Haskell Rosenthal's Research

I. The complete isomorphic structure of operator spaces.

An operator space is a closed linear subspace X of B(H) along with the natural spatial tensor norm induced on K X, where K denotes the C*-algebra of compact operators on a separable infinite dimensional Hilbert space H. The fundamental morphisms are then the completely bounded maps, i.e., those linear operators between operator spaces whose tensor product with the identity map on K is bounded. The structure of operator spaces with the Complete Separable Complementation Property (CSCP) is studied in [85] and [86]. In particular, it is shown in [86] that K has the CSCP, and that a locally reflexive operator space X has the CSCP iff it is completely complemented in any locally reflexive separable superspace Y of X, for a fixed complete embedding of X in B(H) with X Y B(H). A profound open question: If X has the CSCP, does $$X completely embed in K? Perhaps a natural starting point, here, is to assume in addition that X completely embeds in a nuclear C*-algebra.

Some structural results concerning the preduals of von Neumann algebras (i.e., non-commutative $$L¹-spaces) are obtained in [96]. Note that these objects form a natural family of operator spaces. The main motivating special case proved there: $$C₁ (the spaces of trace class operators) does not Banach embed in $$R_*, (the predual of the type $II$₁ hyperfinite factor $script\; R$). Further structural problems concerning non-commutative $$L¹-spaces are under current investigation. For example, if script N, script M are factors of type II and III respectively, is it so that their preduals script N$$_* and script M$$_* are not Banach isomorphic? At least not completely isomorphic? If , does every (closed infinite-dimensional) subspace of a non-commutative $$L^p-space contain a further subspace isomorphic to for some q? (The commutative analogue is valid; a deep discovery of D. Aldous.)

II. Invariant subspaces for certain classes of operators.

I'm currently investigating the following deep open problem: Let X be a separable (infinite dimensional) reflexive Banach space and $script\; A$ a weak-operator-topology closed proper subalgebra of $$B(X). Does $script\; A$ have a non-trivial invariant subspace? In a different direction, the work in [94] initiates the study of operators with substantial invariant subspaces, that is, invariant (closed) subspaces of infinite dimension and codimension.

III. Complemented subspaces of $$C([0,1])$and$ $$L¹([0,1]); quantized analogues.

The main open problems here (long-standing and quite famous):

1. Is every complemented subspace of $$C([0,1]) isomorphic to $$C(K) for some compact metric space $$K?

2. Is every complemented infinite-dimensional subspace of $$L¹ isomorphic to or $$L¹?

I have no idea about the answer to 1., but strongly conjecture the answer to 2. is negative.

The quantized (more general!) analogues go as follows:

1'. Is every completely complemented subspace of a separable nuclear C*-algebra completely isomorphic to a completely contractively complemented subspace?

2'. Same as 1', but replace ``nuclear C*-algebra'' by ``predual of a hyperfinite von Neumann algebra (which acts on separable Hilbert space)''.

IV. The structure of super-complete Banach spaces.

A Banach space X is called super complete if every Banach space finitely represented in X is weakly sequentially complete. The ``-Theorem'' (see [30],[41]) and the recent work on ``$$l-wide-(s) sequences'' (see [83]) yield an intrinsic characterization which is not entirely of local character ([94]). Call X locally super-complete if for all $$l> 1, there is a $$b >1 so that for all k, there is an $$n(k)> k, so that every ``$$l-wide-(s)-wide-(s) sequence in X of length n(k) contains a subsequence of length k. I conjecture this concept is not equivalent to super-completeness in general, but is equivalent for Banach lattices X.

V. Further characterizations of Banach spaces containing $$c₀, and the structure of Banach spaces $$X containing $$c₀.

Two apparently rather deep open problems go as follows.

1. Assume X is non-reflexive and for all non-reflexive subspaces Y of X, there exists a non-reflexive subspace Z of Y with Z* weakly sequentially complete. Does $$c₀ embed in $$X? Note one of the main motivating results in [78]: if X is non-reflexive and every subspace has weakly sequentially complete dual, then $$c₀ embeds in $$X.

2. Assume X is separable, let K denote the ball of X* in its weak* topology, and assume for every $$h < w₁ , there is an f in X** X whose ``non-D oscillation index'' $$i_ND(f|K), is larger than $$h. Does $$c₀ embed in X? Actually, it is not known if these hypotheses imply that X is universal; i.e., does C([0,1]) then embed in X?

Further invariants concerning non D-functions of high index and their associated Banach algebras are studied in [87]; these are clearly profoundly related to problem 2.

Complete bibliography.