Classification of contractively complemented Hilbertian operator spaces
We construct some separable infinite dimensional homogeneous
Hilbertian operator spaces which generalize the row and column spaces R
and C. We show that separable infinite-dimensional Hilbertian JC*-triples
are completely isometric to an element of the set of (infinite)
intersections of these spaces. This set includes the operator spaces R,
C, their intersection, and the space spanned by creation operators on the
full anti-symmetric Fock space. In fact, we show that these new spaces
are completely isometric to the space of creation (resp. annihilation)
operators on the anti-symmetric tensors of the Hilbert space. Together
with the finite-dimensional case studied in our earlier paper (169 on this
list), this gives a full operator space classification of all rank-one
JC*-triples in terms of creation and annihilation operator spaces.
We use the above to show that all contractive projections on a C*-algebra
A with infinite dimensional Hilbertian range are ``expansions'' (which we
define precisely) of normal contractive projections from the second dual
of A onto a Hilbertian space which is completely isometric to one of the
four spaces mentioned above. This generalizes the well known result, first
proved for B(H) by Robertson, that all Hilbertian operator spaces that are
completely contractively complemented in a C*-algebra are completely
isometric to R or C. We also compute various completely bounded
Banach-Mazur distances between these spaces.
(This paper has appeared in J. Funct. Anal. 237 (2006), no. 2, 589-616.
The published version has Éric Ricard as third author)
M. P. Neal < nealm@denison.edu >
B. Russo < brusso@math.uci.edu >