Maps preserving strong 2-Jordan product on some algebras

Let be a unital prime ring with characteristic not 2 and containing a nontrivial idempotent. Assume that is a surjective map. It is shown that f is strong 3-commutativity preserving, that is, f satisfies for all , if and only if for all , where is an element in the extended centroid of with and is a map from into its extended centroid. Applications to prime C-algebras, factor von Neumann algebras, standard operator algebras and matrix algebras are also obtained .

Let A be a unital *-algebra. In this paper, under some mild conditions on A, it is shown that a map ? : A ? Ais a nonlinear mixed bi-skew Lie triple derivation if and only if ? is an additive*-derivation. As applications, nonlinear mixed bi-skew Lie triple derivations on prime *-algebras, von Neumann algebras with no central summands of type I1, factor von Neumann algebras and standard operator algebras are characterized.

Operator algebras come in many flavours. For the purpose of this article, however, the term is only used for one of two kinds of self-adjoint algebras of operators on Hilbert space, viz., C*- algebras (which are norm-closed) or von Neumann algebras (which are closed in the topology of pointwise strong convergence or equivalently, in the weak-* topology it inherits as a result of being a Banach dual space). To be fair, there are a number of people in India (eg., Gadadhar Misra, Tirthankar Bhattacharyya, Jaydeb Sarkar, Santanu Dey, etc.,) who work on non-selfadjoint algebras, mostly from the point of view of connections with complex function theory; but in the interest of restricting the size of this paper, I confine myself here to selfadjoint algebras. I apologise for ways in which my own personal taste and limitations colour this depiction of operator algebras. Another instance of this arbitrary personal taste is a decision to concentrate on the work of younger people. Thus, the work of the more senior people who have worked in operator algebras is only seen via their collaborations with younger people: e.g., KRP via Srinivasan and Rajarama Bhat, Kalyan Sinha via Debashish, Partha, Arup, Raja, etc,. and me via Vijay, Srinivasan and Panchugopal. Not long ago, interest in operator algebras in India was restricted to the three centres of the Indian Statistical Institute. Now, I am happy to note that it has spread to IMSc, some IITs, IISERs, NISER, JNU,.... My role in this article has been merely that of compiling inputs from many active Indian operator algebraists that came to my mind. I wrote soliciting a response from a certain number of them, then put together the responses received. (I apologise to those people who were omitted in this process). My colleague, Partha, with the help of his collaborator Arup, agreed to take care of the C*-related inputs, while I take care of the von Neumann-related ones with the help of my collaborator Vijay. What follows are some areas of ongoing research done in von Neumann algebras in India and some names of people doing such work: (a) subfactors and planar algebras, (Vijay Kodiyalam of IMSc, Chennai); (b) quantum dynamical systems and complete positivity (Rajarama Bhat of ISI, Bengaluru); (c) E0 semigroups (R. Srinivasan of CMI, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar), and (d) Masas in II1 von Neumann algebras and free Araki-Woods factors (Kunal Mukherjee of IIT, Chennai, and Panchugopal Bikram of NISER, Bhubhaneswar).

We study Lie ideals in two classes of triangular operator algebras: nest algebras and triangular UHF algebras. Our main results show that if 2 is a closed Lie ideal of the triangular operator algebra A, then there exist a closed associative ideal IC and a closed subalgebra Or of the diagonal A ni A* so that K C 2 C IC + Oc. INTRODUCTION Let A be an associative complex algebra. Under the Lie multiplication [x, y] xy yx, A becomes a Lie algebra. A Lie ideal in A is a linear manifold Z in A for which [a, k] c ? for every a E A and k c 2. In many instances, there is a close connection between the Lie ideal structure and the associative ideal structure of A. This connection has been investigated for prime rings in [6], in [3] for B($5) the set of bounded operators on a Hilbert space Si, and in [10] for certain von Neumann algebras. (See also [9, 4, 11].) In this paper we pursue this line of investigation for two classes of triangular operator algebras, namely nest algebras anld triangular UHF algebras. The authors would like to thank Frank A. Zorzitto for many helpful conversations. 1. WEAKLY CLOSED LIE IDEALS IN NEST ALGEBRAS Recall that a nest JK on a Hilbert space Si is a chain of closed subspaces of Si which is closed under the operations of arbitrary intersections and closed linear spans, and which includes {O} and S5. The nest algebra T(KJ) is the algebra of all operators on Se leaving every member of JK invariant. This is always closed in the weak operator topology. The diagonal D(KJ) of a nest algebra T(JK) is the von Neumann subalgebra 7(K) n T(K)*. If E, F E KV with E < F, then F E is called an interval of the nest. The nonzero minimal intervals are called atoms. A nest is atomic if the atoms of the nest span 55. We refer the reader to [1] for more information on nest algebras. Our main result, Theorem 12, shows that for every weakly closed Lie ideal Z in 7(JK), there exist a corresponding weakly closed associative ideal IC and a von Neumann subalgebra SK of O(A/) such that

The present paper presents a survey of some recent results devoted to derivations, local derivations and 2-local derivations on various algebras of measurable operators affiliated with von Neumann algebras. We give a complete description of derivation on these algebras, except the case where the von Neumann algebra is of type II 1 _1 . In the latter case the result is obtained under an extra condition of measure continuity of derivations. Local and 2-local derivations on the above algebras are also considered. We give sufficient conditions on a von Neumann algebra M M , under which every local or 2-local derivation on the algebra of measurable operators affiliated with M M is automatically becomes a derivation. We also give examples of commutative algebras of measurable operators admitting local and 2-local derivations which are not derivations.

Nonlinear maps preserving Lie products on factor von Neumann algebras

The axioms of measurements introduced by Ludwig are formulated and studied in the framework of operator algebras. It is shown that a concrete C*-algebra with identity satisfies the axiom of sensitivity increase of effects if and only if it is a von Neumann algebra; although a von Neumann algebra satisfies the axiom of decompossability of ensembles, however, the axiom of components of the mixtures of two ensembles is true only if a von Neumann algebra is a factor of type In (n &amp;lt; + ∞). It is also verified that the set of decision effects, which is proved to be a subset of projections of a von Neumann algebra, has similar lattice structure of quantum mechanics, and its connection with quantum logic in the sense of Varadarjan is also figured out.

Let be a unital ∗‐algebra with the unit I and be the symmetric center of . Assume that contains a nontrivial projection P such that implies A = 0 and implies A = 0. In this paper, we proved that if a map satisfies [ φ ( A ), B ] ∗ = −[ φ ( B ), A ] ∗ for all , then there exists a map such that φ ( A ) = Z A + f ( A ) for all , where . As applications, we obtain the concrete form of nonadditive skew commuting maps on prime ∗‐algebra, factor von Neumann algebras and standard operator algebras.

In this paper, we prove that every multiplicative anti-derivation map of an arbitrary generalized [Formula: see text]-matrix ring is additive. Even more, as consequence, we get the same conclusion for the class of triangular [Formula: see text]-matrix rings, unital prime rings with a nontrivial idempotent, standard operator algebras and factor von Neumann algebras.

Let A be a unital ∗-algebra containing a nontrivial projection. In this paper, we show that every nonlinear mixed bi-skew Jordan-type higher derivation on A is an additive higher ∗-derivation. Moreover, we apply the above result to prime ∗-algebras, factor von Neumann algebras and standard operator algebras.

A non-commutative version of probability theory is outlined, based on the concept of aΣ*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory ofΣ*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. TheΣ*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering theΣ*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.

Let r be a normal semifinite trace on a von Neumann algebra, and let f be a continuous convex function on the interval [0, oo) with f (0) = 0. For a positive element a of the algebra and a positive contraction a on the algebra, the following inequality is obtained: r(f (c((a))) Zf((Ari,r,i)), where f is a convex function on the real line, A is a selfadjoint operator, and {rqi} is an orthonormal set (see [11], for example). Slightly more general, if {Pi} is a pairwise orthogonal family of projections and we set a(A) = >nPiAPi for a selfadjoint operator A, then tr a (f (A)) > tr f (a (A)) (see [2, 6, and 9]). Going further in this direction Brown and Kosaki [3] have recently proved that r(f (v*av)) < r(v* f (a)v), where r is a faithful normal semifinite trace on a von Neumann algebra, v is a contraction, and a is a positive element of the algebra. (In fact, a may be unbounded and affiliated with the algebra.) In the present paper we obtain a similar inequality for an arbitrary positive contraction a on the algebra and show that (*) r(f (a (a))) < r(a(f (a))). Here f is supposed to be a continuous convex function with f(O) = 0. (Note that (*) holds without r if f is operator convex; cf. Theorem IV.1 in [1].) The inequality (*) supports the experience that inequalities involving operators are more simply treatable provided both sides are inside of a trace (see [3 and 8] for further evidences). The idea of our proof is to approximate the operators a and a (a) by diagonal ones by means of their spectral resolution and to apply the classical Jensen's inequality in Received by the editors November 19, 1984 and, in revised form, June 24, 1985 and December 17, 1985. 1980 Mathematics Subject Cai Secondary 26A51, 47B15.

Let [Formula: see text] be a factor von Neumann algebra with dim[Formula: see text]. In this paper, we prove that a map [Formula: see text] satisfies [Formula: see text] for all [Formula: see text] if and only if there exists a real number [Formula: see text] such that [Formula: see text], where [Formula: see text] is an additive ∗-derivation and [Formula: see text].

&lt;abstract&gt;&lt;p&gt;In this article, we proved that each nonlinear higher anti-derivable mapping on generalized matrix algebras is automatically additive. As for its applications, we find a similar conclusion on triangular algebras, full matrix algebras, unital prime rings with a nontrivial idempotent, unital standard operator algebras and factor von Neumann algebras respectively.&lt;/p&gt;&lt;/abstract&gt;

Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.