Re: Is Hol(D) a nice subset of B(H)?

Ali Taghavi <alitghv@xxxxxxxxx> writes:


>Is Hol(D) a nice subspace of B(H)? Can Fred(H) be deformed into Hol(D)??!!:
>Let D be the Unit Compact Disk, Hol(D) is the space of all holomorphic
>maps in a neighborhood of D,

By "a neighborhood", I take you to mean that the neighborhood can vary
with the map, so that more precisely Hol(D) is the set of all *germs*
of holomorphic maps from D to C. If that is the case, how precisely
do you propose to make this set (or algebra) a *space* (presumably,
topological vectorspace or topological algebra)? The last time I
looked into this, I think I learned (but I may remember wrong, or
I could have been mistaken at the time) that if you endow Hol(D)
with its most natural topology, it isn't even metrizable, so it
is not a Frechet space much less (say) a Banach algebra; my notes
say it's a "so-called Silva algebra" and that therefore "its dual
is a Frechet algebra", but I won't swear to either claim (particularly
since at the time I was interested in the analogue where D is replaced
by the closed unit ball in C^2; perhaps here, as often happens,
one complex variable is very different from "several" or even "two"
complex variables in a relevant way).

Alternatively, you might be willing to let Hol(D) be a non-closed
non-Banach subalgebra of the space of all continuous maps from
D to C that are holomorphic on the interior of D, so that Hol(D)
would be metric but very far from complete.

>An element of Hol(D) can be considered as an element of
>B(H), where H=l^2, Can Fred(H) be deformed into Hol(D),
>i.e Put W=Hol(D) intersection Fred(H) is W a deformation
>retract of Fred(H)?

Now you've lost me completely (my fault, not yours).

What end do you have in view?

Lee Rudolph
.