[A free electronic version of this thesis is available from the author's home page.
Individual parts will appear on arXiv a few months after May 2011.]
\let\sqtimes\odot\def\L{{\rm L}}\def\C{{\bf C}}
\def\Hom{\mathop{\rm Hom}}\def\End{\mathop{\rm End}}
\def\op{{\rm op}}\def\Z{{\rm Z}}
The classical $\L_p$-spaces were introduced by Riesz in~1910.
Even earlier Rogers in~1888 and H\"older in~1889
proved the fundamental inequality for $\L_p$-spaces:
If $f\in\L_p$ and $g\in\L_q$, then $fg\in\L_{p+q}$ and $\|fg\|\le\|f\|\cdot\|g\|$.
(Here we denote $\L_p:=\L^{1/p}$, in particular $\L_0=\L^\infty$, $\L_{1/2}=\L^2$, and $\L_1=\L^1$.
The necessity of such a change is clear once we start exploring
the algebraic structure of $\L_p$-spaces.
In particular, the above result essentially states that $\L_p$-spaces form a graded algebra,
which would be completely wrong in the traditional notation.)
Even though the original definition of $\L_p$-spaces depends on the choice of a measure,
we can easily get rid of this choice by identifying $\L_p$-spaces for different
measures using the Radon-Nikodym theorem.
A variant of the Gelfand-Neumark theorem says that the category of measurable spaces
is contravariantly equivalent to the category of commutative von Neumann algebras
via the functor that sends a measurable space to its algebra of bounded measurable functions.
Measure theory can be reformulated exclusively in terms of commutative von Neumann algebras.
We can now drop the commutativity condition and ask what theorems of
measure theory can be extended to the noncommutative case.
This was first done by von Neumann in 1929.
Even though some aspects of measure theory such as the theory of $\L_1$-space
(also known as the predual) were worked out fairly quickly,
and $\L_p$-spaces were defined for some special cases
like bounded operators on Hilbert spaces (Schatten-von Neumann classes),
it took 50 years before in~1979 Haagerup defined noncommutative $\L_p$-spaces
for arbitrary von Neumann algebras.
Just as in the commutative case,
noncommutative $\L_p$-spaces form a unital *-algebra
graded by complex numbers with a nonnegative real part.
In particular, every noncommutative $\L_p$-space of a von Neumann algebra~$M$
is an $M$-$M$-bimodule because $\L_0(M)=M$.
The operations of the unital *-algebra mentioned above
together with the appropriate form of functional calculus can be used
to define a norm (or a quasi-norm if $\Re p>1$) on $\L_p$-spaces,
which coincides with the usual norm in the commutative case.
In~1984 Kosaki extended the classical inequality by Rogers and H\"older
to Haagerup's noncommutative $\L_p$-spaces equipped with the (quasi-)norm mentioned above.
Even though Kosaki's theorem closed the question of extending H\"older's inequality
to the noncommutative case, many algebraic questions remained.
For example, consider the multiplication map $\L_p(M)\otimes_M\L_q(M)\to\L_{p+q}(M)$.
What is its kernel and cokernel?
What kind of tensor product should we use and should we complete it?
Amazingly enough it turns out that this map is an isomorphism if we use the usual
algebraic tensor product without any kind of completion.
In particular, the algebraic tensor product $\L_p(M)\otimes_M\L_q(M)$
is automatically complete.
If we equip it with the projective (quasi-)norm, then the multiplication map
becomes an isometry.
A similar result is true for another form of multiplication map:
The map $\L_p(M)\to\Hom_M(\L_q(M),\L_{p+q}(M))$ ($x\mapsto(y\mapsto xy)$)
is an isometric isomorphism,
where $\Hom_M$ denotes the space of $M$-linear algebraic homomorphisms
without any kind of continuity restrictions.
In particular, every element of $\Hom_M(\L_q(M),\L_{p+q}(M))$ is automatically bounded.
The above two theorems form the first main result of the dissertation.
(The result for Hom for the case of {\it bounded\/} homomorphisms was proved
by Junge and Sherman in 2005, but the automatic continuity part is new.)
We summarize the above results as follows:
\proclaim Theorem.
For any von Neumann algebra~$M$ and for any complex numbers $a$~and~$b$
with a nonnegative real part
the multiplication map $\L_a(M)\otimes_M\L_b(M)\to\L_{a+b}(M)$
and the left multiplication map $\L_a(M)\to\Hom_M(\L_b(M),\L_{a+b}(M))$
are isometric isomorphisms of (quasi-)Banach $M$-$M$-bimodules.
Here $\otimes_M$ denotes the algebraic tensor product (without any kind of completion)
and $\Hom_M$ denotes the algebraic inner hom (without any kind of continuity restriction).
The second main result of this dissertation is concerned with extension of the above
results to {\it $\L_p$-modules}, which were defined by Junge and Sherman in~2005.
An $\L_p(M)$-module over a von Neumann algebra~$M$ is an algebraic $M$-module
equipped with an inner product with values in $\L_{2p}(M)$.
A typical example is given by the space $\L_p(M)$ itself with the inner product $(x,y)=x^*y$.
The phenomena of automatic completeness and continuity extend to $\L_p$-modules.
In particular, if $X$ is an $\L_p(M)$-module, then $X\otimes_M\L_q(M)$
is an $\L_q(M)$-module.
Similarly, if $Y$ is an $\L_{p+q}(M)$-module, then $\Hom_M(\L_q(M),Y)$ is an $\L_p(M)$-module.
Of particular importance are the cases $p=0$ and $p=1/2$.
$\L_0(M)$-modules are also known as Hilbert W*-modules.
The category of $\L_{1/2}(M)$-modules is equivalent to the category of representations
of~$M$ on Hilbert spaces and the equivalence preserves the underlying algebraic $M$-module.
We can summarize these results as follows:
\proclaim Theorem.
For any von Neumann algebra~$M$ and for any nonnegative real numbers $d$~and~$e$
the category of right $\L_d(M)$-modules is equivalent
to the category of right $\L_{d+e}(M)$-modules.
The equivalences are implemented by the algebraic tensor product
and the algebraic inner hom with~$\L_e(M)$.
In particular, all categories of right $\L_d(M)$-modules are equivalent
to each other and to the category of representations of~$M$ on Hilbert spaces.
This theorem can also be extended to bimodules.
An $M$-$\L_d(N)$-bimodule is a right $\L_d(N)$-module~$X$
equipped with a morphism of von Neumann algebras $M\to\End(X)$.
Here $\End(X)$ denotes the space of all continuous $N$-linear endomorphisms of~$X$.
An $\L_d(M)$-$N$-bimodule is defined similarly.
In particular, the category of $M$-$\L_{1/2}(N)$-bimodules is equivalent
to the category of commuting representations (birepresentations) of $M$~and~$N$ on Hilbert spaces.
The latter category is also equivalent to the category of $\L_{1/2}(M)$-$N$-bimodules.
These equivalences preserve the underlying algebraic $M$-$N$-bimodules,
in particular, every $M$-$\L_{1/2}(N)$-bimodule is also an $\L_{1/2}(M)$-$N$-bimodule.
\proclaim Theorem.
The categories of $\L_d(M)$-$N$-bimodules, $M$-$\L_d(N)$-bimodules,
and commuting representations of $M$~and~$N$ on Hilbert spaces
are all equivalent to each other.
The equivalences for different values of~$d$ are implemented as usual
by the algebraic tensor product and the algebraic inner hom with the relevant space~$\L_e(M)$.
The equivalence between $\L_d(M)$-$N$-bimodules and $M$-$\L_d(N)$-bimodules
is implemented by passing from an $\L_d(M)$-$N$-bimodule
to an $\L_{1/2}(M)$-$N$-bimodule, then reinterpreting the latter module
as an $M$-$N$-birepresentation,
then passing to an $M$-$\L_{1/2}(N)$-bimodule, and finally passing to an $M$-$\L_d(N)$-bimodule.
(A weaker form of this result relating $\L_0$-bimodules
and birepresentations was proved earlier by Baillet, Denizeau, and Havet,
who used the completed tensor product and the continuous inner hom.)
Note that passing from $\L_d(M)$-$N$-bimodule to $M$-$\L_d(N)$-bimodule can completely
change the underlying algebraic bimodule structure.
For example, take $d=0$, $M=\C$ (the field of complex numbers)
and $N=B(H)$ for some Hilbert space~$H$.
Then $B(H)$ is a $\C$-$\L_0(B(H))$-bimodule.
The corresponding $\L_0(\C)$-$B(H)$-bimodule is $\L_{1/2}(B(H))$
(the space of Hilbert-Schmidt operators on~$H$), which is completely different from~$B(H)$.
The above equivalences allow us to pass freely between different categories,
choosing whatever category is the most convenient for the current problem.
For example, Connes fusion can be most easily defined for $M$-$\L_0(N)$-bimodules,
where it is simply the completed tensor product.
In fact, the easiest way to define the ``classical'' Connes fusion
(Connes fusion of birepresentations) is to pass from birepresentations of $M$~and~$N$
to $M$-$\L_{1/2}(N)$-modules, then to $M$-$\L_0(N)$-modules,
then compute the completed tensor product, and then pass back to birepresentations.
One of the reasons for studying $\L_d$-bimodules is that they form a target
category for $2|1$-dimensional Euclidean field theories,
which conjecturally describe the cohomology theory known as TMF (topological modular forms).
More precisely, a Euclidean field theory is a 2-functor from a certain
2-category of $2|1$-dimensional Euclidean bordisms to some algebraic 2-category,
which in this case should consist of algebras, bimodules, and intertwiners of some sort.
Thus we are naturally forced to organize
von Neumann algebras, right $\L_d$-bimodules, and their morphisms
into some sort of a 2-category (more precisely, a framed double category):
\proclaim Theorem.
There is a framed double category whose category of objects is the category
of von Neumann algebras and their isomorphisms
and the category of morphisms is the category of right $\L_d$-bimodules (for all values of~$d$)
and their morphisms.
The composition of morphisms is given by the Connes fusion (i.e., the completed tensor product)
of bimodules.
However, one important aspect of Euclidean field theories is still missing
from our description.
Namely, Euclidean field theories are {\it symmetric monoidal\/} functors,
where the symmetric monoidal structure on bordisms is given by the disjoint union
and on the target category it should come from some kind of tensor product.
Thus we are naturally led into the question of constructing a suitable symmetric monoidal
structure on the double category of von Neumann algebras and bimodules.
This involves constructing a tensor product of von Neumann algebras
and an {\it external\/} tensor product of bimodules (which should not be confused
with the {\it internal\/} tensor product of bimodules, i.e., the Connes fusion).
In terms of pure algebra, the external tensor product takes an $M$-$N$-bimodule~$X$ and
a $P$-$Q$-bimodule~$Y$ and spits out an $M\otimes P$-$N\otimes Q$-bimodule~$X\sqtimes Y$.
(The internal tensor product takes an $L$-$M$-bimodule~$X$ and an $M$-$N$-bimodule~$Y$
and spits out an $L$-$N$-bimodule~$X\otimes_MY$.)
Na\"\i vely, one might expect that the usual spatial tensor product of von Neumann algebras
combined with the spatial external tensor product of bimodules should suffice.
Unfortunately, the resulting symmetric monoidal structure is not flexible enough.
In particular, we often need to move actions around, i.e.,
we want to be able to pass from an $L\otimes M$-$N$-bimodule to an $L$-$M^\op\otimes N$-bimodule
and vice versa.
(Here for simplicity we suppress $\L_d$ from our notation.)
This is not possible with the spatial monoidal structure.
For example, the algebra $M$ itself is an $M$-$M$-bimodule,
but it almost never is a $\C$-$M^\op\otimes M$-bimodule.
Thus we are forced to look for a different monoidal structure.
It turns out that the relevant tensor product
on the level of algebras was defined by Guichardet in~1966.
We call it the {\it categorical tensor product},
because it has good categorical universal properties.
We construct a new external tensor product of bimodules,
whose properties can be summarized as follows:
\proclaim Theorem.
The symmetric monoidal category of von Neumann algebras and their isomorphisms
equipped with the categorical tensor product
together with the symmetric monoidal category of $\L_0$-bimodules and their morphisms
equipped with the categorical external tensor product
form a symmetric monoidal framed double category.
This symmetric monoidal structure has good properties,
in particular, we can move actions around.
In fact, every von Neumann algebra is {\it dualizable\/} in this monoidal structure:
\proclaim Theorem.
In the above symmetric monoidal framed double category
every von Neumann algebra~$M$ is dualizable, with the dual von Neumann algebra being~$M^\op$,
the unit morphism being $\L_{1/2}(M)$ as a $\C$-$M^\op\otimes M$-birepresentation
(more precisely, we take the corresponding $\L_0$-bimodule)
and the counit morphism being $\L_{1/2}(M)$ as an $M\otimes M^\op$-$\C$-birepresentation.
We can compute categorified traces (shadows)
of arbitrary endomorphisms of any dualizable object in a symmetric monoidal double category
(or a bicategory)
in the same way we compute the trace of a dualizable object in a symmetric monoidal category.
In our case we can compute the shadow of any $A$-$A$-bimodule,
which turns out to be a $\C$-$\C$-bimodule, i.e., a complex vector space.
Of particular importance are the shadows of identity bimodules:
\proclaim Theorem.
For any von Neumann algebra~$M$ the shadow of~$M$ as an $M$-$M$-bimodule
is isomorphic to $\L_{1/2}(\Z(M))$, where $\Z(M)$ denotes the center of~$M$.
The general theory developed by Ponto and Shulman allows us to take traces
of arbitrary endomorphisms of dualizable 1-morphisms in any bicategory equipped with a shadow.
It is a well-known fact that dualizable 1-morphisms in the bicategory
of von Neumann algebras, $\L_0$-bimodules, and intertwiners
are precisely {\it finite index bimodules}.
If $f$ is an endomorphism of an $M$-$N$-bimodule,
then the left trace of~$f$ is a morphism $\L_{1/2}(\Z(M))\to\L_{1/2}(\Z(N))$
and the right trace of~$f$ is a morphism in the opposite direction.
Since the right trace is the adjoint of the left trace,
we concentrate exclusively on the left trace.
If $M$ and $N$ are factors, then $\L_{1/2}(\Z(M))=\L_{1/2}(\Z(N))=\C$,
thus the left trace is a number and we recover the classical Jones index
as the trace of the identity endomorphism:
\proclaim Theorem.
If $M$ and $N$ are factors and $X$ is a dualizable $M$-$N$-bimodule,
then the trace of the identity endomorphism of~$X$ is equal to the Jones index of~$X$.
In the general case, the trace of the identity endomorphism is a refinement of the Jones index.
One should think of $M$~and~$N$ as direct integrals of factors (von Neumann
algebras with trivial centers) over the measurable spaces corresponding
to $\Z(M)$~and~$\Z(N)$ respectively.
Then an $M$-$N$-bimodule~$X$ can be decomposed as a direct integral
of bimodules over the corresponding factors
over the product~$W$ of measurable spaces corresponding to $\Z(M)$~and~$\Z(N)$.
Now for every point of~$W$ compute the index of the bimodule over this point,
obtaining thus a function on~$W$.
We should think of this function as the Schwartz kernel of the left trace,
which is an operator $\L_{1/2}(\Z(M))\to\L_{1/2}(\Z(N))$.
The above theorem provides a rigorous foundation for this intuitive picture.
