An \(L^2\)-Künneth formula for tracial algebras (Q2891146)

If \(A\) is a weakly dense \(\ast\)-subalgebra of a tracial von Neumann algebra \((M,\tau)\), the Connes-Shlyakhtenko \(L^2\)-Betti numbers associated to \(A\) are denoted by \(\beta_n^{(2)}(A,\tau)\). The aim of the paper is to prove a Künneth formula for these numbers, by showing that for weakly dense \(\ast\)-subalgebras \(A\) and \(B\) of tracial von Neumann algebras \((M,\tau)\) and \((N,\rho)\) one has \(\beta_n^{(2)}(A\bigodot B,\tau\otimes\rho)=\sum_{k+l=n}\beta_k^{(2)} (A,\tau)\beta_l^{(2)}(B,\rho)\), where \(\bigodot\) denotes the algebraic tensor product. Using this formula, examples of compact quantum groups with a non-vanishing first Betti number are constructed.