SimpleL*-algebras of classical type

Abstract

Dedicated to Professor C. T. Rajagopal on his 65 th birthday Introduction J. R. Schue has shown in [4] that a separable infinite-dimensional simple L*-algebra is necessarily of one of three classical types A, B, C (the L*-types B, D coinciding). These L*-algebras are the analogues of the corresponding classes of simple Lie algebras. Each of the classical type L*-algebras occurs as an L*-subalgebra of the H*-algebra ofalt Hilbert-Schmidt operators on a separable Hilbert space 90-By replacing 9o by a Hilbert space ~ of arbitrary (infinite) dimension, we obtain analogously the general (i.e. not necessarily separable) simple L*-algebra of type A, or B, or C. However, in the non-separable case, it is not known whether these types exhaust all simple L*-algebras.The object of this paper is to study these general algebras which we shall refer to as "standard simple L*-algebras of classical type". We show that the types A, B, C are distinct in the sense of isomorphism, and prove that there exists, for each infinite cardinal N precisely one standard simple L*-algebra of (orthogonal) dimension N belonging to each of the three types. We also investigate the conjugacy problem for Cartan subalgebras in the case of these algebras. It turns out that for an L*-algebra of type A, or one of type C, any two Caftan subalgebras are conjugate under an L*-automorphism (which is even spatial in the sense of being implemented bY a unitary operator of the underlying Hilbert space). On the other hand, for a type B algebra, the Cartan subalgebras fall into two (conjugacy) classes such that any two in the same class are conjugate while no two from different classes are. The reason for this occurrence of two conjugacy classes in a type B algebra is related to the fact that a conjugation in an infinite-dimensional Hilbert space admits two kinds of representation (see Proposition 2). Finally, we apply the above results to determine the different conjugacy classes for all semi-simple L*-algebras belonging to a certain class which includes all separable algebras. We also deduce that, in the separable case, the conjugacy index (i.e. the number of conjugacy classes) is ~c + I, where ~ is the (cardinal) number of infinite-dimensional components of type B in the decomposition of L into its simple components.

Infinite-dimensional hyperkähler manifolds associated with Hermitian-symmetric affine coadjoint orbits

“…(This result can be found in [4], [13] and [34].) An Hermitian-symmetric space is a smooth strong Riemannian manifold (M, g) endowed with a g-orthogonal complex structure and which admits, for every x in M , a globally defined isometry s x (the symmetry with respect to x) preserving the complex structure, such that x is a fixed point of s x , and such that the differential of s x at x is minus the identity of T x M .…”

On the classification of infinite-dimensional irreducible Hermitian-symmetric affine coadjoint orbits

“…In the sequel, g will denote an infinite-dimensional separable simple L * -algebra of compact type. According to [1], [6] or [17], it can be realized as a subalgebra of the L * -algebra gl 2 (H) consisting of Hilbert-Schmidt operators on a separable complex Hilbert space H. Let G be the connected L * -group with Lie algebra g, and G C the connected complex L * -group with Lie algebra g C := g ⊕ ig. By the duality g ′ = g given by the trace, we can identify affine adjoint and affine coadjoint orbits of G. Let D be a derivation of g such that the affine (co-)adjoint orbit O of 0 in g associated to the affine adjoint action of G defined by D is strongly Kähler (see [10]).…”

Iwasawa decompositions of some infinite-dimensional Lie groups

“…We shall need the following generalization of Proposition 3 in [Ba69]. Then there exists an orthonormal basis {ξ…”

“…and dim R (H R ∩H ± ) = dim C (H ± ) (see for instance Lemma 1 in [Ba69]). Then there exist countable orthonormal bases in the real Hilbert spaces H R ∩ H ± , which we denote by {x ±l } l≥1 , respectively.…”