The Laplacian subalgebra of $\mathcal {L}(\mathbb {F}_N)^{\overline {\otimes }_k}$ is a strongly singular masa

Abstract:

In this paper, we present a new class of strongly singular maximal abelian subalgebras living inside the $k$-folded tensor product of the free group factor $\mathcal {L}(\mathbb {F}_N)$ with itself ($N\ge 2$). The notions of strongly singular masas in type $\mathrm {II}_1$ factors and that of asymptotic homomorphism were introduced by A. Sinclair and R. Smith. One of their first examples was the Laplacian subalgebra of the free group factor, generated by the sum of words of length 1 in $\mathbb {F}_N$. This subalgebra was known to be a singular masa. Using the results of A. Sinclair and R. Smith, we show that the unique trace-preserving conditional expectation onto the Laplacian subalgebra of $\mathcal {L}(\mathbb {F}_N)^{\overline {\otimes }_k}$ is an asymptotic homomorphism, and hence the Laplacian subalgebra is a strongly singular masa for every $k\ge 1$.