Martin Mathieu

The structure of Lie derivations on C*-algebras

We prove that every Lie derivation on a C � -algebra is in standard form, that is, it can be

Commutativity preserving mappings on semiprime rings

We obtain a representation theorem for bijective additive mappings preserving commutativity in both directions on 2-torsion free unital centrally closed semiprime rings such that the ideal associated with the second Kaplansky polynomial is essential. The same methods also yield descriptions of Lie automorphisms and Lie derivations.

Where to find the image of a derivation

With this paper, we intend to provide an overview of some recent work on a problem on unbounded derivations of Banach algebras that still defies solution, the noncommutative Singer–Wermer conjecture. In particular, we discuss several global as well as local properties of derivations entailing quasinilpotency in the image. 1. Where to look for . . . Derivations may serve as the generators of reversible evolutions of a physical system, say, if this is modelled by a Banach algebra. Not only historically, this point of view gave a strong impetus to the investigation of derivations and of how their properties relate to the structure of Banach algebras. The easiest examples one encounters are the inner derivations δa : x 7→ xa − ax, where a is a given element in the Banach algebra A, and one may be tempted to think that there are no others. Indeed, one is morally right: a non-vanishing first cohomology group is generally considered rather as an obstacle than a delight, as well as philosophically: often derivations become inner in a larger Banach algebra. Here is an example. Let us suppose that A is unital. (Since every derivation δ vanishes on the identity, this is no restriction of generality and will therefore be tacitly assumed henceforth.) Identifying A with a closed subalgebra of the algebra B of all bounded linear operators on A via the left regular representation a 7→ La, one has [La, δ](x) = aδ(x)− δ(ax) = −δ(a)x = L−δ(a)(x) (x ∈ A) , that is, [La, δ] = L−δ(a) or, under the above identification, δ(a) = δ−δ(a) for all a ∈ A. Of course, to this end we have to assume that δ is bounded. This already indicates that the actual problem is with unbounded derivations, but as is well known, even bounded derivations in general need not be inner in the strict sense 1991 Mathematics Subject Classification: Primary 47B47; Secondary 46-02, 46H99, 47-02. The paper is in final form and no version of it will be published elsewhere.

ON THE CENTRAL HAAGERUP TENSOR PRODUCT

For a large class of C *-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

Symmetric amenability and Lie derivations

We prove that every Lie derivation on a symmetrically amenable semisimple Banach algebra can be uniquely decomposed into the sum of a derivation and a centre-valued trace.

Spectrally bounded generalized inner derivations

We characterize the generalized inner derivations on a unital Banach algebra which are spectrally bounded. In particular, a simplified argument for the recent result due to Bresar that every spectrally bounded inner derivation maps into the radical is given. In [3], a linear mapping T on a complex unital Banach algebra A was defined to be spectrally bounded if r(Tx) 0O where r(.) denotes the spectral radius. This concept arises in connection with the non-commutative Singer-Wermer conjecture stating that the image 3A of any derivation 3 on A is contained in the Jacobson radical rad(A) if and only if x6x (3x)x E rad(A) for all x E A. Although some progress on this conjecture has been made lately (see [6] and the references in [3]), it is still open and seems hard to verify. A related, also open, conjecture is the following: The image 3A of a derivation 3 on A is contained in rad(A) if and only if 3 is spectrally bounded. This was recently affirmed for inner derivations by Bresar [2]. His argument rests in an essential way on two results due to Ptak [5], in particular on [5, Proposition 2.1] stating that a spectrally bounded inner derivation 3 has the property that 652A C d(A), the set of quasinilpotent elements of A. In an earlier paper, however, Ptak obtained a number of equivalent conditions for the image of an inner derivation to lie in the radical. For our purposes, the following suffices. Theorem A [4, Proposition 2.1]. Let A be a unital Banach algebra and a e A. The following conditions are equivalent. (a) ax xa E rad(A) for all x E A. (b) r(ax) 0. While (a) => (b) is immediate, the implication (b) => (a) is a consequence of the subharmonicity of the spectral radius. (But see the note at the end of this paper.) Let La and Ra denote the left and right multiplications by a E A, respectively. Combining Theorem A with Bresars result we have that La Ra is Received by the editors July 8, 1993 and, in revised form, November 28, 1993; presented by the first-named author at the Special Session on Nonselfadjoint Operator Algebras, 886th Regional Meeting of the AMS, College Station, Texas, October 22, 1993. 1991 Mathematics Subject Classification. Primary 46H99; Secondary 47B47. The first-named authors research was partially supported by an NSF grant. ? 1995 American Mathematical Society

More properties of the product of two derivations of a C *-algebra

It is proved that the product of two non-zero derivations of a prime C *-algebra is bounded only if both of them are bounded.

The norm problem for elementary operators

Abstract Among the outstanding problems in the theory of elementary operators on Banach algebras is the task to find a formula which describes the norm of an elementary operator in terms of the norms of its coefficients. Here we report on the state-of-the-art of the knowledge on this problem along the lines of our talk at the Functional Analysis Valencia 2000 Conference in July 2000 .

Spectrally bounded operators on simple C*-algebras

A linear mapping T from a subspace E of a Banach algebra into another Banach algebra is called spectrally bounded if there is a constant M > 0 such that r(Tx) < Mr(x) for all x ∈ E, where r(.) denotes the spectral radius. We prove that every spectrally bounded unital operator from a unital purely infinite simple C*-algebra onto a unital semisimple Banach algebra is a Jordan epimorphism.

A simple local multiplier algebra

We present a class of approximately finite dimensional C *-algebras whose local multiplier algebras are simple with real rank zero and stable rank one.