Re: George Mackey, 1916-2006

[Michael Barr <mbarr@math.mcgill.ca>]

I had not heard that Mackey had died and am also saddened, although I
never met him.

Bill does not mention that Mackey's thesis, I think it was published in
the same volume of the Transactions as "General Theory of natural
equivalances", was the direct source of the Chu construction.  As Bill
mentioned the category of pairs embeds the category of Banach spaces
(and continuous linear maps).  It is in fact equivalent to the category of
what are now called Mackey spaces, which are characterized as the locally
convex topological vector spaces that have the finest possible topology
for their set of continuous linear functionals.

I once wrote to Mackey asking him if he had had any intention of
considering a space of linear functionals as a _replacement_ for a
topology or merely an adjunct to it.  Unfortunately, he did not reply,
even to a snail mail "letter" (as we used to call them).

As it happens it was just yesterday that I sent off the followin abstract
of the talk I will give in the category session of the Canadian Math. Soc.
meeting in Calgary in early June:


A standard theorem says that any locally convex topological vector space
has a finer topology, its \emph{Mackey topology} with the same set of
continuous linear functionals and that is the finest possible topology
with that property.  If $E$ and $F$ are two such spaces topologize the
space $\mbox{Hom(E,F)}$ of continuous linear transformations $E\to F$
with the weak topology induced by the algebraic tensor product $E\otimes
F'$ and then let $[E,F]$ denote the associated Mackey topology.  Let
$F^*$ denote the dual $F'$ topologized by the Mackey topology on the
weak dual and let $E\otimes F=[E,F^*]^*$ (whose underlying vector space
is the algebraic tensor product).  Then for any Mackey spaces $E$, $F$,
and $G$,
 \begin{enumerate}
 \item $[E\otimes F,G]\cong [E,[F,G]]$
 \item $E\cong E^*{}^*$
 \item $[E,F]\cong (E\otimes F^*)^*$
\end{enumerate}
 which is summarized by saying that the category of Mackey spaces and
continuous linear transformations is $*$-autonomous.

This category is equivalent to the category of weakly topologized
locally convex topological vector spaces (which have the coarsest
possible topology for their set of continuous linear functionals) which
is therefore also $*$-autonomous.  They are also equivalent to the chu
category of vector spaces (which will be explained).

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On Thu, 23 Mar 2006, F W Lawvere wrote:

>
> Dear friends,
>
>     Together with many others, I am deeply saddened by the death of
> George Mackey. For far too long, I had been delaying the trip to see him
> and continue our discussions which had started in the "Weyl'sche Kammer"
> at the ETH in Zurich and continued in the physics center in Trieste. Long
> before I met him, his insights into mathematics and into quantum mechanics
> had been informing my own thinking. It was the study of his book on
> quantum mechanics in 1967 which led directly to the joint course by
> Saunders Mac Lane and me at the University of Chicago. But his relation to
> category theory goes back much further than that, as Saunders and Sammy
> had explained to me.
>
>     George Mackey's Ph.D. thesis displayed remarkable thinking of a
> categorical nature, even before categories had been defined. Specifically,
> the fact that the category of Banach spaces and continuous linear maps is
> fully embedded into a category of pairings of abstract vector spaces,
> together with the definition and use of "Mackey convergence" of a sequence
> in a "bornological" vector space were discovered there and have played a
> basic role in some form in nearly every book on functional analysis since.
> What is perhaps unfortunately not clarified in nearly every book on
> functional analysis, is that these concepts are intensely categorical in
> character and that further enlightenment would result if they were so
> clarified.
>
>    And who, despite initial skepticism, permitted the first paper giving
> an exposition of the theory of categories to see the light of day in the
> Transactions of the AMS in 1945? None other than the referee, George
> Whitelaw Mackey.
>
>    Sincerely,
> 		F. William Lawvere
>
>
> ************************************************************
> F. William Lawvere
> Mathematics Department, State University of New York
> 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA
> Tel. 716-645-6284
> HOMEPAGE:  http://www.acsu.buffalo.edu/~wlawvere
> ************************************************************
>
>
>
>
>