Souvenirs from Bangalore

by Orr Shalit

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 0th 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.