With the following
operator space structures , is completely isometrically
isomorphic to an operator algebra:
,
,
,
,
.
More generally: The space is completely isometrically isomorphic to an operator algebra, if endowed with an operator space
structure which
dominate
dominates both
and
.
With the following operator space structures, is not completely isomorphic to an operator algebra:
,
.
More generally: The space is not completely isomorphic to an operator algebra,
if endowed with an operator space structure which is
dominate
dominated by both
and
.
In the extreme cases resp.
, we have two opposite results [BLM95, Thm. 3.1]:
For
the following holds true (cf. [BLM95, Thm. 3.4]):
On the contrary, the operator space structure on the spaces obtained
via complex
interpolation
always defines an operator algebra structure.
More precisely [BLM95, Cor. 3.3]:
For each
,
is completely isometrically
isomorphic to an operator algebra.
We write for the operator space structure defined on
by G. Pisier. This operator space structure is obtained by
complex interpolation between
and
.
Caution is advised: The space (and likewise
), whether endowed with the usual or the Schur product,
is not completely isomorphic to an operator algebra [BLM95, Thm. 6.3].