### Souvenirs from Bangalore

I recently returned from the two week long workshop and conference Recent Advances in Operator Theory and Operator Algebras which took place in ISI Bangalore. As I promised myself before going, I was on the look-out for something new to be excited about and to learn. The event (beautifully organized and run) was made of two parts: a workshop, which was a one week mini-school on several topics (see here for topics) and a one week conference. It was very very broad, and there were several talks (or informal discussions) which I plan to pursue further.

In this post and also perhaps in a future one I will try to work out (for my own benefit, mostly) some details of a small part of the research presented in two of the talks. The first part is the Superproduct Systems which arise in the theory of E_0-semigroups on type II_1 factors (following the talk of R. Srinivasan). The second (which I will not discuss here, but perhpas in the future) is the equivalence between the Baby Corona Theorem and the Full Corona Theorem (following the mini-course given by B. Wick). In neither case will I describe the most important aspect of the work, but something that I felt was urgent for me to learn.

#### Superproduct systems from E_0-semigroups

An E_0-semigroup is (for the limited purposes of this post) a family $\{\alpha_t\}_{t\geq 0}$ of unital, normal *-endomorphisms on a von Neumann algebra $M$ such that

1. $\alpha_s \circ \alpha_t = \alpha_{s+t}$
2. $\alpha_0 = id_M$
3. For all $a \in M$, the path $t \mapsto \alpha_t(a)$ is continuous (in the sigma weak topology).

Perhaps the most natural notion of equivalence of E_0 semigroup is the following.

Definition 1: Two E_0-semigroups $\alpha, \beta$ on two von Neumann algebras $M$ and $N$ are said to be conjugate if there is a *-isomorphism $\theta : M \rightarrow N$ such that $\alpha_t = \theta^{-1} \circ \beta_t \circ \theta$ for all t.

Though this is a very natural notion of equivalence, the task of classifying E_0-semigroups up to this equivalence is considered hopeless. A coarser equivalence relation is given along the following lines.

Definition 2: Let $\alpha$ be an E_0-semigroup on M. A strongly continuous family of unitaries $\{U_t \}_{t \geq 0}$ (in M) is said to be a cocycle for $\alpha$ (or an $\alpha$ cocycle) if for all $s,t$,

$U_s \alpha_s(U_t) = U_{s+t} .$

Given a cocycle $U$ for $\alpha$, we may define another E_0-semigroup $\alpha^U$ by

$\alpha^U_t(a) = U_t \alpha_t(a) U_t^* .$

It is easy to check that $\alpha^U$ is also an E_0-semigroup.

Definition 3: Two E_0-semigroups $\alpha, \beta$ are said to be cocycle conjugate (or cocycle equivalent) if there exists an $\alpha$ cocycle $U$ such that $\beta$ is conjugate to $\alpha^U$.

The theory of E_0-semigroups started in 1988 with this paper of Powers, and since then tremendous progress has been made in understanding E_0-semigroups which act type I factors (see this monograph for much of the theory on type I factors). Within some limited and special classes of E_0-semigroups acting on type I factors, the classification up to cocycle conjugacy is understood. There has also been much work on E_0-semigroups on general von Neumann and even C*-algebras, however, there has not been much work which has been specific to a certain type of von Neumann algebras other than type I. The paper “Invariants for E_0-semigroups on II_1 factors” by Oliver Margetts and R. Srinivasan that was presented in the conference concentrates (as it’s name suggests) on E_0-semigroups on II_1 factors (the paper cites some previous work on this, like this work of Alevras, I do not know if all is covered).

The authors introduce several new invariants for E_0-semigroups. The one that drew my attention and spurred me to take a closer look at their paper is the superproduct systems (to be described below). A superproduct system is a family of Hilbert spaces that behaves in a nice way under tensor products. Similar constructions — product systems and subproduct systems — have appeared in the past, and I also used them in my research. Moreover, I have already seen such a structure (superproduct system) crop  up in various places: in a conference talk by Claus Kostler (he suggested this terminology) or in my joint work with Michael Skeide. However, I personally did not feel that I understand their role and how to use them, so I am happy that for the first time they are treated in a systematic way and put to good effect. It is also interesting that the way that superproduct systems appear in Margett and Srinivasan’s paper is different from how I saw it arise in different situations. Let me re-iterate that this is not the main goal of their paper and much more is contained in it, but this is one thing that I would like to understand quite urgently.

Roughly speaking, consider a system $E = \{E_t\}_{t\geq 0}$ of Hilbert spaces. It is said to be a product system if

(*) $E_s \otimes E_t = E_{s+t}$

for all s,t. If $E_s \otimes E_t \supseteq E_{s+t}$ then $E$ is said to be a subproduct system; and if $E_s \otimes E_t \subseteq E_{s+t}$ then it is said to be a superproduct system.

Now let me be more precise. The following constructions follow closely Arveson’s original definition of the concrete product systems associated with an E_0-semigroup on a type I factor, and were developed in the type II_1 setting in the paper of Alexis Alevras (here) and, in a setting of greater generality, in the paper of Muhly and Solel (here). We will assume that the II_1 factor $M$ is represented standardly on $L^2(M)$. Given an E_0-semigroup $\alpha$ on M, we define, for every t, the intertwining space:

$E_t = \{T \in B(L^2(M)) : \forall m \in M . \alpha_t(m) T = T m \}.$

Note that $E_t$ is taken to be a subspace of the bounded operators on $L^2(M)$, not a subspace of $M$. If $S,T \in E_t$, then we compute for all $m \in M$:

$S^*T m = S^* \alpha_t(m) T = m S^* T.$

Thus $S^* T \in M'$, the commutant of $M$. Thus $E_t$ carries an $M'$-valued “inner product” $\langle S, T \rangle = S^* T$. It turns out that $E_t$ is in fact what is known as a W*-correspondence, which is an $M'$ valued inner product space which is also a bimodule over $M'$.

There is more structure. If $S \in E_s$ and $T \in E_t$ then

(**) $ST \alpha_{s+t}(m) = ST \alpha_t(\alpha_s(m)) =S \alpha_s(m) T = m ST,$

so $ST \in E_{s+t}$. We may define a tensor product $E_s \otimes E_t$ by completing the algebraic tensor product $E_s \otimes_{alg} E_t$ with respect to the $M'$ valued inner product

$\langle S_1 \otimes T_1, S_2 \otimes T_2 \rangle = \langle T_1, \langle S_1, S_2 \rangle T_2 \rangle .$

(Actually, one completes further by a weaker topology, but that’s a detail I don’t want to go into).

The map $U_{s,t} : (S,T) \mapsto ST$ extends to an isomorphism from $E_s \otimes E_t$ onto $E_{s+t}$, and we get equation (*) from above, only that we are dealing with a product system of $M'$ correspondences, rather than Hilbert spaces. Note: it was a simple computation (eq. (**) above) that showed that this map goes into $E_{s+t}$, but onto requires a little more work.  The maps $U_{s,t}$ also compose in an associative way, making the family $\{E_t \}_{t\geq 0}$ behave somewhat like a semigroup.

The product system $E$ constructed above from an E_0-semigroup is a cocycle conjugacy invairant: two E_0-semigroups are  cocycle conjugate if and only if their product systems are isomorphic (this is proved in the paper by Alevras mentioned above). Here, isomorphism of product systems is defined in the obvious way: $E = (\{E_t\}, U_{s,t})$ and $F = (\{F_t\}, V_{s,t})$ are said to be isomorphic if there is a family of inner product preserving bijective bimodule maps $W_t : E_t \rightarrow F_t$ which respect the product, meaning that $V_{s,t}(W_s \otimes W_t) = W_{s+t}U_{s,t}$.

Thus, one can say that the problem of classification of E_0-semigroups on $M$ up to cocycle conjugacy is reduced to that of classifying product systems of $M'$-correspondences. However, by definition this problem is (at least) just as hard. To be able to distinguish between various concrete examples one often wishes for invariants that are weaker than complete invariants.

Now comes the (I mean one-of-the) innovation introduced in the Margetts–Srinivasan paper. By Tomita-Takesaki theory, there is a surjective antilinear isometry $J$ on $L^2(M)$ which satisfies $JMJ = M'$. This allows the authors to define, given an E_0-semigorup $\alpha$ on $M$, the complementary E_0-semigroup $\alpha'$ on $M'$ by

$\alpha'_t(m') = J\alpha_t(Jm'J)J .$

One can see that $\alpha$ and $\beta$ are cocycle conjugate if and only if $\alpha'$ and $\beta'$ are cocycle conjugate. Now, just as the product system of intertwining spaces $E = \{E_t\}_{t\geq 0}$ was defined above for $\alpha$, one may define a product system $E' = \{E'_t \}_{t\geq 0}$ corresponding to $\alpha'$. The spaces $E'_t$ are W*-correspondences over $(M')' = M$. Then Margetts and Srinivasan go on to define

$H_t = E_t \cap E'_t .$

If $S,T \in H_t$, then $\langle S,T \rangle \in M \cap M' = \mathbb{C} 1$. Therefore, $H_t$ can be endowed with a scalar valued inner product $\langle \cdot,\cdot \rangle_{H_t}$ by way of $S^*T = \langle S,T\rangle_{H_t} 1$. Since $\|T\|^2 = \|T^*T\| = \langle T, T\rangle_{H_t} = \|T\|_{H_t}$, we see that the hilbert space norm and operator norms on $H_t$ coincide, and in particular $H_t$ is a Hilbert space (because $H_t$ is clearly closed in the norm topology).

Now assume that $S \in H_s, T \in H_t$. Then $ST$ is both in $E_{s+t}$ and in $E'_{s+t}$, so $ST \in H_{s+t}$. Moreover, $\|ST\|^2_{H_{s+t}} 1 = T^*S^*ST = T^* \|S\|^2_{H_s} T = \|S\|^2_{H_s} \|T\|_{H_t} = \|S \otimes T\|^2_{H_s \otimes H_t}$. Thus the map $V_{s,t} : H_s \otimes H_t \rightarrow H_{s+t}$ given by $V_{s,t}(S \otimes T) = ST$ is an isometry of $H_s \otimes H_t$ into $H_{s+t}$. The maps $V_{s,t}$ obviously compose in an associative manner, because the operator multiplication is associative. A structure such as this is called a superproduct system.

The authors show that the superproduct system is a cocycle conjugacy invariant. They then go on to compute the superproduct systems for some concrete examples of E_0-semigroups: Clifford flows, even Clifford flows, and free flows. They show that the superproduct system of the Clifford and even Clifford flows are not product systems. They also show that the superproduct system of the free flow is one dimensional (in particular it is a product systems). Thus, these superproduct systems can be used to distinguish between concretely arising E_0-semigroups. The superproduct systems can also be used to calculate another invariant introduced by the authors, the coupling index.

Let me finish this by writing down an example of a proper superproduct system (that is, one which not a product system). These superproduct systems turn out to be the ones arising as the superproduct systems of the Clifford and the even Clifford flows. Let $H$ be a Hilbert space of dimension $n$. Let $L^2((0,\infty);H)$ be the Hilbert space of square integrable $H$-valued functions over the half line, and form the anti-symmetric Fock space $\Gamma_a(L^2((0,\infty);H)$. Now define

$H^{e,n}_t = \overline{span}\{f_1 \wedge f_2 \wedge \cdots \wedge f_{2m} : f_i \in L^2((0,t);H), m \in \mathbb{N}_0\} .$

Define $V_{s,t} : H^{e,n}_s \otimes H^{e,n}_t \rightarrow H^{e,n}_{s+t}$ by

$V_{s,t}((f_1 \wedge \cdots \wedge f_{2k}) \otimes (g_1 \wedge \cdots \wedge g_{2m})) = S_t f_1 \wedge \cdots \wedge S_t f_{2k} \wedge g_1 \wedge \cdots \wedge g_{2m}.$

Here $S_t$ is the unilateral shift on $L^2((0,\infty);H)$, with obvious restriction $S_t : L^2((0,s);H) \rightarrow L^2((0,s+t);H)$. It is readily verifiable that $V_{s,t}$ compose associatively. Also, the definition of the inner product on the antisymmetric Fock space (with the correct normalization $\langle f_1 \wedge \cdots \wedge f_n , g_1 \wedge \cdots \wedge g_n \rangle = det (\langle f_i, g_j \rangle)$) shows that this is an isometry.

$V_{s,t}$ is not onto. Indeed, consider the first summand of $H^{e,n}_1$, call it $F := \overline{span}\{f_1 \wedge f_2 : f_1, f_2 \in L^2((0,1);H)$. Elements in this summand which are in the image of $V_{1/2,1/2}$ must be limits of sums of the form $V_{1/2,1/2}(\xi \otimes \eta)$ where either $\xi$ or $\eta$ is from the $0$th summand in $H^{e,n}_{1/2}$ and the other is from the first summand. So $F \cap Im V_{1/2,1/2}$ is equal to the sum of the spaces $\overline{span}\{f_1 \wedge f_2 : f_i \in L^2((0,1/2);H)\}$ and $\overline{span}\{f_1 \wedge f_2 : f_i \in L^2((1/2,1);H)\}$. But each of this spaces is orthogonal to the element $\chi_{(0,1/2)} \wedge \chi_{(1/2,1)} \in F$, so $V_{1/2,1/2}$ is not onto.