Isometries and isomorphism classes of Hilbert modules^††thanks: Thanks for financial support are due to NSERC (Canada), the Fredrik and Catherine Eaton Visitorship (Canada/UK), and IMPAN (Poland).

Hilbert modules have many remarkable properties, and as pointed out by Lance lance1994 and others, apparently quite weak notions of isometry of Hilbert modules imply isomorphism of Hilbert modules. We apply this basic fact in several different settings. As a guide to possible applications, we consider briefly the Cuntz semigroup and the K-theory group. These sometimes rather technical objects become particularly attractive in the setting of C*-algebraic quantum groups, where we find that there exists a product operation on Hilbert modules that is quite nice and seems to be distinct from, although related to, the usual interior and exterior tensor products of Hilbert modules.

We then study isomorphism classes of Hilbert modules. These have a natural semigroup structure, under direct sum. Our most advanced result is roughly as follow:

In the above, by an invariant state for a left co-action $\delta$ of a compact or discrete C*-algebraic quantum group $(A,\varDelta)$ on $B$ we mean a state $\eta$ on $B$ such that $(\mbox{\rm Id}\otimes\eta)\delta=\eta(\cdot)1.$ By a well-behaved action of the semiring we mean a multiplicative action where the product on the ring distributes over the direct sum in the semigroup. As well as providing some kind of invariant associated with a co-action, it is quite possible that the above can be viewed as an algebraic topology type of obstacle to the existence of certain co-actions. In other words, we could show non-existence of certain co-actions by showing that one of the implied equations for the action of the semiring on the semigroup must fail.

The work of Goswami Goswami on non-existence of certain co-actions provides evidence that the question of nonexistence may be interesting. Our methods are however completely different than Goswami’s. It is possible that eventually our work may find application to showing noncommutative Borsuk-Ulam type theorems BDH2015 , which roughly speaking assert the lack of existence of a co-action on a special kind of C*-algebra, a noncommutative join.

The paper is organized as follows. In section 2 are the main results on Hilbert modules and the details of the already mentioned product operation. In section 3 we compute this product in several cases. In section 4 we apply our theory to the case of K-theory. In section 5 we consider the mostly finite-dimensional theory of Hopf ideals. In section 6 we give our deepest results, pertaining to co-actions.

We recall that the standard Hilbert module ${H}_{B}$ over a C*-algebra $B$ is by definition the set of sequences $(b_{i})$ such that $\sum b_{i}^{*}b_{i}$ converges in the C*-norm on $B.$ It is a subtle¹¹1There is an amusing discussion on page 239 of WeggeOlsen about various pitfalls in finding the right definitions to make this work. fact that (see for example, (lance1994, , pg. 34-35)) this is isomorphic to the (exterior) tensor product $B\otimes{H}$ where ${H}$ denotes the usual countably generated complex Hilbert space.

We now make some remarks leading to a pleasant geometric interpretation of the above lemma.

We remark first that if $A$ is a C*-algebra with a faithful state, $f,$ we may construct the GNS Hilbert space ${H}$ associated with it. The usual direct sum over all states can be omitted if we are given a faithful state. This construction can be generalized somewhat: it is sufficient to have a countably generated Hilbert module over the given C*-algebra $A,$ and then the given faithful state together with the usual GNS construction provides a GNS Hilbert space ${H}$ that envelopes the given Hilbert module. We clarify our statements by giving the construction briefly. Thus, we regard the elements of $E$ as belonging to a pre-Hilbert space with inner product $f(\left\langle e_{1},e_{2}\right\rangle)$. Since $f$ is a faithful state and $\left\langle e_{1},e_{2}\right\rangle$ is a Hilbert module inner product, the pre-inner product is faithful, and we don’t need to take quotients by a subspace of indefinite vectors as in the usual GNS construction. The upshot is that we obtain a Hilbert space $H$ inside which the given Hilbert module $E$ is a subspace. Generally the subspace will not be closed. We should clarify that this construction is not the induced representation (see (RW, , 24)) construction. Instead of forming a tensor product and building an imprimitivity bimodule, we are simply using the elements of the given Hilbert module $E$ as the elements for the GNS construction.

Now, we combine our lemma with some known results on Hilbert modules (MF, , Prop 3.3).

Now the promised striking geometrical interpretation of the above facts can be stated:

To specify which faithful state is being used to construct the Hilbert space (using the method discussed on page 2) we will say “enveloping GNS Hilbert space of ${H}_{A}$ with respect to $g$” or language to a similar effect.

In the setting of C*-algebraic quantum groups, we will observe a situation where a Hilbert space unitary acts on a Hilbert $A$-module but instead of being $A$-linear it has a skew-linear property that will be explained later. This skew linear property will give us the opportunity to make an interesting application of the above theorem.

Actually, there exist in the literature a few different conditions that are ultimately equivalent to isomorphism for Hilbert modules, and we can state a corollary clarifying the differences and rounding out our result by adding a few of these conditions:

Hopf algebras are bi-algebras with an antipode map $S.$ See abe for information on Hopf algebras in the algebraic setting. If one tries to frame the nice theory of Hopf algebras in a C*-algebraic setting, there are several slightly different approaches that can be taken.

The theory of Kats and Pal’yutkin KP1966 , its further development by Enock and Schwartz EnockSchwartz ; the theory of Woronowicz woronowicz1987 ; PW1990 , of Baaj and Skandalis BS ; BSk.Hopf ; BBS , and of Kustermans and Vaes KustermansVaes should especially be mentioned. The presence or absence of multiplicative units makes a difference in the theory, and generally the unital case, also known as the compact case, is simpler.

For example, in the especially attractive Baaj–Skandalis framework of C*-algebraic quantum groups provided by BS , a compact C*-algebraic quantum group is unital as an algebra, and has structure maps that are compatible with the C*-algebraic structure. In this picture, a compact quantum group $(A,\varDelta)$ is first of all a unital C^∗-algebra $A$ with a coproduct map. The coproduct map is a unital $*$-homomorphism $\varDelta\colon A\rightarrow A\otimes A$ such that $(\varDelta\otimes\iota)\varDelta=(\iota\otimes\varDelta)\varDelta$ and such that $\varDelta(A)(A\otimes 1)$ and $\varDelta(A)(1\otimes A)$ are dense in $A\otimes A$.

For any compact quantum group there exists a unique Haar state $\tau$ of $A$ which is left- and right-invariant, i.e., $(\iota\otimes\tau)\varDelta=\tau(\cdot)1$ and $(\tau\otimes\iota)\varDelta=\tau(\cdot)1$, respectively. This functional is called the Haar functional and is very usually assumed to be faithful.

Multiplicative unitaries were introduced by Baaj and Skandalis in BS (Chapter 4), see also BBS .

Given a compact quantum group $A$ with coproduct $\varDelta\colon A\rightarrow A\otimes_{min}A,$ and faithful Haar state $\tau\colon A\rightarrow\mathbb{C},$ let ${H}$ be the Hilbert space $L^{2}(A,\tau)$ that comes from the GNS construction applied to $A$ via $\tau.$ Let $e\in{H}$ be the image $e:=\pi_{\tau}1$ of $1$ in ${H}.$ Then, $Ae$ is dense in ${H}$ and there is a multiplicative unitary $V\in\mathcal{L}({H}\otimes{H})$ by

A different definition, $W,$ satisfying

was used in a more general setting (the locally compact case) by Kustermans and Vaes KustermansVaes , see also MaesVanDaele .

The next lemma reminds one of the formula for coproducts in terms of multiplicative unitaries: $\varDelta(a)=V(x\otimes 1)V^{*}.$ See, e.g. (BS, , Th. 3.8).

See Lemma 2 on page 2 for a more constructive and explicit proof of the above.

An operator $V$ having the property $V(\varDelta(a)x)=aV(x)$ will be referred to as having the skew-linear property. The skew-linear property implies ordinary linearity over $\mathbb{C},$ because the coproduct $\varDelta$ is a unital homomorphism and thus maps $\lambda\mbox{\rm Id}_{A}$ to $\lambda\mbox{\rm Id}_{A\otimes A}.$

We will see that there exists therefore a product on the Cuntz semigroup.

In the case of stable rank 1, which is all we need here, we may as well define the Cuntz semigroup of a C*-algebra as being given by the isomorphism classes of the countably generated Hilbert modules over that algebra, together with direct sum as a semigroup operation, and inclusion as a relation giving an order structure on that semigroup CEI . See CEI for the adaptations that would be needed in the case of higher stable rank.

We now verify that the above definition 1 does respect the equivalence relation on Hilbert modules in the Cuntz semigroup.

Compact C*-algebraic quantum groups have a well-behaved Fourier transform, which can be most briefly defined by, following Van Daele VanDaele1994 , see also kahng :

where the elements $a$ and $b$ belong to a C*-algebraic quantum group $A,$ $\tau$ is the left Haar weight, and $\beta(\cdot\,,\cdot)$ is the pairing with the dual algebra. Following (kahng, , Def. 3.10), an operator-valued convolution product, $\diamond,$ can then be defined by the property $\mbox{$F$}(a\diamond b)=\mbox{$F$}(a)\mbox{$F$}(b),$ where $a$ and $b$ are elements of a C*-algebraic quantum group $A,$ and $F$ is the Fourier transform defined previously.

This convolution operation takes, as is well-known, a pair of positive operators to a positive operator (EnockSchwartz, , Theorem 1.3.3.i). One could wonder if the product defined in the previous section could, at the level of generators, be viewed as some kind of generalized convolution. We now show that this is indeed the case, at least under the technical condition, related to Kac algebras ²²2Conventional transliteration of Cyrillic suggests the spelling Kats algebra, however, the spelling Kac has become standard., of having a tracial Haar state.

We now assume tracial Haar state for the next Proposition only. In the statement of the next Proposition, $\tau$ denotes the extension of the Haar state to the canonical trace related to the GNS Hilbert space associated with the Haar state.

In Theorem 2.2, if we take the product of two Hilbert modules $\overline{\ell_{1}A}$ and $\overline{\ell_{2}A},$ the product module will be some submodule of the standard Hilbert module $H_{A}.$ By Cohen’s theorem cohen we expect this submodule to be singly generated as a Hilbert module, and we may ask if there is some nice description of some particular choice of that generator.

In the case of stable rank one, everything comes together very nicely, as follows. Clearly, we can consider the equivalence classes of the projective modules within the Cuntz semigroup. It is known that equivalence of projective modules in the Cuntz semigroup is isomorphic to the ordinary equivalence of operator projections, in matrix algebras over $A$. We note that by the first part of Proposition 2, the product $\,\ast\,$ will in fact take a pair of operator projections to a projection. Thus, the product $\,\ast\,$ gives a product on the semigroup of equivalence classes of projections, $V(A).$

Stable rank one implies cancellation (blackadar, , Prop. 6.5.1) in this semigroup of projections. K-theory is defined, in the unital case, as the enveloping Grothendieck group of this semigroup $V(A)$ of projections. Products on semigroups do not always give products on the enveloping Grothendieck group, but will do so exactly in the case that the semigroup is cancellative Dale .

In fact, equation (1) seems to let us define a product on K-theory even without stable rank one.

The existence of the product then can be used to generalize our earlier results for the discrete case kuce.classify.Hopf.by.Ktheory , with a real rank zero condition and the existence of the product replacing discreteness.

The topic of Hopf ideals is mostly of interest in the finite-dimensional case, but we attempt to state the definitions and first result more generally. A Hopf ideal of a Hopf algebra is an algebra ideal in the kernel of the co-unit that is also a co-algebra co-ideal and is stable under the antipode. Equivalently, a Hopf ideal is the intersection of the kernel of the co-unit with a normal and unital Hopf subalgebra. We make the assumption that the antipode is bijective, and hence that left normality equals right normality.

We also point out that in general, we may obtain from our product construction two possibly inequivalent product structures, one associated with the left Haar state and the other associated with the right Haar state.

The main examples of Hopf ideals seem to be in the finite-dimensional case, and in this case, as is well-known, algebraic modules are automatically projective and the Cuntz semigroup in essence becomes $K$-theory. For example, see kuce.classify.Hopf.by.Ktheory0 for more information on the $K$-theory of finite-dimensional C*-algebraic quantum groups. If we consider the finite-dimensional case, then:

The above is possibly self-evident, but it is interesting to obtain it as a special case of a more general result.

The reason why the above is possibly self-evident is as follows: since the product of Hilbert modules that we have defined was constructed to be an analytical version of the algebraist’s operation of restriction of rings (by the co-product homomorphism), being moreover compatible with the inner products that a Hilbert module must necessarily have, one would expect that in the finite-dimensional case, and indeed in the AF case, the closure operation that appears in our definition of the product should have no real significance.

In these cases, we could presumably have worked entirely with algebraic modules and the algebraic restriction of rings operation, making use of the already mentioned fact that over such an algebra, finitely generated algebraic modules are automatically projective.

Suppose $(A,\varDelta)$ is a compact or discrete quantum group. A left coaction $\delta$ of $(A,\varDelta)$ on a C^∗-algebra $B$ is a $*$-homomorphism $\delta\colon B\to M(A\otimes B)$ such that $\delta(B)(A\otimes 1)$ is dense in $A\otimes B$ and that $(\mbox{\rm Id}\otimes\delta)\delta=(\varDelta\otimes\mbox{\rm Id})\delta$. For example, the coproduct homomorphism $\varDelta$ can be viewed as a left coaction of a compact quantum group $(A,\varDelta)$ on $A$ and $\widehat{\varDelta}$ as a right coaction of $(\widehat{A},\widehat{\varDelta})$ on $\hat{A}$. Indeed, following (martinboundary, , pg. 8), using a multiplicative unitary $W$ we may by the formulas

for $x\in\hat{A}$ and $a\in A$, define a left coaction $\Phi$ of $(A,\varDelta)$ on $\hat{A}$ and a right coaction $\hat{\Phi}$ of $(\hat{A},\hat{\varDelta})$ on $A$.

In this example, the co-action homomorphisms are injective, but that need not be the case in general.

An invariant state for a left co-action $\delta$ of a compact or discrete C*-algebraic quantum group $(A,\varDelta)$ on $B$ is a state $\eta$ on $B$ such that $(\mbox{\rm Id}\otimes\eta)\delta=\eta(\cdot)1.$ For the next lemma, we assume the existence of such a faithful invariant state, this is a strong but natural assumption.

From the above lemma, we thus obtain a unitary $W\colon{H}_{B}\rightarrow A\otimes B,$ at the level of Hilbert space structure, possessing a skew-linear property. Tensoring on both sides with the classical Hilbert space ${H}$, we can replace $A\otimes B$ with ${H}_{A\otimes B}$. Just as in Theorem 2.2 we obtain, in the stable rank one case, a product $\,\ast\,\colon{C}u(A)\times{C}u(B)\rightarrow{C}u(B).$

We can of course consider the special case where the co-action homomorphism is given by the co-product homomorphism, in other words, the case $A=B,$ and then we recover the theory discussed in section 2. The associativity type property of the co-action,

implies that the product induced in this way from a co-product is compatible with the product induced from a co-action. In other words, products of the form $A_{1}\,\ast\,A_{2}\,\ast\,B_{1},$ where $A_{i}$ is a Hilbert module over $A$ and $B_{1}$ is a Hilbert module over $B,$ are associative and we do not need to write parentheses. (c.f. Theorem 3.) Rephrasing, what we have shown is that the Cuntz semigroup functor takes a co-action to a multiplicative action:

From a desire to simply illustrate the ideas, we have given detailed proofs for the case of stable rank 1 only, but the method of DKAxiomsPreprint removes this restriction. In fact, if we consider Cuntz semigroups without restricting to stable rank 1, this amounts to using an equivalence relation coarser than just isomorphism of Hilbert modules, and it can be shown that the product we have defined is in fact still well-defined with respect to this coarser equivalence relation. Thus we claim that the above Theorem holds for higher stable rank as well.

We omit the details because Cuntz semigroups for the case of higher stable rank are best studied using the rather technical open projection picture of the Cuntz semigroup. The open projection picture resembles K-theory sufficiently that the techniques used kuce.classify.Hopf.by.Ktheory0 for classifying C*-algebraic quantum groups by K-theory may go through when K-theory is replaced by Cuntz semigroups.

It is quite possible that the above can be viewed as an algebraic topology type of obstacle to the existence of certain co-actions. There is considerable evidence that the Cuntz semigroup and sometimes K-theory contain a surprising amount of information about a C*-algebra. This being the case, one could expect the above invariant to also contain a lot of information. Certainly, the above can be used to provide algebraic obstacles to the existence of certain co-actions, and when co-actions do exist, it could be reasonable to in some sense classify them by their invariants in K-theory or the Cuntz semigroup.

The work of Goswami Goswami on non-existence of certain co-actions, again in the compact case, provides evidence that the question of nonexistence may be especially interesting in the compact case that we have been looking at. Eventually, applications may lie in the direction of the noncommutative Borsuk-Ulam theorem BDH2015 .

The author declares no conflict of interest.