On maximal injective subalgebras of tensor products of von Neumann algebras

Let Mi be a von Neumann algebra, and Bi be a maximal injective von Neumann subalgebra of Mi , i = 1, 2. IfM1 has separable predual and the center of B1 is atomic, e.g., B1 is a factor, then B1 ⊗¯ B2 is a maximal injective von Neumann subalgebra ofM1 ⊗¯ M2. This partly answers a question of Popa. © 2006 Elsevier Inc. All rights reserved.