Pure semigroups of isometries on Hilbert C⁎ -modules

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Pure semigroups of isometries on Hilbert C ∗ -modules B.V. Rajarama Bhat a,∗ , Michael Skeide b a

Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, R.V. College Post, Bangalore 560059, India b Dipartimento E.G.S.I., Università degli Studi del Molise, Via de Sanctis, 86100 Campobasso, Italy

a r t i c l e

i n f o

Article history: Received 7 December 2014 Accepted 18 May 2015 Available online xxxx Communicated by S. Vaes MSC: 47D06 46L08 46L55 46L53 Keywords: Hilbert C ∗ -modules Semigroups Isometries Shift

a b s t r a c t We show that pure strongly continuous semigroups of adjointable isometries on a Hilbert C ∗ -module are standard right shifts. By counterexamples, we illustrate that the analogy of this result with the classical result on Hilbert spaces by Cooper, cannot be improved further to understand arbitrary semigroups of isometries in the classical way. The counterexamples include a strongly continuous semigroup of non-adjointable isometries, an extension of the standard right shift that is not strongly continuous, and a strongly continuous semigroup of adjointable isometries that does not admit a decomposition into a maximal unitary part and a pure part. © 2015 Published by Elsevier Inc.

1. Introduction The following classical result characterizes one-parameter semigroups (in the sequel, semigroups) of isometries on Hilbert spaces; see the standard textbook of Sz.-Nagy and Foias [11, Theorem 9.3] or Sz.-Nagy [10] for a short self-contained treatment. * Corresponding author. E-mail addresses: [email protected] (B.V.R. Bhat), [email protected] (M. Skeide). http://dx.doi.org/10.1016/j.jfa.2015.05.012 0022-1236/© 2015 Published by Elsevier Inc.

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Theorem 1.1. (Cooper [4].) Let S = {St }t∈R+ be a strongly continuous semigroup of isometries on a Hilbert space H. Then H = Hu ⊕ Hp where Hu and Hp are unique subspaces such that: 1. Hu reduces S to a semigroup of unitaries. 2. Hp reduces S to a completely nonunitary semigroup. Moreover, the completely nonunitary part is unitarily equivalent to the standard right shift on L2 (R+ , K) for some multiplicity Hilbert space K. It is our aim to prove the appropriately formulated analogue of the last sentence for semigroups of isometries on Hilbert modules (Theorem 1.2, proved in Section 3) and to analyze to what extent we can save statements about decomposition (Section 2). By Ba (E) we denote the C ∗ -algebra of all adjointable maps on a Hilbert module E. Recall that: • A family S = {St }t∈R+ of linear maps on a vector space is a (one-parameter) semigroup if Sr St = Sr+t and if S0 = id, the identity. • A semigroup S on a normed space V is strongly continuous if t → St v is continuous for all v ∈ V . • A linear map St on a (pre-)Hilbert module E (for instance, on a (pre-)Hilbert space) is an isometry if it preserves inner products: St x, St x = x, y for all x, y ∈ E. (It follows that a semigroup of isometries is strongly continuous if and only if t → x, St y is continuous for all x, y ∈ E.) An isometry between Hilbert spaces is, like all bounded linear operators, adjointable, that is, it has an adjoint. An isometry between (pre-)Hilbert modules is adjointable if and only if there exists a projection onto its range, that is, if and only if its range is complemented. • A unitary is a surjective isometry. A semigroup S of isometries is completely nonunitary if there is no nonzero invariant subspace such that the restricted semigroup is unitary; see below. A semigroup S of adjointable isometries is pure if S∗t converges strongly to 0 for t → ∞. (Note that weak convergence to 0 is not enough. Indeed,  the Riemann–Lebesgue lemma asserts that limt→∞ eiλt f (λ) dλ = 0 for every integrable function f . So, the unitary semigroup of multiplication operators eiλt on L2 (R) converges to 0 weakly.) Equivalently, the projections St S∗t converge strongly to 0, respectively, the projections id −St S∗t converge strongly to id. For a semigroup S of isometries on a Hilbert space, the property to be completely nonunitary and the property to be pure are equivalent. See, however, Example 2.5. • A subspace W of a vector space V is invariant for the semigroup S if St W ⊂ W for all t. The semigroup obtained by (co)restriction of S to an invariant subspace W is called the restricted semigroup. (Note that a semigroup of isometries is completely nonunitary if and only if St W = W for all t implies W = {0}.) A (pre-)Hilbert submodule F of a (pre-)Hilbert module E is reducing for the semigroup S on E if

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both F and F ⊥ (see Section 2) are invariant for S. There are two special instances of reducing submodules: Firstly, if S is adjointable and F is invariant also for the adjoint semigroup S∗ . (In general, if F is invariant for S, then S∗t (F ⊥ ), F  = F ⊥ , st F  ⊂ F ⊥ , st F  = {0}, so S∗t (F ⊥ ) ⊂ F ⊥ , that is, F ⊥ is invariant for S∗ . So if F is invariant for S∗ , then F ⊥ is invariant for S = S∗∗ .) Secondly, if the restricted semigroup of a semigroup of isometries S to an invariant subspace F is a unitary semigroup. (Indeed, from St F = F and St being an isometry, we obtain St (F ⊥ ), F  = St (F ⊥ ), St F  = F ⊥ , F  = {0}, so St (F ⊥ ) ⊂ F ⊥ .) For semigroups on Hilbert spaces there are many equivalent characterizations of their closed reducing subspaces, and each closed reducing subspace decomposes the semigroup into a direct sum. (Already for pre-Hilbert spaces that latter statement is false; and regarding geometric aspects like orthogonal complements, Hilbert modules behave much more like pre-Hilbert spaces than like Hilbert spaces.) But only the definition we give here, meets the situations we deal with in Section 2, when we analyze how much of the first part of Theorem 1.1 can be saved. • Let F be a Hilbert module over a C ∗ -algebra B. The standard right shift over F , called multiplicity module, is the semigroup v = {vt }t∈R+ of isometries on L2 (R+ , F ) := L2 (R+ ) ⊗ F defined by  [vt f ](x) =

f (x − t) x ≥ t, 0 else.

Here, the external tensor product L2 (R+ ) ⊗ F of the Hilbert C-module L2 (R+ ) and the Hilbert B-modules F can be identified with the completion of the space   of functions span x → h(x)y | h ∈ Cc (R+ ), y ∈ F (Cc meaning continuous functions with compact support) in the norm arising from the inner product f, g :=  f (x), g(x) dx. Clearly, the standard right shift is strongly continuous. (Indeed, for any bounded strongly continuous functions t → at ∈ B(L2 (R+ )) the function t → at ⊗ idF ∈ Ba (L2 (R+ ) ⊗ F ) is strongly continuous, too, because by boundedness it is sufficient to check strong continuity on the total set of elementary tensors, and because on elementary tensors strong continuity of at ⊗ idF follows from strong continuity of at .) In Section 3, we will prove: Theorem 1.2. Let {St }t≥0 be a pure strongly continuous semigroup of adjointable isometries on a Hilbert module E. Then {St } is unitarily equivalent to the standard right shift on L2 (R+ , F ) over some multiplicity module F . The reader who is interested only in that result may switch to the proof in Section 3, immediately. In the remainder of this introduction, we motivate the proof. We explain briefly why for Hilbert modules we need a fresh proof. In Section 2 we explain by counterexamples, why Theorem 1.2, as compared with Theorem 1.1, is the best we

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may hope for. We explain why we may not hope for a generalization of the Stone–von Neumann theorem to Hilbert modules. In Section 4 we discuss a version von Neumann (or W ∗ -)modules. Motivation of the proof. Suppose we knew that E = L2 (R+ , F ) “in some way” (that is, after choosing a suitable unitary V : L2 (R+ , F ) → E) and that, in this identification, S is the standard right shift (that is, St = V vt V ∗ under the unitary equivalence transform arising from that suitable unitary). The first problem to be faced is, how can we extract F from the abstract space E and the abstract isometry semigroup S? From the solution we propose here, all the other steps will suggest themselves. Note that E contains many copies of F , namely, for each 0 ≤ a < b the submodule II [a,b) F of functions II [a,b) y (where y ∈ F and II S denotes the indicator function of the √ set S), is isomorphic to F via II [a,b) y → y b − a. Moreover, the isomorphism L2 (R+ , F ) ⊃ II [a,b) F −→ F −→ II [a+t,b+t) F ⊂ L2 (R+ , F ) we obtained that way, is nothing but St restricted to II [a,b) F . On the other hand, the elements II [a,b) y (varying also a and b) form a total subset of E. So, understanding how to get them abstractly will, if Theorem 1.2 is true, lead to a proof. First of all, assuming the isometries are adjointable, we easily get the submodules Ea,b := L2 ([a, b), F ) of E. Indeed, the projection St S∗t onto the range of St , is nothing but multiplication with II [t,∞) , and the projection onto Ea,b is ∗ Sa Sa

II [a,b) = II [a,∞) − II [b,∞) =

− Sb S∗b =: pa,b .

So, how to find, inside Ea,b , the elements of the form II [a,b) y? We define ua,b to be the t unitary shift modulo b − a on Ea,b . (This can be done entirely in terms of S; see (3.1).) Then II [a,b) y are precisely those elements that are invariant under that unitary group. We find them by applying the projection 1 qa,b : x −  → b−a

b ua,b t x dt a

to all elements x ∈ Ea,b . (Thanks to strong continuity of S, this integral is a well-defined Riemann integral over a continuous vector-valued function.) In this way, we get elements in Ea,b that behave like elements II [a,b) y. The analytic heart of the proof will be to show that the elements qc,d x (x ∈ Ea,b , [c, d) ⊂ [a, b)) are total in Ea,b . This is done in Lemma 3.8, which asserts that, letting n → ∞, n 

q k−1 , k n

k=1

n

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converges strongly to the identity of E0,1 . The rest is simple reconstruction of the right  ∞ shift on E = k∈N Ek−1,k ∼ out of u0,1 = E0,1 t . (See Lemma 2.4.) The key point in the proof is that we manage (basically by Lemma 3.8) to approximate explicitly in norm an arbitrary element of E by sums over elements that behave like II [a,b) y. No arguments like zero-complement or weakly total = strongly total (which work only for Hilbert spaces) are involved. The original proof for Hilbert spaces involves unbounded operators and adjoints; it does not appear to be generalizable to modules. 2. (Counter)examples and other obstacles The proofs of several statements in the classical Theorem 1.1 for Hilbert spaces rely on several crucial properties and results that are not available for Hilbert modules. The most important are: Self-duality of closed subspaces and, therefore, existence of projections onto them; existence of adjoints; weakly total subsets are norm total. In this section, we explain the consequences of having these pieces missing by counterexamples and prove some weaker statements. The proof of that statement we can confirm fully, Theorem 1.2, has been outlined in the end of the introduction and will be performed in Section 3. Closely related to projections onto closed submodules are the subtleties around orthogonal complements. For convenience, we start by repeating those facts that generalize easily to Hilbert modules. (Of course, orthogonal complements can be defined for arbitrary subsets, and a number of statements remain true also for pre-Hilbert modules and their not necessarily closed submodules, for instance, such as the B-linear span of a subset. We ignore these, here.) • Let F and G be closed submodules of a Hilbert B-module E such that F, G = {0}. Then y + z → y ⊕ z defines an isometry F + G → F ⊕ G. This isometry is, clearly, surjective, that is, a unitary. We denote this situation as F + G = F ⊕ G. Of course, F ∩ G = {0}. • Let F be a closed submodule of a Hilbert B-module E. The orthogonal complement of F is defined as F ⊥ := {x ∈ E: F, x = {0}}. Since, clearly, F, F ⊥  = {0}, we have F + F ⊥ = F ⊕ F ⊥ . (Corollary: F ⊕ F ⊥ = E if and only if there exists a projection p ∈ Ba (E) with pE = F .) Obviously, F1 ⊂ F2 implies F1⊥ ⊃ F2⊥ . • Clearly, F ⊥⊥ ⊃ F . Applying this to F ⊥ we get (F ⊥ )⊥⊥ ⊃ F ⊥ , and by the preceding conclusion we get (F ⊥⊥ )⊥ ⊂ F ⊥ . So, F ⊥⊥⊥ = F ⊥ . (Corollary: If G = F ⊥ for some submodule F , then G⊥⊥ = G. Corollary: An adjointable map a: E → E  is zero on F ⊥⊥ if and only if it is zero on F . Indeed, aF = {0} ⇔ E  , aF  = {0} ⇔ a∗ E  , F  = {0} ⇔ a∗ E  ⊂ F ⊥ = (F ⊥⊥ )⊥ ⇔ a∗ E  , F ⊥⊥  = {0} ⇔ E  , aF ⊥⊥  = {0} ⇔ aF ⊥⊥ = {0}. Note, too, that it appears to be unknown if this statement is true for all bounded right linear maps a. If this was true, one could show that if G

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is a submodule of E containing F and fulfilling F ⊥ ∩ G = {0}, then G⊥ = F ⊥ and, consequently, G⊥⊥ = F ⊥⊥ ⊃ G.1 ) • Since also F ⊥ + F ⊥⊥ = F ⊥ ⊕ F ⊥⊥ , we get (F ⊕ F ⊥ )⊥ = F ⊥ ∩ F ⊥⊥ = {0}. So, F and F ⊥ separate the points of E and the relative orthogonal complement of F in F ⊥⊥ is {0}, that is, F separates the points of F ⊥⊥ . Clearly, F ⊥⊥ is the biggest submodule containing F and orthogonal to F ⊥ (that is, subset of F ⊥⊥ ). So, if we have orthogonal submodules F and G of E and if G = F ⊥ , then F ⊥⊥ ⊕ F ⊥ is the biggest submodule of E that allows the decomposition into a direct sum with summands containing F and G, respectively. (Once more, if the values of bounded right linear maps on F ⊥⊥ should turn out to be determined uniquely by the values on F , it would be possible to show that for two orthogonal submodules F and G separating the points of E (F ⊥ ∩ G⊥ = {0}), F ⊥⊥ ⊕ G⊥⊥ is the unique maximal choice to embed F ⊕ G into a direct sum contained in E.2 Without that, it is only easy to see that also F ⊥⊥ and G⊥⊥ are orthogonal.) As the most general situation, we consider a semigroup S of isometries on a Hilbert module E that is a priori neither adjointable nor strongly continuous. Example 2.1 illustrates, among many peculiarities regarding decomposition, that a strongly continuous semigroup of isometries need not be adjointable. In Observation 2.7 we point out that a ‘surprisingly reasonable’ semigroup of adjointable isometries (namely, an extension of a standard right shift) is not strongly continuous. Example 2.1. Let B be a C ∗ -algebra and let I be a proper closed ideal of B. We mention the following technical result: For any Hilbert space H, the complement of H ⊗I in H ⊗B is (H ⊗ I)⊥ = H ⊗ (I ⊥ ). (Indeed, by means of an ONB (es )s∈S for H, the elements of H ⊗ B are precisely those of the form x = s ee ⊗ bs where the sum s b∗s bs exists. Of course, if all bs are in I ⊥ , then H ⊗ I, x = {0}. If one bs is not I ⊥ , then there exists c ∈ I such that c∗ bs = 0. Consequently, es ⊗ c, x = c∗ bs = 0, so that x is not in the complement of H ⊗ I.) Consequently, H ⊗ I has zero-complement in H ⊗ B if and only if I is essential in B, and H ⊗ I is complemented in H ⊗ B if and only if I is complemented in B. Consider the bilateral right shift ut on L2 (R, B), obviously, a strongly continuous semigroup. Clearly, ut sends E := L2 (R− , I) ⊕ L2 (R+ , B) into E, so that the (co)restrictions St of ut to E define a strongly continuous semigroup of isometries. It is easy to see that an isometry is adjointable if and only if its image is complemented; see, for instance, Skeide [6, Proposition 1.5.13]. So, if I is not complemented 1 Indeed, F ⊂ G ⊂ E, so G⊥ ⊂ F ⊥ (complements in E). Now, F ⊥ ∩ G is the relative complement of F in G. If the last corollary would be true for all a ∈ Br (E, E  ), then Φ ∈ Br (G, B) and ΦF = {0} would imply Φ = 0. So, if for x ∈ E we consider the element x∗  G in Br (G, B), we find that x∗ F = {0} implies x∗ G = {0}, that is, x ∈ F ⊥ implies x ∈ G⊥ . 2 The meaning of this statement is: F, G = {0} and (F + G)⊥ = {0}, then F ⊂ F  , G ⊂ G , and F  , G = {0} imply F  ⊂ F ⊥⊥ and G ⊂ G⊥⊥ . And this statement follows simply by applying the statement proved in the preceding footnote.

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in B, then St E = L2 (R− , I) ⊕ L2 ([0, t], I) ⊕ L2 ([t, ∞), B) is not complemented in E = L2 (R− , I) ⊕ L2 ([0, t], B) ⊕ L2 ([t, ∞), B), so that St is not adjointable. If I is essential in B, then even (St E)⊥ = {0} for all t. It would be an interesting question to examine all strongly continuous semigroups of (not necessarily) adjointable isometries with this property. (If (St E)⊥ = {0} and St is adjointable, then St is a unitary.) On the other hand, if I is complemented in B (so that St is adjointable), then E = L2 (R, I) ⊕ L2 (R+ , I ⊥ ). Both summands are invariant for S. On the first summand, St restricts to a (unitary) bilateral right shift. On the second summand it restricts to a standard right shift over I ⊥ . What about the decomposition stated in Theorem 1.1 in the general situation? Well, let us first speak about the maximal unitary part. Note that St E is a decreasing family of

(closed) submodules of E. Clearly, the (closed!) submodule Eu := t∈R+ St E is invariant for S. Since St E is decreasing, the restriction is a unitary onto Eu . Moreover, if E  is any other invariant closed submodule of E such that the restriction of S is unitary, then E  = St E  ⊂ St E for all t so that E  ⊂ Eu . In other words, Eu carries the unique maximal unitary part of S. In Example 2.1, we get Eu = L2 (R, I), and the maximal unitary part is complementary if and only if I is. Moreover, on the invariant submodule Eu⊥ = L2 (R+ , I ⊥ ) the restriction of S is a pure semigroup of adjointable isometries (already given as a standard right shift). We have E = Eu ⊕ Eu⊥ if and only if I is complemented in B. In particular, if I is essential (and proper), then Eu⊥ is {0} but Eu = E = Eu⊥⊥ . What about Eu⊥ in the general situation? Well, since S restricts to a unitary semigroup on Eu , the submodule Eu is even reducing, that is, also Eu⊥ is invariant for S. We must

have t∈R+ St (Eu⊥ ) = {0}. (Otherwise, this subset of S0 (Eu⊥ ) = Eu⊥ would contribute to

⊥ t∈R+ St E = Eu .) So, the restriction of S to Eu is a completely nonunitary semigroup. Note that, in Example 2.1, the submodule E  := L2 (R+ , I) of Eu fulfills

 ⊥  t∈R+ St E = {0}. So, also the restriction of S to the invariant submodule Eu ⊕ E = 2 ⊥ L (R+ , I ⊕ I ) is a completely nonunitary semigroup. There is no such condition as “the maximal invariant submodule for S such that S restricts to a completely nonunitary semigroup” that can replace pureness. However, (Eu⊥ ⊕ E  )⊥ = L2 (R− , I), which is not invariant for S. So, Eu⊥ ⊕ E  is not reducing. We do not know if Eu⊥ is something like the biggest reducing submodule for S such that the restriction is a completely nonunitary semigroup. (The main problem is to show that if F1 and F2 are invariant submodules such that both restrictions are completely nonunitary semigroups, then F1 + F2 shares this property.) We collect: Proposition 2.2. A semigroup S of isometries on E (neither necessarily adjointable nor necessarily strongly continuous) has a unique maximal unitary part acting on the reducing submodule Eu and the restriction of S to the invariant submodule Eu⊥ is a completely nonunitary semigroup. Eu ⊕ Eu⊥ need not be all of E.

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Note that it is unclear whether Eu⊥⊥ is always invariant, so that both Eu⊥ and Eu⊥⊥ would be reducing, or not. (In Example 2.1, Eu⊥⊥ = L2 (R− , I) ⊕L2 (R+ , I ⊥⊥ ) is invariant for all choices of I. But in an attempt to prove that this is true for all semigroups of isometries, we again bump into the question if a bounded right linear map is determined by its values on a submodule with zero-complement.3 ) However, if Eu⊥⊥ = Eu is invariant for S, then by maximality of Eu , the semigroup restricted to Eu⊥⊥ = Eu is certainly not unitary.  There is an interesting submodule of Eu⊥ , namely, Ep := t (St E)⊥ . (Indeed, since ⊥ St E is not smaller than Eu , the complement (St E) is not bigger than the comple⊥ ment Eu , and this turns over to union and closure.) This submodule is interesting,  because if S is pure so that idE −St S∗t converges strongly to idE , then Ep = t (St E)⊥ =  ∗ t (idE −St St )E = E. And whenever the statements of Theorem 1.1 hold in full, then Eu⊥ = Ep . Generally, also the submodule Ep is invariant for S and the restriction to Ep is a completely nonunitary semigroup of isometries. (Indeed, one easily verifies that  ⊥ ⊥ ⊥ St ((Sr E) ) ⊂ (St+r E) , so that St Ep = r (St (Sr E) ) ⊂ Ep . Of course, the restriction ⊥ of S to any invariant submodule of Eu can only be a completely nonunitary semigroup.) In Example 2.1, we have Ep = Eu⊥ , but in Example 2.5 we will see that this need not be so, not even if S is adjointable. In general, we also do not know, if Ep⊥ = Eu . (Note, however, that if Ep⊥  Eu is invariant for S then the restricted semigroup cannot be unitary, because Eu is maximal unitary.) This situation improves if S is adjointable. In Example 2.1, the restriction of S to Eu⊥ = Ep is a standard right shift. In particular, the restricted St are adjointable independently on whether the original St were adjointable or not. Also here, we do not know if this is true in general for one of the restrictions of ⊥ S to Eu or to Ep . However, if, in the general situation, the St are adjointable from the beginning, then Ep⊥ =

t

(St E)⊥

⊥

=

  (St E)⊥⊥ = St E, t

t

because St E is complemented, so, Ep⊥ = Eu (so that, in particular, Ep is reducing) and, further, Eu⊥⊥ = Ep⊥⊥⊥ = Ep⊥ = Eu . The restriction of S to Ep remains adjointable. (Note that this does not just follow from that Ep is reducing. If the restriction of St is adjointable, then its adjoint has no choice but being the restriction of S∗t . So the question is, whether S∗t leaves Ep invariant, in which case the restriction of S∗t is an adjoint of the restriction of St . Suppose x ∈ (Sr E)⊥ , so that x, Sr y = 0 for all y ∈ E.  So, S∗t x, Sr y = x, Sr (St y) = 0, that is, S∗t x ∈ (Sr E)⊥ . Since r (Sr E)⊥ is dense, S∗t leaves Ep invariant.) The restriction to Ep is pure. (Indeed, to show that S∗t y → 0 for ⊥⊥ Statement: Eu is invariant for St . Proof under the stated hypothesis: We have to show that for all ⊥⊥ ⊥ t ∈ R+ and y ∈ Eu , we get z, St y = 0 for all z ∈ Eu . But since Eu is invariant, the element z, st • of ⊥⊥ Br (E, B) is zero on Eu . By the stated hypothesis, it is zero on Eu . 3

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all y ∈ Ep , by boundedness of S∗t it is sufficient to show that for y from the dense  subset r (Sr E)⊥ . So let y ∈ (Sr E)⊥ for some r. Then 0 = Sr x, y = x, S∗r y for all x ∈ E, hence, S∗r y = 0. It follows that St y = 0 for all t ≥ r.) Moreover, Ep is the unique biggest reducing submodule of E such that the restriction of s to that submodule is a pure semigroup of isometries. (Indeed, since (idE −St S∗t )x ∈ (St E)⊥ , the elements of Ep can be characterized as all those x ∈ E for which (idE −St S∗t )x → x. On the other hand, if F is reducing for S, so that the restriction S has an adjoint S∗ that is the restriction of S∗ , and if the restriction of S to F is pure, then for all x ∈ F we get (idE −St S∗t )x = (idE −St S∗ t )x → x, so F ⊂ Ep .) We collect: Proposition 2.3. For a semigroup S of adjointable isometries on E (not necessarily strongly continuous) Ep is the unique maximal reducing submodule for S such that the restriction is pure, and Ep⊥ = Eu = Eu⊥⊥ (so that Ep⊥ is reducing for S and carries the unique maximal unitary part). Eu ⊕ Ep = Ep⊥ ⊕ Ep has orthogonal complement {0} but (see below) need not be all of E. Clearly, if S is also strongly continuous, then to the part on Ep we may apply Theorem 1.2. It is unclear if, assuming adjointability, Eu ⊕Eu⊥ = Eu⊥⊥ ⊕Ep⊥⊥ ⊃ Eu ⊕Ep = Ep⊥ ⊕Ep can be different from E. (Without adjointability, we know already it need not.) But, we now discuss the promised example that shows that even assuming adjointability, E need not coincide with Eu ⊕Ep . Since in this example Eu = {0}, it also follows that completely nonunitary (Eu = {0}) does not imply pure (Ep = E). We prepare with the following lemma (the second part of which will also be important in the proof in Section 3). Lemma 2.4. Let ut denote the unitary right-shift modulo 1 on L2 [0, 1). Suppose ˘S is an ˘ For t ∈ R+ denote by nt the largest (adjointable) isometry on a Hilbert B-module E. integer ≤ t. 1. The maps St

 n n +1  := (ut ⊗ idF˘ ) II [0,1−(t−nt )) ⊗ ˘S t + II [1−(t−nt ),1) ⊗ ˘S t

˘ define a strongly continuous semigroup of (adjointable) isometries on L2 [0, 1) ⊗ E. ∞ ˘ 2. If E = F and if ˘S: (y1 , y2 , . . .) → (0, y1 , y2 , . . .) is the one-sided right shift, then St , under the canonical isomorphism L2 [0, 1) ⊗ F ∞ ∼ = L2 (R+ ) ⊗ F, = (L2 [0, 1) ⊗ F )∞ ∼ = L2 [0, 1)∞ ⊗ F ∼ is nothing but the standard right shift vt on L2 (R+ ) ⊗ F .

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Proof. Clearly, the St are isometries. They are adjointable if and only if ˘S is adjointable. The function ut is strongly continuous and the projections II [0,1−(t−nt )) and II [1−(t−nt ),1) in B(L2 [0, 1)) depend strongly continuously on t ∈ [n−1, n) for all n ∈ N. So, for the same reason for which the standard right shift vt is strongly continuous as discussed before Theorem 1.2, also St is strongly right continuous. On the other hand, the definition of St does not change (for t > 0), if we replace nt with the largest integer < t. So, St is also strongly left continuous. For showing Part 1 it, therefore, remains to show that the St form a semigroup. n Note that Sn = id ⊗˘S for all n ∈ N0 . (In particular, S0 = id.) Also, St Sn = St+n = Sn St for all t ∈ R+ and n ∈ N0 , is easily verified. So, it is sufficient to verify the semigroup property Sr St = Sr+t for r, t ∈ (0, 1). Note that  II [1−r,1) ut = ut

II [1−r−t,1−t) II [0,1−t) + II [2−r−t,1)

t ≤ 1 − r, t ≥ 1 − r,

hence,  II [0,1−r) ut = (id −II [1−r,1) )ut = ut

II [0,1−r−t) + II [1−t,1) II [1−t,2−r−t)

t ≤ 1 − r, t ≥ 1 − r.

Therefore, Sr St

= (ur ut ⊗ idF˘ ) ⎧ (II [0,1−r−t) + II [1−t,1) )II [0,1−t) ⊗ idF˘ ⎪ ⎪   ⎪ ⎪ + (II [0,1−r−t) + II [1−t,1) )II [1−t,1) + II [1−r−t,1−t) II [0,1−t) ⊗ s˘ ⎪ ⎪ ⎪ ⎨ + II [1−r−t,1−t) II [1−t,1) ⊗ s˘2 × II II [0,1−t) ⊗ idF˘ ⎪ ⎪   ⎪ [1−t,2−r−t) ⎪ ⎪ + II II + (II + II )II ⊗ s˘ [1−t,2−r−t) [1−t,1) [0,1−t) [2−r−t,1) [0,1−t) ⎪ ⎪ ⎩ + (II [0,1−t) + II [2−r−t,1) )II [1−t,1) ⊗ s˘2 = (ur+t ⊗ idF˘ )    II [0,1−r−t) ⊗ idF˘ + II [1−t,1) + II [1−r−t,1−t) ⊗ s˘ + 0 ⊗ s˘2   × 0 ⊗ idF˘ + II [1−t,2−r−t) + II [0,1−t) ⊗ s˘ + II [2−r−t,1) ⊗ s˘2  = (ur+t ⊗ idF˘ )

t ≤ 1 − r,

t≥1−r

t ≤ 1 − r, t≥1−r

II [0,1−r−t) ⊗ idF˘ +II [1−r−t,1) ⊗ s˘ t ≤ 1 − r, II [0,2−r−t) ⊗ s˘ + II [2−r−t,1) ⊗ s˘2 t ≥ 1 − r

= Sr+t in either case. Part 2 is shown best by decomposing L2 (R+ ) ⊗F into a direct sum over L2 [n −1, n) ⊗F (n ∈ N) in which, then, each L2 [n − 1, n) ⊗ F is identified with the n-th summand L2 [0, 1) ⊗ F in the picture (L2 [0, 1) ⊗ F )∞ . It is clear that it is sufficient that St and vt do the same thing for t ∈ [0, 1) and to check it on functions II [a,b) y where either

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[a, b) ⊂ [n − 1, n) remains after shift by t entirely in [n − 1, n) or where it is shifted entirely to [n, n + 1). Since nt = 0, in the first case we remain in the direct summand n corresponding to [n − 1, n) (reflected by ˘S t = idF ∞ in the first summand of St ), while in the second case we end up in the summand corresponding to [n, n + 1) (reflected by ˘Snt +1 = ˘S in the second summand of St ). 2 Example 2.5. The first part of the lemma promises to find an example of a strongly continuous semigroup of adjointable isometries St on a Hilbert module E with Ep⊥ = {0} ˘ such but Ep = E, provided we find an adjointable isometry ˘S on a Hilbert module E  n ˘ ˘p := ˘⊥ ˘ ˘ ⊥ that E n (S E) = E but Ep = {0}. (Indeed, defining E and St as in the lemma, n ˘p = E. we conclude from Sn = id ⊗˘S and from St S∗ decreasing, that Ep = L2 [0, 1) ⊗ E t

The statement about the complement is shown precisely as for I and B in the beginning of Example 2.1.) ˘ := Cb (N, H), the bounded Let v˘ be a proper isometry on a Hilbert space H. Define E H-valued functions on N. It is routine to show that the inner product f, g defined by ˘ into a Hilbert module over B := Cb (N). Then the setting f, g(k) := f (k), g(k) turns E ˘ ˘ [˘Sf ](k) := v˘f (k), is operator ˘S on E defined by pointwise action of v˘ on a function f ∈ E, ∗ an isometry and pointwise action of v˘ is an adjoint. Since v˘ is proper, we may choose an n−1 n ˘ orthonormal family {en }n∈N in H such that en ∈ (˘ v n−1 v˘∗ − v˘n v˘∗ )H. Define f ∈ E n n ∗ ˘p . On the other by f (k) = ek . Then, clearly, limn→∞ ˘S ˘S f does not exist, so f ∈ / E ∗n ˘ hand, if v˘ is pure (that is v˘ converges strongly to 0), then, clearly, Ep⊥ = {0}. Observation 2.6. Note that in the preceding example with pure v˘, it is easy to see that Ep may be identified with H ⊗ B. Since every infinite-dimensional Hilbert space admits a pure isometry, this also shows that in this case Cb (N, H)  H ⊗ Cb (N). One may ask, why in the example, not taking immediately a pure strongly continuous semigroup of isometries v˘t on H (necessarily unitarily equivalent to the standard right shift on some L2 (R+ , K))? In fact, there is no problem to define, then, on the same ˘ = Cb (N, H) a semigroup of adjointable isometries ˘St by pointwise action of v˘t on E ˘ defined by ˘ However, the action of ˘S on the function g ∈ E functions on E. II 1 , 1  1 1 g(k) := y  k+1 k = y k(k + 1)II k+1 ,k 1 1 − k k+1 for some non-zero vector y ∈ K, shows that the semigroup ˘St is not strongly continuous. ˘p = H⊗Cb (N)  It is strongly continuous when restricted to the submodule E / g. The same ˘ ˘ g shows that St is not even B-weakly continuous, that is, t → g, St g is not continuous (in any standard topology of B). Observation 2.7. Note that every semigroup of (not necessarily adjointable) isometries  (‘Dilating’ means that St on E may be ‘dilated’ to a semigroup  St of unitaries on E.  E ⊃ E and that  St (co)restricts to St on E.) Indeed, if we put Et := E and define the

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maps βt,s : Es → Et for all t ≥ s to be St−s , then the Et and βt,s form an inductive  denote the inductive limit and denote by it the canonical embeddings of system. Let E  Et into E. It is not difficult to show that  St : ks+t x → ks x determines a unitary for each t S. (See the appendix of Bhat and Skeide [2] for and that these unitaries form a semigroup   and restriction to E gives inductive limits of Hilbert modules.) Of course, E ∼ = i0 E ⊂ E back S. (The inductive limit for the semigroup of not necessarily adjointable isometries in Example 2.1 is L2 (R, B) with canonical embeddings it sending L2 (R− , I) ⊕ L2 (R+ , B) onto the subset L2 ((−∞, −t], I) ⊕ L2 ([−t, ∞), B) of L2 (R, B).)  if S is strongly continuous if and only if S is; E is complemented in E One may show:  and only if the βt,s are adjointable, that is, if the St are adjointable; if S is even pure, then ⊥ the projections pt onto ( St E) converge strongly to idE  and 0 for t → ∞ and t → −∞, respectively. So, if S is a strongly continuous semigroup of adjointable isometries, then the family pλ is a spectral measure that is strongly continuous. So, it follows that the integrals  T t :=

 eitλ dpλ

 Extending  exist strongly and define a strongly continuous group of unitaries on E. S to negative times, the two strongly continuous unitary groups  S and  T fulfill the Weyl commutation relations ist  Ss  Tt = e  Tt  Ss .

Now, if we apply our theorem to the pure strongly continuous semigroup S, so that 2  = L2 (R, F ) with it E = S is given as a standard right shift on some L (R+ , F ), then E 2 L ([−t, ∞), F ), and itx [ T t (f )](x) = e f (x).

The Stone–von Neumann theorem asserts that every pair of strongly continuous semigroups of unitaries on a Hilbert space H that fulfills the Weyl relations, is unitarily equivalent to the pair  S and  T for some Hilbert space F . This can be interpreted in two directions. On the one hand, whenever the Stone–von   Neumann theorem holds, we can use this to prove Theorem 1.2. (Simply construct E, S,  and T , apply the Stone–von Neumann theorem, and remember how E and S sit inside L2 (R, F ).) On the other hand, if, starting with strongly continuous unitary groups  S  that fulfill the Weyl relations, we would succeed to find a submodule E and  T on E itλ Tt back as e , then turning the pλ (defined as above) into a spectral measure giving  Theorem 1.2 would allow prove the Stone–von Neumann theorem for that pair.  itλ T t as e dpλ by means of Stone’s For Hilbert spaces the latter can be done, writing  R  does the job.) For Hilbert theorem, and showing that  Ss dpλ = dpλ+s  Ss . (Then E := p0 E modules there is no spectral theorem, consequently, there is no Stone theorem for unitary

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groups. We, thus, do not know how to restrict one of the unitary groups fulfilling the Weyl relations to a pure semigroup of isometries to which our theorem could be applied. Consequently, we do not get a Stone–von Neumann theorem for unitary groups on Hilbert modules. Observation 2.8. Note that for von Neumann (or W ∗ -modules), Theorem 1.1 generalizes verbatim, if we understand that “strongly continuous” is now referring to the strong operator topology of the von Neumann module E ⊂ B(G, H); see Section 4 for details. The fact that the semigroup in Theorem 1.2 is a C0 -semigroup, which is much stronger a continuity condition, plays a crucial role in the proof in Section 3. While Hilbert modules frequently behave rather like pre-Hilbert spaces, von Neumann modules behave very much like Hilbert spaces. In fact, the reduction of the von Neumann module version of Theorem 1.1 to Theorem 1.1 itself that we sketch in Section 4, amounts to not much more than checking a certain commutant property. Of course, in the very same way we do get a Stone–von Neumann theorem for strongly continuous semigroups of unitaries on a von Neumann module. Observation 2.9. Most of the things in this section work for semigroups of isometries {St }t∈S indexed by more general monoids S, provided S is suitably directed. In the very definition of pure, it is necessary to make sense out of limt∈S S∗t x = 0. But also in

showing that the restriction to the invariant subspace Eu := t∈S St E is unitary, a cofinal property is needed. In the definition of Ep we would have to add a linear span under the closure; also here cofinal properties are used to establish the ‘nice’ properties of Ep . What we need is a direction ≥ on S that is reflected by the subspaces St E of E as t ≥ r ⇒ St E ⊂ Sr E. By Srr E = Sr Sr E ⊂ Sr E, we see that the definition t ≥ r :⇔ t ∈ rS does the job. This (reflexive and transitive) relation is a direction if and only if tS ∩ rS = ∅ for all r, t ∈ S, that is, if and only if the monoid S is left-reversible. If S is cancellative, then the opposite monoid Sop is what is known as Ore monoid. (See the standard reference Clifford and Preston [3].) Let S be a left-reversible monoid. Then the positive statements in Proposition 2.2 about the maximal unitary part of a general semigroup of isometries and in Proposition 2.3 about maximal unitary and maximal pure parts of semigroups of adjointable isometries remain true for semigroups over S. The negative statement in Proposition 2.2 is documented by Example 2.1. To preserve that example, we simply replace L2 with 2 , R+ with S and R with the non-abelian Grothendieck cover of S; see [3, Theorems 1.23–1.25]. However, for that the Grothendieck cover contains S, it is necessary that S be cancellative, so, S has to be the opposite of an Ore monoid. The negative statement in Proposition 2.3, Eu ⊕ Ep need not be E, is Example 2.5. However, the heart in Example 2.5 is the semigroup over N0 generated by the single isometry ˘S, the amplification of which, then, is “interpolated” continuously by means of Lemma 2.4(1). Without a topology on S we are not interested in continuity issues, and as already explained in Observation 2.6 for the case R+ , also for a general left-reversible monoid S

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we get an example on Cb (N, H) as soon as we get a pure semigroup of isometries v˘ over S acting on a Hilbert space H = {0}. (Since H is not {0}, for at least one t ∈ S the isometry v˘t is proper, and the construction of the sequence of vectors en in H goes as in Example 2.5 if we look just at the subsemigroup {˘ vtn }n∈N0 .) Existence of such a pure semigroup can, again, be guaranteed if S is the opposite of an Ore monoid and not a  group. (Simply take the Hilbert space t∈S C of the semigroup algebra, on which the elements of S act as left shift. The action is isometric if and only if S is cancellative. And if S is cancellative then the left shift by t is a proper isometry for all noninvertible t.) Ore monoids (containing no invertible elements but 0) include the discrete and continuous multi-parameter cases Nd0 and Rd+ (which are abelian and, therefore, coincide with their opposite monoids), but also nonabelian examples. (For every k ≥ 2, the free monoid generated by two indeterminates a, b subject to the relation ak b = ba is an Ore monoid with no invertible elements but 0.) Of course, the isometry ˘S constructed in Example 2.5, covers S = N0 ; it also illustrates that the classical Wold decomposition of an arbitrary isometry on a Hilbert space into a unitary part and a purely isometric part, may fail for Hilbert modules. 3. Proof of Theorem 1.2 From now on we fix on the situation in Theorem 1.2. So, B is a C ∗ -algebra and E is a Hilbert C ∗ -module over B. We fix a pure strongly continuous one-parameter semigroup of adjointable isometries {St }t≥0 on E. The operators St S∗t form a decreasing family of projections with S0 S∗0 = id and, by pureness, St S∗t → 0, strongly. So, for 0 ≤ a < b < ∞ putting pa,b := Sa S∗a − Sb S∗b , the pa,b are projections, too, and form a spectral measure on the reals half-line. We also put pc,d := 0 if c > d, and pa,∞ := Sa S∗a . We take Ea,b := pa,b E. Recall the standard notation c ∨ d := max{c, d}, and c ∧ d := min{c, d}. Proposition 3.1. Let 0 ≤ t < ∞, 0 ≤ a < b ≤ ∞ and 0 ≤ c < d ≤ ∞. (i) pa,b pc,d = pa∨c,b∧d . (ii) St pa,b = pa+t,b+t St ; pa,b St = St p(a−t)∨0,(b−t)∨0 . (iii) S∗t pa,b = p(a−t)∨0,(b−t)∨0 S∗t ; pa,b S∗t = S∗t pa+t,b+t . Proof. (i) is a standard computation for spectral measures. The first formula in (ii) follows from St Sa S∗a = Sa+t S∗a = Sa+t S∗a+t St . The second formula in (ii) follows by taking also into account that for t ≥ a we get Sa S∗a St = ∗ Sa St−a = St = S0 S0 St . The formulae in (iii) are adjoints of (ii). 2 The Ea,b are our candidates for L2 ([a, b), F ). A typical behavior is: Corollary 3.2. From St pa,b = pa+t,b+t St and pa,b S∗t = S∗t pa+t,b+t we infer that stricts to a unitary Ea,b → Ea+t,b+t with inverse S∗t restricted to Ea+t,b+t .

St

(co)re-

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We are now going to define the unitary groups ua.b that simulate the shift modulo b − a on L2 ([a, b), F ). For 0 ≤ a < b < ∞ and 0 ≤ t < (b − a), define ua,b by t ∗ ua,b t = St pa,b−t + Sb−a−t pb−t,b .

Making use of Proposition 3.1, we also have: ∗ ua,b t = pa+t,b St + pa,a+t Sb−a−t .

(3.1)

We extend the definition of ua,b periodically to all t ∈ R by setting t a,b ua,b t = ut−n(b−a) ,

for t ∈ [n(b − a), (n + 1)(b − a)), with n ∈ Z. is a unitary on Ea,b and Proposition 3.3 (Unitarity and group property). Each ua,b t a,b a,b a,b ur ut = ur+t for all r, t ∈ R. Proof. Since ut is periodic with period b − a, it is sufficient if we do computations for times in [0, b − a). By Corollary 3.2, ua,b t , as the direct sum of two unitaries from Ea,b = Ea,b−t ⊕ Eb−t,b onto Ea+t,b ⊕ Ea,a+t = Ea,b , is unitary. For the semigroup property, let 0 ≤ r, t < b − a. Then a,b ∗ ∗ ua,b r ut = (Sr pa,b−r + Sb−a−r pb−r,b )(pa+t,b St + pa,a+t Sb−a−t )

= Sr pa∨(a+t),(b−r)∧b St + Sr pa,(b−r)∧(a+t) S∗b−a−t + S∗b−a−r p(b−r)∨(a+t),b St + S∗b−a−r pb−r∨a,b∧(a+t) S∗b−a−t = Sr pa+t,b−r St + Sr pa,(b−r)∧(a+t) S∗b−a−t + S∗b−a−r p(b−r)∨(a+t),b St + S∗b−a−r pb−r,a+t S∗b−a−t . We have to distinguish two cases. Firstly, a + t < b − r, that is, r + t < b − a. So, a,b ∗ ∗ ua,b r ut = Sr pa+t,b−r St + Sr pa,a+t Sb−a−t + Sb−a−r pb−r,b St + 0

= Sr+t pa,b−r−t + pa+r,a+r+t Sr S∗b−a−t + S∗b−a−r St pb−r−t,b−t = Sr+t pa,b−r−t + pa+r,a+r+t pr,∞ S∗b−a−r−t + S∗b−a−r−t pb−r−t,b−t = Sr+t pa,b−r−t + pa+r,a+r+t S∗b−a−r−t + S∗b−a−r−t pb−r−t,b−t = Sr+t pa,b−r−t + S∗b−a−r−t pb−t,b + S∗b−a−r−t pb−r−t,b−t = Sr+t pa,b−r−t + S∗b−a−r−t pb−r−t,b = ua,b r+t .

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Secondly, a + t ≥ b − r, that is, r + t ≥ b − a. So, a,b ∗ ∗ ∗ ∗ ua,b r ut = 0 + Sr pa,b−r sb−a−t + sb−a−r pa+t,b St + sb−a−r pb−r,a+t sb−a−t

= Sr+t−(b−a) pb−t,2b−a−r−t + St−(b−a−r) pa,b−t + s∗2(b−a)−r−t p2b−a−r−t,b = Sr+t−(b−a) pa,2b−a−r−t + s∗2(b−a)−r−t p2b−a−r−t,b a,b = ua,b r+t−(b−a) = ur+t .

Now, a semigroup homomorphism between groups is a group homomorphisms. So, the ua,b t , effectively, form a unitary one-parameter group. 2 is strongly continuous. Proposition 3.4 (Continuity). t → ua,b t Proof. Observe that for x ∈ E and t ≥ 0, S∗t x −x = S∗t x − S∗t St x ≤ S∗t (x − St x) = x − St x. Hence t → S∗t is strongly continuous. If t → At and t → Bt are strongly continuous, then t → At Bt is also strongly continuous. (Indeed, by the principle of uniform boundedness, At is locally uniformly bounded. By At Bt x − Ar Br x ≤ At  (Bt − Br )x + (At − Ar )Br x, we see that for fixed r ∈ R and x ∈ E, At Bt x converges to Ar Br x for t → r.) Therefore, maps like t → St+α S∗t+β and t → S∗t+α St+β are strongly continuous. Consequently, t → ua,b is strongly continuous on [0, (b − a)). t Further, ∗ lim ua,b t = lim [St pa,b−t + sb−a−t pb−t,b ]

t↑(b−a)

t↑(b−a)

= lim [St (Sa S∗a − Sb−t S∗b−t ) + s∗b−a−t (Sb−t S∗b−t − Sb S∗b )] t↑(b−a)

= 0 + id(Sa S∗a − Sb S∗b ) = pa,b = ua,b 0 . Therefore, also the periodic extension to the real line is strongly continuous. 2 The functions of the form II [a,b) y in L2 ([a, b), F ) are precisely those which are invariant under the unitary shift modulo b −a. Taking the mean should provide us with a projection onto that invariant subspace. Therefore, for x ∈ E, and 0 ≤ a < b < ∞, we define 1 qa,b x = b−a

b−a  ua,b t x dt 0

in the sense of Riemann integral over the continuous function ua,b t x. Lemma 3.5. ua,b r qa,b = qa,b . = ua,b Proof. It is enough to show the statement for 0 ≤ r < (b − a). As ua,b t t−(b−a) for b − a ≤ t < 2(b − a), we get

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ua,b r qa,b x

1 = b−a

=

1 b−a

1 = b−a

17

b−a  a,b ua,b r ut x dt 0 (b−a)+r 

ua,b t x dt r b−a 

ua,b t x dt r

1 + b−a

r ua,b t x dt 0

2

= qa,b x.

Corollary 3.6. qa,b is a subprojection of pa,b on E. Proof. As ∗ ua,b t = pa+t,b St + pa,a+t Sb−a−t ,

it is clear that the range of qa,b is contained in the range of pa,b . To show that a linear map q is a projection, it is sufficient to check qx, qy =  b−a 1 ∗ x, qy. For qa,b that formula follows from qx, qy = b−a x, (ua,b t ) qa,b y and the 0 lemma. 2 The following simple corollary of the lemma shows that pieces of qa,b x in subintervals behave nicely with respect to the shift. Corollary 3.7. For 0 < r < b − a, Sr pa,b−r qa,b

= pa+r,b Sr qa,b = pa+r,b ua,b r qa,b = pa+r,b qa,b .

We now come to the analytical heart of our proof of Theorem 1.2. The following lemma will guarantee that we may approximate everything by linear combinations of elements qa,b x (x ∈ E, 0 ≤ a < b < ∞). Lemma 3.8. For x ∈ E0,1 , lim

n→∞

n 

q k−1 , k x = x. n

n

k=1

Proof. Note that n 

1

q k−1 , k x = n n

k=1

n  n

n

0 k=1

k−1

ut n

k ,n

x dt.

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18

  For  > 0, choose N such that for all t ∈ 0, N1 ,   , and S∗t x − x < . 2 2

St x − x < Consider n ≥ N . We get

n n n         k−1 k ,    ut n n x − x =  p k−1 +t, k St x + p k−1 , k−1 +t s∗1 −t x  n n n n n

k=1

k=1

−

n 

k=1

p k−1 +t, k x + n

n 

n

k=1

  p k−1 , k−1 +t x  n n

k=1

n n         ≤ p k−1 +t, k (St x − x) +  p k−1 , k−1 +t (s∗1 −t x − x) n n n n n

k=1

k=1

n n         ≤ p k−1 +t, k  St x − x +  p k−1 , k−1 +t  (S∗1 −t x − x) n n n n n

k=1

≤

k=1

  + = . 2 2

Hence, 1

 n

n  n

1

k−1 k n ,n

ut

  x dt − x = n

0 k=1

n   n 0

1

k−1 k n ,n

ut

  − id x dt ≤ n

k=1

n  dt = .

2

0

From now on we prepare for proving in Proposition 3.13 that E0,1 ∼ = L2 ([0, 1), q0,1 E), making q0,1 E our ‘hot’ candidate for being the F we seek, and for proving in Corollary 3.14 that u0,1 , indeed, transforms under this isomorphism into the shift modulo 1. Proposition 3.9. For 0 ≤ a < c < d < b < ∞, pc,d qa,b pc,d =

d−c qc,d . b−a

Proof. For 0 ≤ t ≤ (b − a), pc,d (pa+t,b St + S∗t pa+t,b )pc,d = pc∨a+t,d pc+t,d+t St + S∗t pc+t,d+t pc∨a+t,d = pc+t,d St + S∗t pc+t,d , which is non-zero only if 0 < t < (d − c). Hence, by an application of change of variable

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1 qa,b x = b−a

b−a  (pa+t,b St + pa,a+t S∗b−a−t )x dt 0

1 = b−a

=

1 b−a

19

b−a b−a    pa+t,b St x dt + pa,b−t S∗t x dt 0

(z = b − a − t)

0

b−a  (pa+t,b St + S∗t pa+t,b )x dt, 0

we get 1 pc,d qa,b pc,d x = b−a

d−c  (pc+t,d St + S∗t pc+t,d )x dt 0

d−c qc,d x. = b−a Proposition 3.10. qa+r,b+r E.

Sr qa,b

= qa+r,b+r Sr , that is,

2 Sr

(co)restricts to a unitary qa,b E →

Proof. 1 Sr qa,b x = b−a

=

=

=

1 b−a 1 b−a 1 b−a

b−a  Sr [St pa,b−t

+ pa,a+t S∗b−a−t ]x dt

0 b−a 

[St pa+r,b+r−t Sr x + pa+r,a+r+t Sr s∗b−a−t x] dt

0 b−a 

[St pa+r,b+r−t Sr x + pa+r,a+r+t pr,∞ s∗b−a−t Sr x] dt

0 b−a  [St pa+r,b+r−t + pa+r,a+r+t s∗(b+r)−(a+r)−t ]Sr x dt 0

= qa+r,b+r Sr x.

2

Proposition 3.11. Suppose z ∈ q0,1 E. Then for 0 < r, t, with r + t < 1, p0,r+t z = p0,r z + Sr p0,t z. Proof. We have p0,r+t z = p0,r z + pr,r+t z = p0,r z + pr,r+t pr,1 z. Now Corollary 3.7, with a = 0, b = 1 yields pr,1 z = pr,1 Sr z. Hence, p0,r+t z = p0,r z + pr,r+t pr,1 Sr z = p0,r z + pr,r+t Sr z = p0,r z + Sr p0,t z. 2

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Proposition 3.12. Suppose z, w ∈ q0,1 E. Then for 0 < c1 < d1 < 1 and 0 < c2 < d2 < 1,

pc1 ,d1 z, pc2 ,d2 w = μ([c1 , d1 ] [c2 , d2 ])z, w, where μ denotes the Lebesgue measure. Proof. For 0 < t < 1, set f (t) = z, p0,t w. Then from the previous proposition, for 0 < r + t < 1, f (r + t) = p0,r+t z, p0,r+t w = p0,r z, p0,r w + Sr p0,t z, Sr p0,t w = p0,r z, p0,r w + p0,t z, p0,t w = f (r) + f (t). Then by strong continuity of t → p0,t , it follows that f (t) = tz, w for all t. Hence the result. 2 Recall that L2 ([0, 1], F ) := L2 [0, 1] ⊗ F is the external tensor product. The subset of functions of the form II [c,d) z (0 ≤ c < d ≤ 1) is total. Proposition 3.13. Take F = q0,1 E. Define M : L2 ([0, 1], F ) → E0,1 by M (II [c,d) z) = pc,d z, for z ∈ F and 0 ≤ c < d ≤ 1 (II denotes the indicator function), and extending linearly. Then M extends to a unitary map. Proof. The isometry property of M has been proved in the previous proposition. Now for x ∈ E0,1 , and 1 ≤ k ≤ n, by Proposition 3.9, q (k−1) , k x is in the range of p (k−1) , k q0,1 . n n n n Then from Proposition 3.8,

lim

n→∞

n  k=1

1

q (k−1) , k x = lim n n

n

n→∞

n  n

k−1

ut n

k ,n

x dt = x.

0 k=1

This shows that the range of M is whole of E0,1 .

2

Let πt denote the periodic shift on L2 ([0, 1], F ). Corollary 3.14. M ∗ u0,1 = πt M ∗ . t Proof. It suffices to check this on pc,d z with either 0 ≤ c < d < 1 − t or 1 − t ≤ c < d < 1, as every other choice is a sum of such. For the stated cases, the statement follows directly from the definitions of u0,1 t , πt , and M , taking also into account Propositions 3.10 and 3.11. 2

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21

Remark 3.15. Of course, the proposition and its corollary remain true, replacing the interval [0, 1) with any other nonempty interval [a, b) ⊂ R+ . Proof of Theorem 1.2. By pureness, E=

∞ 

Ek−1,k =

k=1

∞ 

Sk E0,1 .

k=0

Like in the proof of Lemma 2.4(2), it is convenient to keep in mind that Ek−1,k corresponds to L2 [k − 1, k) ⊗ F , but that these via the (adjoints of the) shifts Sk and vk , are identified with E0,1 and L2 [0, 1) ⊗ F , respectively. Proposition 3.10 (and its analogue for v on the L2 -side) assures that shifting back [k − 1, k) to [0, 1) does no harm. The rest can be done precisely as in the proof of Lemma 2.4(2), namely, checking only for t < 1 that the two cases, one a piece not shifted out of the interval [k − 1, k) and another piece sent to the next interval [k, k + 1), end up in the right direct summands. 2 4. The case of von Neumann modules Let B ⊂ B(G) be a von Neumann algebra acting (nondegenerately) on a Hilbert space G. Skeide [9, Definition 2]: A concrete von Neumann B-module is a strongly closed subspace E of B(G, H) (H another Hilbert space) such that EB ⊂ E, E ∗ E ⊂ B, and span EG = H. Then E inherits the structure of a Hilbert B-module by defining the inner product x, y := x∗ y; the right multiplication is (necessarily!) given by composition of maps. (Every W ∗ -module E over B (that is, a self-dual Hilbert module over a W ∗ -algebra) can be turned into a concrete von Neumann module in the following way: Define H := E  G (this is the internal tensor product over B); identify E ⊂ B(G, H) by letting x ∈ E act as x: g → x  g. There is nothing arbitrary in this procedure in that whenever a pre-Hilbert B-module E is identified as an operator submodule of B(G, H), then H and E G are canonically isomorphic via xg → x g. In fact, von Neumann B-modules have been first defined in Skeide [5, Definition 4.4] (taking also into account [5, Proposition 4.5]) as pre-Hilbert B-modules such that E is strongly closed in B(G, E  G).) Von Neumann modules are self-dual and, therefore, every bounded right linear map is adjointable; see [5, Theorem 4.16] or [8]. Also, every von Neumann submodule of a (pre-)Hilbert B-module is complemented and, therefore, the image of a projection; see [6, Proposition 1.5.9]. The picture E  G suggests two representations, a representation of Ba (E)  a acting as a  idG and a representation of B   b acting as idE b . In the picture of concrete von Neumann modules (to which we stick here), the first representation acts as a: xg → (ax)g. It is faithful, and we use it to identify Ba (E) ⊂ B(H). The second representation, to which we refer as commutant lifting ρ (with reference to the section entitled “lifting commutants” in Arveson [1]), acts as ρ (b ): xg → xb g. It is normal and nondegenerate, but not necessarily faithful. Now by Skeide [8], the subspace E ⊂ B(G, H) is characterized as

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E =

  x ∈ B(G, H): ρ (b )x = xb (b ∈ B  )

and Ba (E) = ρ (B  ) . The latter shows that Ba (E) ⊂ B(H) is a von Neumann algebra in its own right. On the other hand, given any normal nondegenerate representation ρ of B  , defining E as the preceding intertwiner space, we get a von Neumann B-module. As observed in Skeide [7], the correspondence is functorial. When restricted to the category of concrete von Neumann modules (with adjointable maps as morphisms), [9, Theorem 7] asserts that we obtain a bijective functor (not just an equivalence) with the category of normal nondegenerate representations of B  (with intertwiners as morphisms). We need two more notions: A family at of elements in a von Neumann algebra A ⊂ B(H) is point-strongly continuous if t → at h is continuous for all h ∈ H. In the case of the von Neumann algebra Ba (E) ⊂ B(H), this should not be confused with strong continuity in Ba (E). If F ⊂ B(G, K) is a concrete von Neumann B-module, then by ¯ s F we mean the von Neumann B-module obtained by strong L2,s (R+ , F ) = L2 (R+ ) ⊗ 2 closure of L (R+ , F ) in B(G, L2 (R+ , F )  G) = B(G, L2 (R+ , K)) where the element f ⊗ y ∈ L2 (R+ ) ⊗ F = L2 (R+ , F ) acts as g → f ⊗ yg ∈ L2 (R+ ) ⊗ K = L2 (R+ , K). If σ  : B  → B(K) is the commutant lifting of F , then idL2 (R+ ) ⊗σ  is the commutant lifting of L2,s (R+ , F ). The standard right shift on L2 (R+ , F ) extends to a point-strongly continuous semigroup of (necessarily adjointable) isometries referred to as the standard right on L2,s (R+ , F ). Theorem 4.1. Let S = {St }t∈R+ be a point-strongly continuous semigroup of isometries on a concrete von Neumann B-module E ⊂ B(G, H). Then E = Eu ⊕ Ep where Eu ⊂ B(G, Hu ) and Ep ⊂ B(G, Hp ) are unique reducing von Neumann submodules (with H = Hu ⊕ Hp ) such that: 1. The restriction of 2. The restriction of

S S

to Eu is unitary. to Ep is completely nonunitary.

Moreover, the completely nonunitary part is unitarily equivalent to the standard right shift on L2,s (R+ , F ) ⊂ B(G, L2 (R+ , K)) for some concrete multiplicity von Neumann B-module F ⊂ B(G, K). Proof. The decomposition into Eu and Ep := Eu⊥ is seen best in the setting of Section 2, which is eased by the fact that the St are adjointable automatically and that ide −St S∗t and St S∗t converge (one increasingly, the other decreasingly) point-strongly in the von Neumann algebra Ba (E) to a pair of orthogonal projections summing up to idE . The



image of the limit of St S∗t is, clearly Eu := t∈R+ St E = t∈R+ St S∗t E. On the complement S∗t converges point-strongly to 0; this leaves no space for a unitary subspace, so the restriction to Eu⊥ is completely nonunitary. The discussion also shows, it is “pointstrongly pure” showing that Eu⊥ merits to be named Ep . (The whole discussion could have been done in terms of Hu and Hp from Theorem 1.1. But this would have left us

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23

with the question if the projections onto these subspaces of H belong to Ba (E) ⊂ B(H).) Now, we simply put Hu = span Eu G and Hp = span Ep G and by construction the restriction of S to Hu is unitary and the restriction of S to Hp is completely nonunitary—and pure. For the supplement, assume now that S is completely nonunitary (both as semigroup in Ba (E) and in B(H)). By the supplement of Theorem 1.1, as semigroup in B(H) we obtain that H = L2 (R+ , K) and St ∈ B(H) is given by the right shift. Moreover, on H = L2 (R+ , K) = L2 (R+ ) ⊗ K there is a normal nondegenerate representation ρ of B  such that Ba (E) = ρ (B  ) . All St are in Ba (E), so this representation has to commute with the von Neumann algebra generated by the right shifts St . It is well-known that commutant of the right shift on L2 (R+ ) is trivial. (The von Neumann subalgebra L∞ (R+ ) generated by the indicator functions of intervals is maximal abelian and, therefore, coincides with its commutant. And no element of L∞ (R+ ) that is not a multiple of the identity commutes with all right shifts.) Therefore, ρ has to map into idL2 (R+ ) ⊗B(K), that is, there is a (necessarily normal and nondegenerate) representation σ  of B  on K such that ρ = idL2 (R+ ) ⊗σ  . Putting F :=





y ∈ B(G, K): σ  (b )y = yb (b ∈ B  )

finishes the proof. 2 Of course, the theorem holds for all W ∗ -modules, replacing the continuity condition on ∗ S by σ-weak continuity. (We just have to represent the W -algebra B as a von Neumann algebra on a Hilbert space, and transform E into a concrete von Neumann B-module as explained in the beginning of this section. The σ-weak topology of Ba (E) is intrinsic and does not depend on how we obtained the identification in B(H). On the other hand, the set {St } is bounded and on bounded subsets the σ-weak topology coincides with the weak topology and weak continuity of a semigroup of isometries implies strong continuity.) Acknowledgments We started thinking about Theorem 1.2 during a Research in Pairs project at Mathematisches Forschungsinstitut Oberwolfach in 2007; hospitality at MFO during this RiP is gratefully acknowledged. MS is grateful to BVRB and the Indian Statistical Institute, Bangalore Center for kind hospitality during several stays, and for travel support from the Dipartimento E.G.S.I. of University of Molise. A big thank you goes to the referee for careful reading and valuable suggestions that improved the final form of this paper. References [1] W. Arveson, Subalgebras of C ∗ -algebras, Acta Math. 123 (1969) 141–224.

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[2] B.V.R. Bhat, M. Skeide, Tensor product systems of Hilbert modules and dilations of completely positive semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 3 (2000) 519–575, Rome, Volterra-Preprint 1999/0370. [3] A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups I, Mathematical Surveys, vol. 7, American Mathematical Society, 1961. [4] J.L.B. Cooper, One-parameter semigroups of isometric operators in Hilbert space, Adv. Math. 48 (1947) 827–842. [5] M. Skeide, Generalized matrix C ∗ -algebras and representations of Hilbert modules, Math. Proc. R. Ir. Acad. 100A (2000) 11–38, Cottbus, Reihe Mathematik 1997/M-13. [6] M. Skeide, Hilbert modules and applications in quantum probability, Habilitationsschrift, Cottbus, 2001, available at http://web.unimol.it/skeide/. [7] M. Skeide, Commutants of von Neumann modules, representations of Ba (E) and other topics related to product systems of Hilbert modules, in: G.L. Price, B.M. Baker, P.E.T. Jorgensen, P.S. Muhly (Eds.), Advances in Quantum Dynamics, Cottbus 2002, in: Contemporary Mathematics, vol. 335, American Mathematical Society, 2003, pp. 253–262, preprint, arXiv:math.OA/0308231. [8] M. Skeide, Von Neumann modules, intertwiners and self-duality, J. Operator Theory 54 (2005) 119–124, arXiv:math.OA/0308230. [9] M. Skeide, Commutants of von Neumann correspondences and duality of Eilenberg–Watts theorems by Rieffel and by Blecher, Banach Center Publ. 73 (2006) 391–408, arXiv:math.OA/0502241. [10] B. Sz.-Nagy, Isometric flows in Hilbert space, Math. Proc. Cambridge Philos. Soc. 60 (1964) 45–49. [11] B. Sz.-Nagy, C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland Publishing Company, 1970.