We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialised case where the higher-rank graph consists of one vertex, we are able to use the Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

We prove that a finite index regular inclusion of $II_1$ -factors with commutative first relative commutant is always a crossed product subfactor with respect to a minimal action of a biconnected weak Kac algebra. Prior to this, we prove that every finite index inclusion of $II_1$ -factors which is of depth 2 and has simple first relative commutant (respectively, is regular and has commutative or simple first relative commutant) admits a two-sided Pimsner–Popa basis (respectively, a unitary orthonormal basis).

We study unitary quotients of the free product unitary pivotal category ${{A}_{2}}\,*\,{{T}_{2}}$. We show that such quotients are parametrized by an integer $n\,\ge \,1$ and an $2n$–th root of unity $\omega $. We show that for $n\,=\,1,\,2,\,3$, there is exactly one quotient and $\omega \,=\,1$. For $4\,\le \,n\,\le \,10$, we show that there are no such quotients. Our methods also apply to quotients of ${{T}_{2}}\,*\,{{T}_{2}}$, where we have a similar result.

The essence of our method is a consistency check on jellyfish relations. While we only treat the specific cases of ${{A}_{2}}\,*\,{{T}_{2}}$ and ${{T}_{2}}\,*\,{{T}_{2}}$, we anticipate that our technique can be extended to a general method for proving the nonexistence of planar algebras with a specified principal graph.

During the preparation of this manuscript, we learnt of Liu's independent result on composites of ${{A}_{3}}$ and ${{A}_{4}}$ subfactor planar algebras (arxiv:1308.5691). In 1994, Bisch–Haagerup showed that the principal graph of a composite of ${{A}_{3}}$ and ${{A}_{4}}$ must fit into a certain family, and Liu has classified all such subfactor planar algebras. We explain the connection between the quotient categories and the corresponding composite subfactor planar algebras. As a corollary of Liu's result, there are no such quotient categories for $n\,\ge \,4$.

This is an abridged version of arxiv:1308.5723.

In this paper, we study uniform perturbations of von Neumann subalgebras of a von Neumann algebra. Let $M$ and $N$ be von Neumann subalgebras of a von Neumann algebra with finite probabilistic index in the sense of Pimsner and Popa. If $M$ and $N$ are sufficiently close, then $M$ and $N$ are unitarily equivalent. The implementing unitary can be chosen as being close to the identity.

We construct the quantum $s$-tuple subfactors for an AFD $\text{I}{{\text{I}}_{1}}$ subfactor with finite index and depth, for an arbitrary natural number $s$. This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

We determine when an automorphism of a subfactor of type $\text{II}{{\text{I}}_{0}}$ with finite index is nonstrongly free in the sense of $\text{C}$. Winsløw in terms of the modular endomorphisms introduced by M. Izumi.

We show that the multi-sided inclusion ${{R}^{\otimes l}}\,\subset \,R$ of braid-type subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$, introduced in Multi-sided braid type subfactors$[\text{E}3]$, contains a sequence of intermediate subfactors: ${{R}^{\otimes l\,}}\subset \,{{R}^{\otimes l-1\,}}\subset \,\,\cdots \,\,\subset \,{{R}^{\otimes 2\,}}\subset \,R$. That is, every $t$-sided subfactor is an intermediate subfactor for the inclusion ${{R}^{\otimes l}}\,\subset \,R,\,\text{for 2}\,\le \,t\,\le \,l$. Moreover, we also show that if $t\,>\,m$ then ${{R}^{\otimes t}}\,\subset \,{{R}^{\otimes m}}$ is conjugate to ${{R}^{\otimes t-m+1\,}}\subset \,R$. Thus, if the braid representation considered is associated to one of the classical Lie algebras then the asymptotic inclusions for the Jones-Wenzl subfactors are intermediate subfactors.

We generalise the two-sided construction of examples of pairs of subfactors of the hyperfinite $\text{I}{{\text{I}}_{1}}$ factor $R$ in $[\text{E1}]$—which arise by considering unitary braid representations with certain properties—to multi-sided pairs. We show that the index for the multi-sided pair can be expressed as a power of that for the two-sided pair. This construction can be applied to the natural examples—where the braid representations are obtained in connection with the representation theory of Lie algebras of types $A,\,B,\,C,\,D$. We also compute the (first) relative commutants.

We show that finite dimensional subfactors do not have subdiagonal algebras unless the Jones index is one.