathcal{M}$ and ${{(1+D_{0}^{2})}^{-1/2}}\in E(M,\tau )$, where $E(\mathcal{M},\tau )$ is a symmetric operator space associated with $\mathcal{M}$. In particular, we prove that $\phi \left( D \right)-\phi \left( {{D}_{0}} \right)$ belongs to the non-commutative ${{L}_{p}}$-space for some $p\in (1,\infty )$, provided ${{(1+D_{0}^{2})}^{-1/2}}$ belongs to the noncommutative weak ${{L}_{r}}$-space for some $r\in [1,p)$. In the case $\mathcal{M}\,=\,\mathcal{B}\left( \mathcal{H} \right)$ and $1\,\le \,p\,\le \,2$, we show that this result continues to hold under the weaker assumption ${{(1+D_{0}^{2})}^{-1/2}}\in {{C}_{p}}$. This may be regarded as an odd counterpart of A. Connes’ result for the case of even Fredholm modules.