3 We consider commuting squares of nite dimensional von Neumann algebras having the algebra of complex numbers in the lower left corner. Examples include the vertex models, the spin models (in the sense of subfactor theory) and the commuting squares associated to nite dimensional Kac algebras. To any such commuting square we associate a compact Kac algebra and we compute the corresponding subfa...

A von Neumann algebra is the commutant in B(H) (the algebra of bounded operators on a Hilbert space H) of a selfadjoint subset of B(H), in other words, a *-subalgebra of B(H) that is equal to its own bicommutant. These objects, whose investigation began with the work of Murray and von Neumann in the 30’s and 40’s on “rings of operators”, are vastly numerous and diverse, but there is a good sche...

Hilbert(ian) A-modules over finite von Neumann algebras with a faithful normal trace state (from global analysis) and Hilbert W*-modules over A (from operator algebra theory) are compared, and a categorical equivalence is established. The correspondence between these two structures sheds new light on basic results in L-invariant theory providing alternative proofs. We indicate new invariants fo...

counterpart , these new observables in A∈A ( ) ′′ A A ω π A have no abstract counterpart in . However, they will have an abstract counterpart in . There is also a natural embedding of A into ∗∗ ∗∗ ∗∗ ⊆ A A A such that . The most common way to construct a von Neumann algebra is to start with a C*-algebra and an abstract state A ω , construct a representation ( , ω ) π ω π H ( via the GNS theorem...

In this paper we prove the existence of finite systems G of generators, with special properties, for the von Neumann algebras L(Γ)⊗B(H), obtained by tensoring the group von Neumann algebra of a fuchsian group Γ with the bounded linear operators on a infinite dimensional, separable, Hilbert space H . The results show that the linear span of ordered monomials that are products of powers of adjoin...

ABSTRACT. Suppose M is a hyperfinite von Neumann algebra with a tracial state φ and {a1, . . . , an} is a set of self-adjoint generators for M. We calculate δ0(a1, . . . , an), the modified free entropy dimension of {a1, . . . , an}. Moreover we show that δ0(a1, . . . , an) depends only on M and φ. Consequently δ0(a1, . . . , an) is independent of the choice of generators for M . In the course ...

Let A be a C∗-algebra. We prove that every absolutely summing operator from A into l2 factors through a Hilbert space operator that belongs to the 4-Schatten-von Neumann class. We also provide finite dimensional examples that show that one can not replace the 4-Schatten-von Neumann class by p-Schatten-von Neumann class for any p < 4. As an application, we show that there exists a modulus of cap...

We prove some structure results for isometries between noncommutative L spaces associated to von Neumann algebras. We find that an isometry T : L(M1) → L(M2) (1 ≤ p < ∞, p 6= 2) can be canonically expressed in a certain simple form whenever M1 has variants of Watanabe’s extension property [W2]. Conversely, this form always defines an isometry provided that M1 is “approximately semifinite” (defi...

Let %? be a separable Hubert space and !T c 3§(J%f) be an algebra of bounded operators. Say F is triangular if ^ n ^ * is a maximal abelian self-adjoint subalgebra (m.a.s.a.) of 3B{%?) and call this m.a.s.a. the diagonal of J7". A triangular algebra is maximal triangular if it is not properly contained in any triangular algebra. Triangular algebras of operators have been studied for 30 years no...