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 $latex {alpha_t}_{tgeq 0}$ of unital, normal *-endomorphisms on a von Neumann algebra $latex M$ such that
- $latex alpha_s circ alpha_t = alpha_{s+t}$
- $latex alpha_0 = id_M$
- For all $latex a in M$, the path $latex t mapsto alpha_t(a)$ 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 $latex alpha, beta$ on two von Neumann algebras $latex M$ and $latex N$ are said to be conjugate if there is a *-isomorphism $latex theta : M rightarrow N$ such that $latex alpha_t = theta^{-1} circ beta_t circ theta$ 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 $latex alpha$ be an E_0-semigroup on M. A strongly continuous family of unitaries $latex {U_t }_{t geq 0}$ (in M) is said to be a cocycle for $latex alpha$ (or an $latex alpha$ cocycle) if for all $latex s,t$,
$latex U_s alpha_s(U_t) = U_{s+t} .$
Given a cocycle $latex U$ for $latex alpha$, we may define another E_0-semigroup $latex alpha^U$ by
$latex alpha^U_t(a) = U_t alpha_t(a) U_t^* .$
It is easy to check that $latex alpha^U$ is also an E_0-semigroup.
Definition 3: Two E_0-semigroups $latex alpha, beta$ are said to be cocycle conjugate (or cocycle equivalent) if there exists an $latex alpha$ cocycle $latex U$ such that $latex beta$ is conjugate to $latex alpha^U$.
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 $latex E = {E_t}_{tgeq 0}$ of Hilbert spaces. It is said to be a product system if
(*) $latex E_s otimes E_t = E_{s+t} $
for all s,t. If $latex E_s otimes E_t supseteq E_{s+t}$ then $latex E$ is said to be a subproduct system; and if $latex E_s otimes E_t subseteq E_{s+t}$ 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 $latex M$ is represented standardly on $latex L^2(M)$. Given an E_0-semigroup $latex alpha$ on M, we define, for every t, the intertwining space:
$latex E_t = {T in B(L^2(M)) : forall m in M . alpha_t(m) T = T m }.$
Note that $latex E_t$ is taken to be a subspace of the bounded operators on $latex L^2(M)$, not a subspace of $latex M$. If $latex S,T in E_t$, then we compute for all $latex m in M$:
$latex S^*T m = S^* alpha_t(m) T = m S^* T.$
Thus $latex S^* T in M’$, the commutant of $latex M$. Thus $latex E_t$ carries an $latex M’$-valued “inner product” $latex langle S, T rangle = S^* T$. It turns out that $latex E_t$ is in fact what is known as a W*-correspondence, which is an $latex M’$ valued inner product space which is also a bimodule over $latex M’$.
There is more structure. If $latex S in E_s$ and $latex T in E_t$ then
(**) $latex ST alpha_{s+t}(m) = ST alpha_t(alpha_s(m)) =S alpha_s(m) T = m ST, $
so $latex ST in E_{s+t}$. We may define a tensor product $latex E_s otimes E_t$ by completing the algebraic tensor product $latex E_s otimes_{alg} E_t$ with respect to the $latex M’$ valued inner product
$latex langle S_1 otimes T_1, S_2 otimes T_2 rangle = langle T_1, langle S_1, S_2 rangle T_2 rangle .$
(Actually, one completes further by a weaker topology, but that’s a detail I don’t want to go into).
The map $latex U_{s,t} : (S,T) mapsto ST$ extends to an isomorphism from $latex E_s otimes E_t$ onto $latex E_{s+t}$, and we get equation (*) from above, only that we are dealing with a product system of $latex M’$ correspondences, rather than Hilbert spaces. Note: it was a simple computation (eq. (**) above) that showed that this map goes into $latex E_{s+t}$, but onto requires a little more work. The maps $latex U_{s,t}$ also compose in an associative way, making the family $latex {E_t }_{tgeq 0}$ behave somewhat like a semigroup.
The product system $latex E$ 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: $latex E = ({E_t}, U_{s,t})$ and $latex F = ({F_t}, V_{s,t})$ are said to be isomorphic if there is a family of inner product preserving bijective bimodule maps $latex W_t : E_t rightarrow F_t$ which respect the product, meaning that $latex V_{s,t}(W_s otimes W_t) = W_{s+t}U_{s,t}$.
Thus, one can say that the problem of classification of E_0-semigroups on $latex M$ up to cocycle conjugacy is reduced to that of classifying product systems of $latex M’$-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 $latex J$ on $latex L^2(M)$ which satisfies $latex JMJ = M’$. This allows the authors to define, given an E_0-semigorup $latex alpha$ on $latex M$, the complementary E_0-semigroup $latex alpha’$ on $latex M’$ by
$latex alpha’_t(m’) = Jalpha_t(Jm’J)J .$
One can see that $latex alpha$ and $latex beta$ are cocycle conjugate if and only if $latex alpha’$ and $latex beta’$ are cocycle conjugate. Now, just as the product system of intertwining spaces $latex E = {E_t}_{tgeq 0}$ was defined above for $latex alpha$, one may define a product system $latex E’ = {E’_t }_{tgeq 0}$ corresponding to $latex alpha’$. The spaces $latex E’_t$ are W*-correspondences over $latex (M’)’ = M$. Then Margetts and Srinivasan go on to define
$latex H_t = E_t cap E’_t .$
If $latex S,T in H_t$, then $latex langle S,T rangle in M cap M’ = mathbb{C} 1$. Therefore, $latex H_t$ can be endowed with a scalar valued inner product $latex langle cdot,cdot rangle_{H_t}$ by way of $latex S^*T = langle S,Trangle_{H_t} 1$. Since $latex |T|^2 = |T^*T| = langle T, Trangle_{H_t} = |T|_{H_t}$, we see that the hilbert space norm and operator norms on $latex H_t$ coincide, and in particular $latex H_t$ is a Hilbert space (because $latex H_t$ is clearly closed in the norm topology).
Now assume that $latex S in H_s, T in H_t$. Then $latex ST$ is both in $latex E_{s+t}$ and in $latex E’_{s+t}$, so $latex ST in H_{s+t}$. Moreover, $latex |ST|^2_{H_{s+t}} 1 = T^*S^*ST = T^* |S|^2_{H_s} T = |S|^2_{H_s} |T|_{H_t} = |S otimes T|^2_{H_s otimes H_t}$. Thus the map $latex V_{s,t} : H_s otimes H_t rightarrow H_{s+t}$ given by $latex V_{s,t}(S otimes T) = ST$ is an isometry of $latex H_s otimes H_t$ into $latex H_{s+t}$. The maps $latex V_{s,t}$ 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 $latex H$ be a Hilbert space of dimension $latex n$. Let $latex L^2((0,infty);H)$ be the Hilbert space of square integrable $latex H$-valued functions over the half line, and form the anti-symmetric Fock space $latex Gamma_a(L^2((0,infty);H)$. Now define
$latex H^{e,n}_t = overline{span}{f_1 wedge f_2 wedge cdots wedge f_{2m} : f_i in L^2((0,t);H), m in mathbb{N}_0} .$
Define $latex V_{s,t} : H^{e,n}_s otimes H^{e,n}_t rightarrow H^{e,n}_{s+t}$ by
$latex V_{s,t}((f_1 wedge cdots wedge f_{2k}) otimes (g_1 wedge cdots wedge g_{2m})) = S_t f_1 wedge cdots wedge S_t f_{2k} wedge g_1 wedge cdots wedge g_{2m}.$
Here $latex S_t$ is the unilateral shift on $latex L^2((0,infty);H)$, with obvious restriction $latex S_t : L^2((0,s);H) rightarrow L^2((0,s+t);H)$. It is readily verifiable that $latex V_{s,t}$ compose associatively. Also, the definition of the inner product on the antisymmetric Fock space (with the correct normalization $latex langle f_1 wedge cdots wedge f_n , g_1 wedge cdots wedge g_n rangle = det (langle f_i, g_j rangle)$) shows that this is an isometry.
$latex V_{s,t}$ is not onto. Indeed, consider the first summand of $latex H^{e,n}_1$, call it $latex F := overline{span}{f_1 wedge f_2 : f_1, f_2 in L^2((0,1);H)$. Elements in this summand which are in the image of $latex V_{1/2,1/2}$ must be limits of sums of the form $latex V_{1/2,1/2}(xi otimes eta)$ where either $latex xi$ or $latex eta$ is from the $latex 0$th summand in $latex H^{e,n}_{1/2}$ and the other is from the first summand. So $latex F cap Im V_{1/2,1/2}$ is equal to the sum of the spaces $latex overline{span}{f_1 wedge f_2 : f_i in L^2((0,1/2);H)}$ and $latex overline{span}{f_1 wedge f_2 : f_i in L^2((1/2,1);H)}$. But each of this spaces is orthogonal to the element $latex chi_{(0,1/2)} wedge chi_{(1/2,1)} in F$, so $latex V_{1/2,1/2}$ is not onto.