Hilbert von Neumann Modules Versus Concrete von Neumann Modules

Apart from presenting some new insights and results, one of our main purposes is to put some records in the development of von Neumann modules straight. The von Neumann or \(W^*\)–objects among the Hilbert (\(C^*\)–)modules are around since the first papers by Paschke [12] and Rieffel [14, 15] that lift Kaplansky’s setting [8] to modules over noncommutative \(C^*\)–algebras. While the formal definition of \(W^*\)–modules is due to Baillet, Denizeau, and Havet [2], the one of von Neumann modules as strongly closed operator spaces started with Skeide [19]. It has been paired with the definition of concrete von Neumann modules in Skeide [23]. It is well-known that (pre-)Hilbert modules can be viewed as ternary rings of operators and in their definition of Hilbert-von Neumann modules, Bikram, Mukherjee, Srinivasan, and Sunder [4] take that point of view.

We illustrate that a (suitably nondegenerate) Hilbert-von Neumann module [4] is the same thing as a strongly full concrete von Neumann module [23]. We also use this occasion to put some things in the papers [4, 19, 23] right. We show that the tensor product of (concrete, or not) von Neumann correspondences is, indeed, (a generalization of) the tensor product of Connes correspondences (claimed in Skeide [24]), viewed in a way quite different from [4]. We also present some new arguments that are useful even for (pre-)Hilbert modules.

1. 1.

   In [23] we omitted to repeat that a module is closed under addition.

2. 2.

   It is to be noted that a von Neumann algebra \(\mathcal {B}\) is, as always(!), acting nondegenerately as \(\mathcal {B}\subset \mathcal {B}(G)\) on a Hilbert space G. Having said this, we recall the well-known (at least, since Rieffel [15]) fact that every pre-Hilbert module over a concrete operator pre-\(C^*\)–algebra \(\mathcal {B}\subset \mathcal {B}(G)\) embeds canonically (and uniquely, up to obvious unitary equivalence) into \(\mathcal {B}(G,H)\), where \(H:=E\odot G\) is the Hausdorff completion of the algebraic tensor product \(E\,\underline{\otimes }\,G\) with the semiinner product determined by \(\langle x\otimes g,x'\otimes g'\rangle :=\langle g,\langle x,x'\rangle g'\rangle \). Now sits naturally in , and it is clear what it means for to be a von Neumann algebra in .

3. 3.

   Calling this an “apparently much simpler manner” avoiding a “two stages completion” in the abstract of [4] seems out of place. The “two stages completion” they are referring to is that of norm closure (to get a \(C^*\)–module) and then strong closure. It is not necessary to go through \(C^*\)–modules in the construction, and while [19, Definition 4.4] is really (for convenience, in order not to frighten people who are afraid of pre-Hilbert modules) formulated only for Hilbert modules, [20, Definition 3.1.1] (which for some reason is the only work quoted in [4]) is formulated for pre-Hilbert modules. As for the (Hausdorff) completion to obtain H, in [4, Sects. 2+3] quite similar constructions are performed. It would appear strange to “allow” such constructions for Stinespring type theorems and the tensor product of correspondences, but not to obtain the tensor product \(E\odot G\). (Anyway, it is well-known (at least, from or Skeide [19], but probably earlier; see also Murphy [11]) how to obtain the Stinespring construction of a CP-map into \(\mathcal {B}\subset \mathcal {B}(G)\) from the tensor product of Pascke’s GNS-correspondence [12] E and the representation space G of \(\mathcal {B}\). In this context, while it is not clear to us why [4, Sect. 2] is restricted to standard Hilbert-von Neumann modules, a comparison with [19, Sect. 7] seems interesting.)

4. 4.

   Attention, a few authors by \(W^*\)–module mean just a Hilbert module over a \(W^*\)–algebra. But this seems not quite enough to merit to be considered an object in a \(W^*\)–category.

5. 5.

   One should note that the various definitions may be compared not only as to what extent they allow to check easily if a given pre-Hilbert module E over a von Neumann algebra is a von Neumann module or not; checking strong closure seems to have considerable advantages over checking self-duality. But they may also be compared as to what extent it is easy to obtain a von Neumann module from E in case it is not yet. Paschke [12] has constructed a self-dual completion. As a space it is simply \(\mathcal {B}^r(E,\mathcal {B})\), the space of all bounded right linear \(\varPhi :E\rightarrow \mathcal {B}\). But while it is actually not a priori easy to find that space, it is even much harder to compute the inner products of its elements. The definitions in [4, 19, 20, 23] suggest: Take the strong closure of E – and it is obvious what the inner product of elements x, y in this closure is, namely, \(\langle x,y\rangle =x^*y\), and that it takes values in the von Neumann algebra \(\mathcal {B}\).

6. 6.

   In [4, Proposition 1.9] it is shown that Hilbert-von Neumann modules are self-dual, and it is claimed earlier that this shows that this is equivalent to being a von Neumann module in the sense of [19]. But, actually the backwards direction of this statement is missing; it would not be easy to obtain it without the procedure of embedding a (pre-)Hilbert module into \(\mathcal {B}(G,H)\) (see Footnote 2), which the authors of [4] are so keen to avoid.

   Anyway, it appears a remarkable idea to show equivalence of Hilbert-von Neumann modules and concrete von Neumann modules appealing to the (not so easy) self-duality, while we have the (fairly obvious) Theorem 1.1 (and the easy observation that the procedure explained in Footnote 2 transforms a von Neumann module into a concrete von Neumann module and the observation that a concrete von Neumann module is, obviously, a von Neumann module).

7. 7.

   See [4, Lemma 1.6].

8. 8.

   See also the proof of [4, Proposition 1.7].

9. 9.

   The result [13, Theorem 1.4] is for tensor products of Banach modules over \(C^*\)–algebras. [19, Lemma 3.9] is for (pre-)Hilbert modules, but (as sketched in [22, Remark 3.3]; see also the proof of [4, Proposition 1.9]) the proof in [19] works correctly only for pre-Hilbert modules over a von Neumann algebra; in the case of \(C^*\)–algebras, there is a gap in the proof. See the appendix for a correct and new proof of this important result.

10. 10.

    See also the proof of [4, Proposition 1.9].

11. 11.

    See also the proof of [4, Proposition 1.7].

12. 12.

    See also the proof of [4, Proposition 1.9].

13. 13.

    The proof of self-duality in [22] has, in fact, a huge overlap with that of [15, Proposition 6.10]. The discussion of intertwiner spaces adds only a little to what was known already to [15]. Only the proof of boundedness of \(\varPhi \odot \textsf {id}_G\) (see also the proof of [4, Proposition 1.9]) seems to be really new—and quite a bit simpler than that of the statement about Banach modules in [13] alluded to in [15].

14. 14.

    It appears that many books or papers with introductions to the topic of Hilbert modules, either look only at amplifications of adjointable operators (where boundedness is an easy consequences of automatic contractivity of homomorphisms between \(C^*\)–algebras) or look at not necessarily adjointable operators but without considering their amplifications. It is likely that Rieffel’s original result in [13] (going beyond our scope) has been reproved in the (more general!) context of operator modules over (not necessarily self-adjoint) operator algebras.

15. 15.

    See also [4, Proposition 1.7]. Of course, for strongly closed submodules of von Neumann modules, this can be proved by constructing the projection with the help of a quasi-orthonormal basis for the submodule (which is a von Neumann module in its own right). But for the simple and elegant (standard) proof from self-duality of the submodule, \(\mathcal {B}\) may even be only a pre-\(C^*\)–algebra.

16. 16.

    Recall that a correspondence from \(\mathcal {A}\) to \(\mathcal {B}\) (or \(\mathcal {A}\)–\(\mathcal {B}\)–correspondence) is a Hilbert \(\mathcal {B}\)–module E with a nondegenerate(!) left action of \(\mathcal {A}\) that induces a homomorphism \(\mathcal {A}\rightarrow \mathcal {B}^a(E)\).

17. 17.

    By simple \(C^*\)–algebraic arguments it follows that the sesquilinear map on the algebraic tensor product \(E\otimes F\) defined as in the first relation above, is positive, hence, defines a semiinner product on \(E\otimes F\). (See the appendix.) Doing Hausdorff completion we obtain the Hilbert \(\mathcal {C}\)–module \(E\odot F\), and since the left action of \(\mathcal {A}\) induced by the second relation is by adjointable elements, it survives the quotient.

    However, already in [20, Sect. 4.2] the following alternative has been observed: It is easy, using cyclic decomposition of the representation of \(\mathcal {B}\) on G (see also the proofs [19, Lemma 3.9] and [4, Proposition 1.9]), to see positivity of the inner product of \(E\odot G\). Using this, (assuming \(\mathcal {C}\subset \mathcal {B}(L)\)) one gets that \(x\odot y\) may be represented as operator \(L_xL_y\in \mathcal {B}(L,E\odot (F\odot L))\) where \(L_y=y\odot \textsf {id}_L\in \mathcal {B}(L,F\odot L)\) and \(L_x=x\odot \textsf {id}_{F\odot L}\in \mathcal {B}(F\odot L,E\odot (F\odot L))\); see, in particular, [20, Eq. (4.2.2)]. (See also the discussion leading to [4, Proposition 3.4].)

18. 18.

    We have heard rumors that the creator of that construction is not too fond of that terminology, so we shall not use it.

19. 19.

    We did not quite understand, if no relation between Hilbert-von Neumann bimodules and Connes correspondences is made in [4], or if the authors think the relation would be the occurrence of two representations in the definition of Hilbert-von Neumann bimodule. The latter would not be applicable, because for a Hilbert-von Neumann bimodule \(E\subset \mathcal {B}(G,H)\) there is the representation \(\pi \) as for Hilbert-von Neumann modules (see the beginning of Sect. 1) acting on G plus an analogue representation acting on H, while a Connes correspondence (see the basic fact 5 below) is a single Hilbert space H with a representation and an anti-representation both acting on H.

20. 20.

    Already in [19, Sect. 7] (see also [3, Example 2.16 and Observation 2.17]) it has been explained that when E is the GNS-correspondence (Paschke [12]) of a CP-map into a concrete \(C^*\)–algebra \(\mathcal {B}\subset \mathcal {B}(G)\), then \(\rho \) is, indeed, the Stinespring representation. See also [4, Sect. 2].

21. 21.

    Arveson’s representation discussed in the section “Lifting Commutants” in [1], is a special case of our commutant liftings.

22. 22.

    Note that the algebraic tensor product of a right \(\mathcal {B}\)–module E and a left \(\mathcal {B}\)–module F over \(\mathcal {B}\) is . In the setting of the theorem, the semiinner product defined on \(E\,\underline{\otimes }\,F\) as usual always survives the quotient but still need not be inner on \(E\,\underline{\odot }_\mathcal {B}F\). (It is inner if \(\mathcal {B}\) and E are complete; see [20, Proposition 4.2.22].) Therefore, it is pointless to divide first out the relations \(xb\otimes y-x\otimes by\) and then a still possible kernel of the semiinner product on \(E\,\underline{\odot }_\mathcal {B}F\), but we divide out immediately the kernel of the semiinner product on \(E\,\underline{\otimes }\,F\). We emphasize, however, that the relation \(xb\odot y=x\odot by\) is satisfied at any stage.

Discussions with several people helped a lot to improve this paper. In particular, I would like to thank Malte Gerhold and Orr Shalit.

Authors and Affiliations

1. Università degli Studi del Molise, Via de Sanctis, 86100, Campobasso, Italy

   Michael Skeide

Corresponding author

Correspondence to Michael Skeide .

Editors and Affiliations

1. Vito Volterra Interdepartamental Centre, Universita degli Studi di Roma, Rome, Italy

   Luigi Accardi

2. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, United Arab Emirates

   Farrukh Mukhamedov

3. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain, Abu Dhabi, United Arab Emirates

   Ahmed Al Rawashdeh

Appendix

The scope of this appendix is to present a new and ‘painless’ proof of the following result.

Theorem A.1

Let E be a (pre-)Hilbert module over a (pre-)\(C^*\)–algebra \(\mathcal {B}\) and let F be a (pre-) \(C^*\)–correspondence from \(\mathcal {B}\) to another (pre-)\(C^*\)–algebra \(\mathcal {C}\). Then for every \(a\in \mathcal {B}^r(E)\) we have

so \(a\odot \textsf {id}_F\in \mathcal {B}^r(E\odot F)\).

Observation A.2

• We have not yet specified what we mean by \(E\odot F\) in case of ‘pre-’. It is simply \(\textsf {span}{\,}\) \(\{x\odot y:x\in E,y\in F\}\subset \overline{E}\odot \overline{F}\).^{Footnote 22} So, since a is bounded we simply may complete the spaces E, F, \(\mathcal {B}\), \(\mathcal {C}\), proving the theorem only without ‘pre-’.

• Note that we need not have equality. Effectively, \(E\odot F\), hence \(a\odot \textsf {id}_F\), may very well be zero independently of \(\left\Vert a\right\Vert \).

• Of course, the theorem remains true for \(a\in \mathcal {B}^r(E_1,E_2)\); simply embed \(\mathcal {B}^r(E_1,E_2)\) into .

Remark A.3

For Banach modules over \(C^*\)–algebras, the theorem is [13, Theorem 1.4]. There should be versions for operator modules over (non-selfadjoint) operator algebras (including similar statements for CB-norms). The theorem is easy, if a is required adjointable. (\(a\mapsto a\odot \textsf {id}_F\) is a homomorphism between \(C^*\)–algebras and, therefore, contractive.)

The proof of the theorem is a typical example for the following strategy to prove statements about (internal) tensor products.

Observation A.4

1. 1.

   Frequently, a certain statement is easy (or easier) to prove on simple tensors \(x\odot y\) instead of general elements \(\sum _{i=1}^nx_i\odot y_i\) of the algebraic tensor product.

2. 2.

   Use the equality of \(E\odot F=E_n\odot F^n\) via \(\sum _{i=1}^nx_i\odot y_i=X_n\odot Y^n\) (with \(X_n=(x_1,\ldots ,x_n)\in E_n\) and , where \(E_n\) is the row-module with inner product \(\langle X_n,X'_n\rangle =\bigl (\langle x_i,x'_j\rangle \bigr )\in M_n(\mathcal {B})\) and the obvious action of \(M_n(\mathcal {B})\) on \(F^n\)) to reduce the problem to simple tensors.

Among the various applications, there is that the inner product on \(E\odot F\) is actually positive (\(\Bigl \langle \sum _{i=1}^nx_i\odot y_i,\sum _{i=1}^nx_i\odot y_i\Bigr \rangle =\langle X_n\odot Y^n,X_n\odot Y^n\rangle =\langle Y^n,\langle X_n,X_n\rangle Y^n\rangle \ge 0\), having established before that the inner product of \(E_n\) is positive; see Skeide [25, Sect. 5]). Another application is the proof in Skeide and Sumesh [26, Corollary] of Blecher’s result [5, Theorem 4.3] that \(E\odot F\) is (completely isometrically) isomorphic to the Haagerup tensor product.

Proof of the theorem. Recall that we may restrict to the case without ‘pre-’. The statement of the theorem actually means that we show the estimate for elements of the form \(\sum _{i=1}^nx_i\odot y_i\) and, then, extend extend \(a\odot \textsf {id}_F\), well defined on that dense subspace, continuously to all of \(E\odot F\). (Recall from Footnote 22 that this subspace is \(E\,\underline{\odot }_\mathcal {B}F\).)

The proof decomposes into four parts.

(i) For all \(a\in \mathcal {B}^r(E)\) we have \(\langle ax,ax\rangle \le \left\Vert a\right\Vert ^2\langle x,x\rangle \) in \(\mathcal {B}\). (This is the easier direction of [12, Theorem 2.8], which asserts that the inequality actually characterizes among all linear maps on E those that are in \(\mathcal {B}^r(E)\). See there for the quick proof of the part we need here; the other direction is much harder.)

(ii) For each \(a\in \mathcal {B}^r(E)\) we have \(\mathcal {B}^a(E)\ni a\ge 0\) if (and only if) \(\langle x,ax\rangle \ge 0\) for all \(x\in E\). (In this form, this is Lance [9, Lemma 4.1], while [12, Proposition 6.1] is the special case we would actually need in the next part. Paschke actually shows in his proof Lance’ more general statement; but here we prefer Lance’ proof.)

(iii) The inequality in Part (i) survives ‘amplification’ to \(E_n\). More precisely, for each \(X_n=(x_1,\ldots ,x_n)\in E_n\), we have \(\Big (\langle ax_i,ax_j\rangle \Big )\le \left\Vert a\right\Vert ^2\Big (\langle x_i,x_j\rangle \Big )\) in \(M_n(\mathcal {B})\) or, equivalently,

Since \(M_n(\mathcal {B})\) acts in the obvious way on the Hilbert \(\mathcal {B}\)–module \(\mathcal {B}^n\), we may apply Part (ii) to see that this element of \(M_n(\mathcal {B})\) is positive. Indeed, for we find

By Part (i), this is positive.

(iv) By Part (iii) (applied in the step where the \(\le \) occurs), for we find

what was to be shown.   \(\square \)

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