Masamichi Takesaki

Conditional Expectations in von Neumann Algebras

Let M be a von Neumann algebra and N its von Neumann subalgebra. Let ϑ be a faithful, semifinite, normal weight on M+ such that the restriction ϑ ¦ N of ϑ onto N is semifinite. The first main result is that N is invariant under the modular automorphism group σtϑ associated with ϑ if and only if there exists a σ-weakly continuous faithful projection ϵ of norm one from M onto N such that ϑ(x) = ϑ ∘ ϵ(x) for every xϵMϑ. The second result is that a von Neumann algebra M is finite if and only if any maximal abelian self-adjoint subalgebra of M is the range of a σ-weakly continuous projection of norm one. This result is an answer for the question which Kadison raised in the authors talk at the International Congress of Mathematicians in Nice, 1970.

Duality for crossed products and the structure of von Neumann algebras of type III

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Preliminary 251 Construction of crossed products . . . . . . . . . . . . . . . . . . . . . . . 253 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 Bi-dual weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 Subgroups and subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 284 The structure of a v o n Neumann algebra of type I I I . . . . . . . . . . . . . . 286 Algebraic invariants S ( ~ ) and T ( ~ ) of A. Connes . . . . . . . . . . . . . . . 294 Induced action and crossed products . . . . . . . . . . . . . . . . . . . . . 297 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

The Radon-Nikodym theorem for von neumann algebras

Let ϕ be a faithful normal semi-finite weight on a von Neumann algebraM. For each normal semi-finite weight ϕ onM, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operatorh, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ(h·). Conversely, each suchh determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result thatM is semi-finite if and only if Σ consists of inner automorphisms.

Duality and Subgroups

The second author proved, in [27], a duality theorem for general locally compact groups as a generalization of the so-called Tannaka duality theorem for compact groups. In the proof, the regular representation plays an essential role. In order to clarify the role of the regular representation in duality theory, the first author gave, in [23], a characterization of group algebras in terms of a Hopf-von Neumann algebra and showed that the abelian von Neumann algebra L-(G) of all essentially bounded functions on a locally compact group G together with the co-multiplication 6, the involution j and the Haar measure f involves the duality principle for the given group G. In this paper we shall examine, by showing a correspondence between closed subgroups and subalgebras of the Hopf-von Neumann algebra {L-(G), a, j}, how this duality principle works for subgroups. For a locally compact abelian group G, the Pontryagin duality theorem shows that there is a beautiful and complete one-to-one correspondence between a closed subgroup H of G and a closed subgroup H I of the dual group G such that

Disjointness of the KMS-states of different temperatures

Disjointness of (KMS)-states of different temperatures is proved.

States and automorphism groups of operator algebras

Suppose that a group of automorphisms of a von Neumann algebraM, fixes the center elementwise. We show that if this group commutes with the modular (KMS) automorphism group associated with a normal faithful state onM, then this state is left invariant by the group of automorphisms. As a result we obtain a “noncommutative” ergodic theorem. The discrete spectrum of an abelian unitary group acting as automorphisms ofM is completely characterized by elements inM. We discuss the KMS condition on the CAR algebra with respect to quasi-free automorphisms and gauge invariant generalized free states. We also obtain a necessary and sufficient condition for the CAR algebra and a quasi-free automorphism group to be η-abelian.

The structure of the automorphism group of an injective factor and the cocycle conjugacy of discrete abelian group actions

We complete classification of discrete abelian or finite group actions on injective type III1 factors up to cocycle conjugacy. We also give a proof for Connes’ characterization of the Ker (mod) and Cnt(M) for an injective factor M of type III. §0 Introduction. The purpose of this paper is to give a proof of Connes’ announcement on approximately inner automorphisms and centrally trivial automorphisms of an injective 1980 Mathematics Subject Classification (1985 Revision). 46L40.

Local normality in quantum statistical mechanics

It is shown that K.M.S.-states are locally normal on a great number ofC*-algebras that may be of interest in Quantum Statistical Mechanics. The lattice structure and the Choquet-simplex structure of various sets of states are investigated. In this respect special attention is payed to the interplay of the K.M.S.-automorphism group with other automorphism groups under whose action K.M.S.-states are possibly invariant. A seemingly weaker notion thanG-abelianness of the algebra of observables, namelyG′-abelianness, is introduced and investigated. Finally a necessary and sufficient condition (on aC*-algebra with a sequential separable factor funnel) for decomposition of a locally normal state into locally normal states is given.

Duality and subgroups, II

Let M(G) be the von Neumann algebra generated by the left regular representation λ of a locally compact group G, and NH be the von Neumann subalgebra of M(G) generated by the image λ(H) of a closed subgroup H. For an element M(G) to fall in NH it is necessary and sufficient that the support of x in the sense of Eymard, [4], is contained in H. This result yields that the correspondence of H and NH is a lattice isomorphism.

Compact abelian group actions on injective factors

We classify compact abelian group actions on injective type III factors up to conjugacy, which completes the final step of classification of compact abelian group actions on injective factors.