Let B be a unital monotone sigma-complete C*-algebra, X a Banach space and (T-n) (n=1, 2,...) a sequence of weakly compact operators from B to X. Our key theorem implies that if limT(n)p exists for each projection p in B, then there exists a weakly compact operator T such that parallel to T ''(n)(b)-T ''(b)parallel to -> 0, for each b in B ''. (An immediate corollary of this is that B is a Grothendieck space.) Let mu be any state of B such that each T-n is strongly absolutely continuous with respect to mu. Then, as a consequence of our key theorem, we obtain the following uniform absolute continuity property. For each epsilon > 0, there exists delta > 0, such that, whenever b is in the unit ball of B and mu(bb*+b*b)<delta, then parallel to T(n)b parallel to <=epsilon for all n. This last result is a far reaching non-commutative generalization of the classical Brooks-Jewett theorem for vector measures.

Let A1,A2,…,Ar be C∗-algebras with second duals A1″,A2″,…,Ar″, and let X be an arbitrary Banach space. Let Γ:A1×A2×⋯×Ar→X be a bounded r-linear map, and denote by Γ″:A1″×A2″×⋯×Ar″→X″ the Johnson–Kadison–Ringrose extension (i.e., the separately weak∗ to weak∗ continuous r-linear extension) of Γ. The problem of characterising those Γ for which Γ″ takes its values in X was solved by Villanueva when the algebras are all commutative. Because the Dunford–Pettis property fails for noncommutative C∗-algebras, the ‘obvious’ extension of Villanueva's characterisation does not give the correct condition. In this paper we solve this problem for general C∗-algebras. This result is then applied to obtaining a multilinear generalisation of the normal-singular decomposition of a bounded linear operator on a von Neumann algebra.

Let B be a monotone σ-complete C∗-algebra. Let (μn) (n=1,2,…) be a sequence in the dual of B such that limμn(p) exists for each projection p. We prove that the sequence must converge weakly. As an application, we obtain a non-commutative generalisation of the Brooks–Jewett Theorem.

Let A be a C*‐algebra and let K be a relatively weakly compact subset of the dual of A. Let ψbe a positive linear functional on A such that, for each φin K, φis strongly absolutely continuous with respect to ψ. Then, for each ε> 0, there exists δ> 0, such that for each x in the closed unit ball of A, ψ(xx* +x*x)1/2 ≤δimplies |φ(x) |≤εfor every φ∈K. This result is extended to the situation where K is a σ‐bounded set of weakly compact operators from A to a Banach space Y.

Let F be the Fermion C-*-algebra and let F-infinity be its (Pedersen) Borel (*)-envelope. Then there exists a sigma-ideal, M, the meagre ideal, such that F-infinity/M is the regular (sigma)-completion of F; a monotone complete type III factor which has no normal states. Also, there exists a sigma-ideal, N, such that F-infinity/N can be identified with the generic dynamics factor; a monotone complete type III factor which has no normal states. It might be considered plausible that M = N. We show here that this is false. There exists a central projection q in F-infinity such that q is in M and 1 - q is in N. Hence F-infinity = qM target (1 - q)N.

Let (phi(n)) be a norm bounded sequence in the pre-dual of a von Neumann algebra M. In general it is not true that this sequence has a weakly convergent subsequence. But given a normal state psi, then, for any epsilon > 0, there exists a projection e such that psi (1 - e) less than or equal to epsilon and the restriction of (phi(n)) to eMe has a subsequence which converges weakly to a normal functional on eMe. (C) 2002 Elsevier Science (USA). All rights reserved.

Let (A, X) and (B, Y) be measurable spaces and let V be a Dedekind sigma -complete vector lattice. Let mu (1) and mu (2) be measures defined on A and B, respectively, and taking their values in the positive cone of V. We define sigma -additivity of V-valued measures with respect to the order structure of V. Let A x B be the sigma -field generated by A and B. It is shown here that classical results of Strassen can be generalized to this situation. In particular, when mu (1)(X) = mu (2)(Y), there exists a V-valued sigma -additive measure mu on A x B such that mu (A x Y) = mu (1)(A) and mu (X x B) = mu (2)(B) if mu (1) is sigma -additive, mu (2) is sigma -compact and V satisfies the lattice condition of being weakly sigma -distributive. When V is Dedekind complete and satisfies the stronger property of weak (sigma, infinity)-distributivity then analogous results hold with mu (2) satisfying the weaker property of being completely compact.

Schaefer and Zhang have recently obtained an analogue, for sequentially order continuous functionals on C ( K ) C(K) , of a much earlier theorem of Dixmier. In this note it is shown that the Schaefer-Zhang Theorem has a natural generalisation to non-commutative C ∗ C^* -algebras. These results are obtained as consequences of our main theorem which is concerned with affine functions on compact convex sets.

Journal Article Admissible dynamics systems for monotone complete C-algebras Get access K Saitô, K Saitô Search for other works by this author on: Oxford Academic Google Scholar JDM Wright JDM Wright Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 50, Issue 198, June 1999, Pages 231–247, https://doi.org/10.1093/qjmath/50.198.231 Published: 01 June 1999

A general quantum history theory can be characterized by the space of histories and by the space of decoherence functionals. In this note we consider the situation where the space of histories is given by the lattice of projection operators on an infinite dimensional Hilbert space H. We study operator representations for decoherence functionals on this space of histories. We first give necessary and sufficient conditions for a decoherence functional being representable by a trace class operator on H⊗H, an infinite dimensional analogue of the Isham–Linden–Schreckenberg representation for finite dimensions. Since this excludes many decoherence functionals of physical interest, we then identify the large and physically important class of decoherence functionals which can be represented, canonically, by bounded operators on H⊗H.

APPROXIMATELY ADDITIVE STATES ON TYPE I C*-ALGEBRAS Get access L. J. BUNCE, L. J. BUNCE University of Reading Department of MathematicsPO Box 220 Reading RG6 2AX, UK Search for other works by this author on: Oxford Academic Google Scholar J. D. MAITLAND WRIGHT J. D. MAITLAND WRIGHT University of Reading Department of MathematicsPO Box 220 Reading RG6 2AX, UK Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 47, Issue 3, September 1996, Pages 269–277, https://doi.org/10.1093/qmath/47.3.269 Published: 01 September 1996 Article history Received: 21 March 1994 Published: 01 September 1996

Let A \mathcal A be a C ∗ C^\ast -algebra, and let μ \mu be a (local) quasi-trace on A \mathcal A . Then μ \mu is linear if, and only if, the restriction of μ \mu to the closed unit ball of A \mathcal A is uniformly weakly continuous.

Let A be a C*-algebra with no quotient isomorphic to the algebra of all two-by-two matrices. Let mu be a quasi-linear functional on A. Then mu is linear if, and only if, the restriction of mu to the dosed unit ball of A is uniformly weakly continuous.

Journal Article STRANGE PROJECTIONS IN TENSOR PRODUCTS OF VON NEUMANN ALGEBRAS Get access KAZUYUKI SAITÔ, KAZUYUKI SAITÔ Mathematical Institue Tôhoku UniversitySENDAI 980 Japan Search for other works by this author on: Oxford Academic Google Scholar J. D. MAITLAND WRIGHT J. D. MAITLAND WRIGHT Isaac Newton Institute for Mathematical Sciences20 Clarkson Road Cambridge CB3 0EH, UK Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 46, Issue 2, June 1995, Pages 197–199, https://doi.org/10.1093/qmath/46.2.197 Published: 01 June 1995 Article history Received: 13 April 1993 Revision received: 04 October 1993 Published: 01 June 1995

When A is a C*-algebra, a function d: A(sa) --> A(sa) is said to be a velocity map if, for each commutative subalgebra B of A(ss), d: B --> A(sa) is a derivation.Let A be a norm closed ideal, or quotient, in a von Neumann algebra without Type I-2 part and let P(A) be the set of projections in A. It is shown that if d : P(A) --> A is a bounded function such that d(ef) = ed(f) + d(e)f whenever ef = fe, then d extends uniquely to a derivation of A. Hence every velocity map of A(sa) bounded on the unit ball extends to a derivation of A.

Journal of the London Mathematical SocietyVolume 49, Issue 1 p. 133-149 Notes and Papers The Mackey-Gleason Problem for Vector Measures on Projections in Von Neumann Algebras L. J. Bunce, L. J. Bunce Analysis and Combinatorics Research Centre, Department of Mathematics, PO Box 220, University of Reading, Reading RG6 2AXSearch for more papers by this authorJ. D. Maitland Wright, J. D. Maitland Wright Analysis and Combinatorics Research Centre, Department of Mathematics, PO Box 220, University of Reading, Reading RG6 2AXSearch for more papers by this author L. J. Bunce, L. J. Bunce Analysis and Combinatorics Research Centre, Department of Mathematics, PO Box 220, University of Reading, Reading RG6 2AXSearch for more papers by this authorJ. D. Maitland Wright, J. D. Maitland Wright Analysis and Combinatorics Research Centre, Department of Mathematics, PO Box 220, University of Reading, Reading RG6 2AXSearch for more papers by this author First published: February 1994 https://doi.org/10.1112/jlms/49.1.133Citations: 23AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Citing Literature Volume49, Issue1February 1994Pages 133-149 RelatedInformation

Journal Article A CONTINUUM OF DISCRETE GROUP ACTIONS ON A FERMIONIC FACTOR Get access KAZUYUKI SAITÔ, KAZUYUKI SAITÔ Mathematical Institute Tôhoku UniversitySendai 980 Japan Search for other works by this author on: Oxford Academic Google Scholar J. D. MAITLAND WRIGHT J. D. MAITLAND WRIGHT Analysis and Combinatorics Research Centre Mathematics Department The UniversityP.O.Box 220 Reading, RG62AX Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 44, Issue 3, September 1993, Pages 339–343, https://doi.org/10.1093/qmath/44.3.339 Published: 01 September 1993 Article history Received: 17 February 1992 Published: 01 September 1993

Journal Article DYNAMIC ACTIONS ON MONOTONE COMPLETE FACTORS Get access KAZUYUKI SAITÔ, KAZUYUKI SAITÔ Mathematical Institute Tôhoku UniversitySendai 980 Japan Search for other works by this author on: Oxford Academic Google Scholar J. D. MATTLAND WRIGHT J. D. MATTLAND WRIGHT Analysis and Combinatorics Research Centre Mathematics DepartmentThe University P.O. Box 220 Reading, RG6 2AX Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 44, Issue 2, June 1993, Pages 239–247, https://doi.org/10.1093/qmath/44.2.239 Published: 01 June 1993 Article history Received: 22 January 1991 Published: 01 June 1993

Journal Article ERGODIC ACTIONS ON THE FERMIONIC FACTOR Get access KAZUYUKI SAITÔ, KAZUYUKI SAITÔ Mathematical Institute Tôhoku UniversitySendai 980 Japan Search for other works by this author on: Oxford Academic Google Scholar J. D. MAITLAND WRIGHT J. D. MAITLAND WRIGHT Analysis and Combinatorics Research Centre Mathematics Department The UniversityP.O. Box 220, Whiteknights Reading, RG6 2AX Search for other works by this author on: Oxford Academic Google Scholar The Quarterly Journal of Mathematics, Volume 44, Issue 4, December 1993, Pages 493–496, https://doi.org/10.1093/qmath/44.4.493 Published: 01 December 1993 Article history Received: 27 March 1992 Published: 01 December 1993

A problem of Araki concerning the characterization of orthogonality preserving positive maps between preduals of von Neumann algebras is solved in a general setting.