Commutative quotients of finite W-algebras

In this paper we reduce the problem of 1-dimensional representations for the finite W-algebras and Humphreys' conjecture on small representations of reduced enveloping algebras to the case of rigid nilpotent elements in exceptional Lie algebras. We use Katsylo's results on sections of sheets to determine the Krull dimension of the largest commutative quotient of the finite W-algebra U(g,e).     

Normal weights on W1-algebras

Let ? be a weight on a W¹-algebra. If ? is normal, in the sense that it respects monotone increasing limits, then ? is the sum of positive normal functionals. This provides the complete solution to a problem raised by J. Dixmier.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.     

Derivations of non-separable C1-algebras

The question of which C¹-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C¹-tensor product of a von Neumann algebra and an abelian C¹-algebra has only inner derivations. Other special types of C¹-algebras are shown to have only inner derivations as well such as the C¹-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C¹-algebra having only inner derivations. Derivations from a smaller C¹-algebra into a larger one are also considered, and this concept is generalized to include derivations between C¹-algebras connected by a ¹-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C¹-algebra which converges to zero (in norm) when restricted to any abelian C¹-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general.     

Ein operatorwertiger Hahn-Banach Satz

We generalize Arveson's extension theorem for completely positive mappings [1] to a Hahn-Banach principle for matricial sublinear functionals with values in an injective C¹-algebra or an ideal in B($\text{H}$). We characterize injective W¹-algebras by a matricial order condition. We illustrate the matricial Hahn-Banach principle by three applications: (1) Let $\text{A, B, b}$ be unital C¹-algebras, $\text{b}$ a subalgebra of $\text{A}$ and $\text{B, B}$ injective. If ?: $\text{A}$ → $\text{B}$ is a completely bounded self-adjoint $\text{b}$-bihomomorphism, then it can be expressed as the difference of two completely positive $\text{b}$-bihomomorphism. (2) Let $\text{M}$ be a W¹-algebra, containing 1_$\text{H}$, on a Hilbert space $\text{H}$. If $\text{M}$ is finite and hyperfinite, there exists an invariant expectation mapping P of B($\text{H}$) onto $\text{M}$′. P is an extension of the center trace. (3) Combes [7] proved, that a lower semicontinuous scalar weight on a C¹-algebra is the upper envelope of bounded positive functionals. We generalize this result to unbounded completely positive mappings with values in an injective W¹-algebra.     

Representability and Specht problem for G-graded algebras

Let W be an associative PI-algebra over a field F of characteristic zero, graded by a finite group G. Let idG(W) denote the T-ideal of G-graded identities of W. We prove: 1. [G-graded PI-equivalence] There exists a field extension K of F and a finite-dimensional Z/2Z×G-graded algebra A over K such that idG(W)=idG(A^∗) where A^∗ is the Grassmann envelope of A. 2. [G-graded Specht problem] The T-ideal idG(W) is finitely generated as a T-ideal. 3. [G-graded PI-equivalence for affine algebras] Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite-dimensional algebra A over K such that idG(W)=idG(A).     

Die struktur endlicher schwach abgeschlossener Jordanalgebren Teil II. Diskrete Jordanalgebren

In the preceding note [6] we reduced the study of continuous finite weakly closed Jordan algebras to real associative W^*-algebras of type II₁. Here we treat the remaining case of discrete finite weakly closed Jordan algebras and describe them completely by finite dimensional simple formally real Jordan algebras and by simple formally real Jordan algebras of quadratic forms of real Hilbert spaces. Jacobsons theory of Jordan algebras with minimum condition combined with W^*-algebra techniques constitutes an essential tool in the proof.     

Noncommutative topological dynamics and compact actions on C1-algebras

The classical notions of topological transitivity and minimality of a topological dynamical system are extended and analyzed in the context of C¹-dynamical systems. These notions are compared with other notions naturally arising in noncommutative ergodic theory. As an application, a C¹-algebra version of a theorem of Araki, Haag, Kastler, and Takesaki (Comm. Math. Phys.53 (1977), 97–134) about the correspondence between a compact automorphism group (here assumed to be abelian) and its fixed-point subalgebra is proved in the presence of a commuting topologically transitive action. A variation of this theorem in the setting of standard W¹-inclusions is also presented.     

Measures on projections in a W *-algebra of type I 2

In the paper we present two results for measures on projections in a W ^*-algebra of type I ₂. First, it is shown that, for any such measure m, there exists a Hilbert-valued orthogonal vector measure µ such that ‖µ(p)‖² = m(p) for every projection p. In view of J. Hamhalter’s result (Proc. Amer. Math. Soc., 110 (1990), 803–806), this means that the above assertion is valid for an arbitrary W ^*-algebra. Secondly, a construction of a product measure on projections in a W ^*-algebra of type I ₂ (an analogue of the product measure in classical Lebesgue theory) is proposed.     

One-Parameter Groups Arising from Real Subspaces of Self-Dual Hilbert W*-Moduli

The purpose of this paper is to generalize the results of M. A. Rieffel and A. van Daele [8, §§ 1, 2, 3] for Hilbert W^*-moduli over commutative W^*-Algebras. Some special real subspaces of such Hilbert W^*-moduli and the related operators are investigated. Particularly, the relation is established between ^*weakly continuous unitary one-parameter groups of operators arising from them and the generalized K.M.S. condition. All key definitions are formulated without any commutativity supposition for the underlying W^*-algebra. The interpretation of these results is given for sets of continuous sections of “self-dual” locally trivial Hilbert bundles over hyperstonian compact spaces. At the end of this paper some aspects of the general noncommutative case are discussed.     

Traces on simple AF C1-algebras

The relations between the set of traces on a simple approximately finite dimensional C¹-algebra A and the algebraic and geometric properties of the Elliott dimension group K₀(A) are studied. It is shown that every metrizable Choquet simplex occurs as the set of normalized traces of a simple unital AF algebra. A simple AF algebra can have both finite and infinite traces, so a finite simple AF algebra need not be algebraically simple. It is shown that a simple AF algebra is algebraically simple if and only if it has no infinite traces, and is stable if and only if it has no finite traces.     

Die Struktur endlicher schwach abgeschlossener Jordanalgebren Teil I. Stetige Jordanalgebren

On the basis of the authors previous paper [6] the properties of the lattice of idempotents in a finite weakly closed Jordan algebra A are examined. The results then yield a unique direct decomposition A=A_d⊕A_c where A_d is a discrete and A_c is a continuous Jordan algebra. An application of Jacobsons coordinatization theorem shows A_c=H(D), D associative, since there exist sufficiently many strongly connected idempotents. Then we prove that D is a uniquely determined continuous real W^* -algebra with finite normal faithful trace.     

The Banach algebra l1(S)

Let S be the free semigroup with a finite or countably infinite set of generators plus an identity. It is shown that there is a natural involution 1 on the convolution Banach algebra l¹(S) such that (l¹(S), 1) has a separating family of finite-dimensional star representations. The star representations of the l¹-algebra of some other semigroups are also considered. The spectrum of every element of l¹(S) which is not a scalar multiple of the identity is shown to be a connected set with interior.     

Locally C1-equivalent algebras

It is shown that a Banach star algebra is a C¹-algebra in an equivalent norm, if each of its commutative closed star subalgebras is a C¹-algebra in an equivalent norm. This theorem has several interesting consequences.     

K-theoretic amenability for SL2(Qp), and the action on the associated tree

It is proved that if a locally compact group G acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable, then G is K-amenable. In particular, the canonical map from the full C¹-algebra onto the reduced C¹-algebra of G induces isomorphisms in K-theory. The main corollary of our result is that SL₂(Q_p) and some other groups over local fields are K-amenable.     

On the continuity of the map ϑ → ¦ϑ¦ from the predual of a W1-algebra

Let ?, ψ be elements in the predual of a W¹-algebra. For their absolute value parts ¦?¦, ¦ψ¦, the estimate $\text{∥¦?¦ ? ¦ψ¦∥ ? (2 ∥? + ψ∥ ∥? ? ψ∥)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is obtained.     

Equivalence and KMS states on periodic C1-dynamical systems

Using ideas from the earlier work [2], we show that for a C¹-algebra with periodic dynamics there is for each β > 0 an equivalence relation on the fixed-point algebra, which regulates the β-KMS states on the algebra. In particular, the algebra has a separating family of β-KMS states if and only if the fixed-point algebra is finite with respect to the equivalence.     

On operator monotone and operator convex functions

We establish monotonicity and convexity criteria for a continuous function f: R⁺ → R with respect to any C*-algebra. We obtain an estimate for the measure of noncompactness of the difference of products of the elements of a W*-algebra. We also give a commutativity criterion for a positive τ-measurable operator and a positive operator from a von Neumann algebra.     

Isomorphism of Hilbert modules over stably finite C-algebras

It is shown that if A is a stably finite C^∗-algebra and E is a countably generated Hilbert A-module, then E gives rise to a compact element of the Cuntz semigroup if and only if E is algebraically finitely generated and projective. It follows that if E and F are equivalent in the sense of Coward, Elliott and Ivanescu (CEI) and E is algebraically finitely generated and projective, then E and F are isomorphic. In contrast to this, we exhibit two CEI-equivalent Hilbert modules over a stably finite C^∗-algebra that are not isomorphic.