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Let $A$ be an algebra with identity and $\Delta:A\to A\otimes A$ a coproduct that admits a counit. If there exist a faithful left integral and a faithful right integral, one can construct an antipode and $(A,\Delta)$ is a Hopf algebra. This is the Larson-Sweedler theorem. There are generalizations of this result for multiplier Hopf algebras, weak Hopf algebras and weak multiplier Hopf algebras. In the case of a multiplier Hopf algebra, the existence of a counit can be weakened and can be replaced by the requirement that the coproduct is full. A similar result is true for weak multiplier Hopf algebras. What we show in this note is that in fact the result for multiplier Hopf algebras can still be obtained without the condition of fullness of the coproduct. As it turns out, this property will already follow from the other conditions. Consequently, also in the original theorem for Hopf algebras, the existence of a counit is a consequence of the other conditions. This slightly generalizes the original result. The situation for weak multiplier Hopf algebras seems to be more subtle. We discuss the problems and see what is still possible here. We consider these results in connection with the development of the theory of locally compact quantum groups. This is discussed in an appendix.

Let $A$ be a non-degenerate algebra over the complex numbers and $\Delta$ a homomorphism from $A$ to the multiplier algebra $M(A\otimes A)$. Consider the linear maps $T_1$ and $T_2$ from $A\otimes A$ to $M(A\otimes A)$ defined by \begin{equation*} T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\text{and}\qquad T_2(c\otimes a)=(c\otimes 1)\Delta(a). \end{equation*} The pair $(A,\Delta)$ is a multiplier Hopf algebra if these two maps have range in $A\otimes A$ and are bijections from $A\otimes A$ to itself. In our recent paper on the Larson-Sweedler theorem, single sided multiplier Hopf algebras emerge in a natural way. For this case, instead of requiring the above for the maps $T_1$ and $T_2$, we now have this property for the maps $T_1$ and $T_4$ or for $T_2$ and $T_3$ where \begin{equation*} T_3(a\otimes b)=(1\otimes b)\Delta(a) \qquad\text{and}\qquad T_4(c\otimes a)=\Delta(a)(c\otimes 1). \end{equation*} As it turns out, also for these single sided multiplier Hopf algebras, the existence of a unique counit and antipode can be proven. In fact, rather surprisingly, using the properties of the antipode, one can actually show that for a single sided multiplier Hopf algebra all four canonical maps are bijections from $A\otimes A$ to itself. In other words, $(A,\Delta)$ is automatically a regular multiplier Hopf algebra. We take the advantage of this approach to reconsider some of the known results for a regular multiplier Hopf algebra.

A coproduct on a vector space $A$ is defined as a linear map $\Delta:A\to A\otimes A$ satisfying coassociativity $(\Delta\otimes\iota)\Delta=(\iota\otimes\Delta)\Delta$. We use $\iota$ for the identity map. If $G$ is a finite group and if $A$ is the space of all complex functions on $G$, a coproduct on $A$ is defined by $\Delta(f)(p,q)=f(pq)$ where $p,q\in G$. We identify $A\otimes A$ with complex functions on the Cartesian product $G\times G$. Coassociativity follows from the associativity of the product in $G$. Unfortunately, sometimes this notion of a coproduct is not the appropriate one. In this note, we consider the case of an algebra $A$, not necessarily unital but with a non-degenerate product. Now a coproduct is a linear map from $A$ to $M(A\otimes A)$, the multiplier algebra of $A\otimes A$. Unfortunately, it is no longer possible to express coassociativity in its usual form as the maps $\Delta\otimes\iota$ and $\iota\otimes \Delta$, defined on $A\otimes A$, may no longer be defined on the multiplier algebra $M(A\otimes A)$. Similar problems occurs when we want to define a useful notion of a coaction in the case of non-unital algebras. We discuss this in another paper. Not all the results we present in this paper are new. We provide a number of references to the original papers where some of this material is treated. However, in the original papers, results are not always found in an organized form and we hope to improve that here. Further a few solutions to some open questions are included as well as some more peculiar examples. Finally, we discuss some open problems and possible further research.

This is the last part of a series of three papers on the subject. In the first part we have considered the duality of algebraic quantum groups. In that paper, we use the term algebraic quantum group for a regular multiplier Hopf algebra with integrals. We treat the duality as studied in that theory. In the second part we have considered the duality for multiplier Hopf $^*$-algebras with positive integrals. The main purpose of that paper is to explain how it gives rise to a locally compact quantum group in the sense of Kustermans and Vaes. In a preliminary section of the second part we have mentioned the analytic structure of such a $^*$-algebraic quantum group. We have not gone into the details, except for that part needed to obtain that the scaling constant is trivial and for proving that the composition of a positive left integral with the antipode gives a right integral that is again positive. Also the construction of the square root. of the modular element is discussed and used to obtain the two Haar weights on the associated locally compact quantum group. In this paper we treat the analytic structure of the $^*$-algebraic quantum group in detail. The analytic structure is intimately related with the modular structure. We obtain a collection of interesting formulas, relating the two structures. Further we compare these results with formulas obtained in the framework of general locally compact quantum groups. As for the first two papers in this series, also here no new results are obtained. On the other hand the approach is different, more direct and instructive. It might help the reader for understanding the more general theory of locally compact quantum groups.

In the paper Algebraic quantum groups and duality I, we consider a pairing $(a,b)\mapsto\langle a,b\rangle$ of regular multiplier Hopf algebras $A$ and $B$. When $A$ has integrals and when $B$ is the dual of $A$, we can describe the duality with an element $V$ in the multiplier algebra $M(B\otimes A)$ satisfying and defined by $\langle V, a\otimes b\rangle=\langle a,b\rangle$ for all $a,b$. Properties of the dual pair are formulated in terms of this multiplier $V$. It acts, in a natural way, on $A\otimes A$ as the canonical map $T$, given by $T(a\otimes a')=\Delta(a)(1\otimes a')$. In this second paper on the subject, we assume that the pairing is coming from a multiplier Hopf $^*$-algebra with positive integrals. In this case, the positive right integral on $A$ can be used to construct a Hilbert space $\mathcal H$. The duality $V$ now acts as a unitary operator on the Hilbert space tensor product $\mathcal H\otimes\mathcal H$. This eventually makes it possible to complete the algebraic quantum group to a locally compact quantum group. The procedure to pass from the algebraic quantum group to the operator algebraic completion has been treated in the literature but the construction is rather involved because of the necessary use of left Hilbert algebras. In this paper, we give a comprehensive, yet concise and somewhat simpler approach. It should be considered as a springboard to the more complicated theory of locally compact quantum groups.

Let $(A,\Delta)$ be a finite-dimensional Hopf algebra. The linear dual $B$ of $A$ is again a finite-dimensional Hopf algebra. The duality is given by an element $V\in B\otimes A$, defined by $\langle V,a\otimes b\rangle=\langle a,b\rangle$ where $a\in A$ and $b\in B$. We use $\langle\,\cdot\, , \,\cdot\,\rangle$ for the pairings. In the introduction of this paper, we recall the various properties of this element $V$ as sitting in the algebra $B\otimes A$. More generally, we can consider an algebraic quantum group $(A,\Delta)$. We use the term here for a regular multiplier Hopf algebra with integrals. For $B$ we now take the dual $\widehat A$ of $A$. It is again an algebraic quantum group. In this case, the duality gives rise to an element $V$ in the multiplier algebra $M(B\otimes A)$. Still, most of the properties of $V$ in the finite-dimensional case are true in this more general setting. The focus in this paper lies on various aspects of the duality between $A$ and its dual $\widehat A$. Among other things we include a number of formulas relating the objects associated with an algebraic quantum group and its dual. This note is meant to give a comprehensive, yet concise (and sometimes simpler) account of these known results. This is part I of a series of three papers on this subject. The case of a multiplier Hopf $^*$-algebra with positive integrals is treated in detail in part II and part III.

Quantum groups have been studied within several areas of mathematics and mathematical physics. This has led to different approaches, each of them with their own techniques and conventions. Starting with Hopf algebras, where there is a general consensus, moving in the direction of topological quantum groups, where there is no such consensus, it is easy to get lost. Not only many difficulties have to be overcome, but also several choices must be made. The way this is done is often confusing. Some choices even turn out to be rather annoying. As an introductory lecture at the conference on Topological quantum groups and Hopf algebras in 2016, we have explained these choices, difficulties and annoyances, encountered on the road from Hopf algebras to topological quantum groups. In these notes, we discuss more aspects of the development of locally compact quantum groups. We not only explain some of these difficulties in greater detail, but we also give background information about the different steps, combined with some historical comments. We start with finite quantum groups and continue with discrete quantum groups, compact quantum groups and algebraic quantum groups. Multiplicative unitaries are an important side track before we finally arrive at locally compact quantum groups. Along the way, we also formulate some interesting remaining problems in the theory.

Let $B$ and $C$ be non-degenerate idempotent algebras and assume that $E$ is a regular separability idempotent in $M(B\otimes C)$. Define $A=C\otimes B$ and $\Delta:A\to M(A\otimes A)$ by $\Delta(c\otimes b)=c\otimes E\otimes b$. The pair $(A,\Delta)$ is a weak multiplier Hopf algebra. Because we assume that $E$ is regular, it is a regular weak multiplier Hopf algebra. There is a faithful left integral on $(A,\Delta)$ that is also right invariant. Therefore, we call $(A,\Delta)$ a unimodular algebraic quantum groupoid. By the general theory, the dual $(\widehat A,\widehat \Delta)$ can be constructed and it is again an algebraic quantum groupoid. In this paper, we treat this algebraic quantum groupoid and its dual in great detail. The main purpose is to illustrate various aspects of the general theory. For this reason, we will also recall the basic notions and results of separability idempotents and weak multiplier Hopf algebras with integrals. The paper is to be considered as an expository note., Comment: arXiv admin note: text overlap with arXiv:1701.04951

Let $(A,\Delta)$ be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra $A$, with or without identity, and a coproduct $\Delta$ on $A$, satisfying certain properties. The main difference with multiplier Hopf algebras is that now, the canonical maps $T_1$ and $T_2$ on $A\otimes A$, defined by $$T_1(a\otimes b)=\Delta(a)(1\otimes b) \qquad\quad\text{and}\qquad\quad T_2(c\otimes a)=(c\otimes 1)\Delta(a),$$ are no longer assumed to be bijective. Also recall that a weak multiplier Hopf algebra is called regular if its antipode is a bijective map from $A$ to itself. In this paper, we introduce and study the notion of integrals on such regular weak multiplier Hopf algebras. A left integral is a non-zero linear functional on $A$ that is left invariant (in an appropriate sense). Similarly for a right integral. For a regular weak multiplier Hopf algebra $(A,\Delta)$ with (sufficiently many) integrals, we construct the dual $(\widehat A,\widehat\Delta)$. It is again a regular weak multiplier Hopf algebra with (sufficiently many) integrals. This duality extends the known duality of finite-dimensional weak Hopf algebras to this more general case. It also extends the duality of multiplier Hopf algebras with integrals, the so-called algebraic quantum groups. For this reason, we will sometimes call a regular weak multiplier Hopf algebra with enough integrals an algebraic quantum groupoid. We discuss the relation of our work with the work on duality for algebraic quantum groupoids by Timmermann. We also illustrate this duality with a particular example in a separate paper. In this paper, we only mention the main definitions and results for this example. However, we do consider the two natural weak multiplier Hopf algebras associated with a groupoid in detail and show that they are dual to each other in the sense of the above duality.

Let $(A,\Delta)$ be a regular weak multiplier Hopf algebra. Denote by $E$ the canonical idempotent of $(A,\Delta)$ and by $B$ the image of the source map. Recall that $B$ is a non-degenerate algebra, sitting nicely in the multiplier algebra $M(A)$ of $A$ so that also $M(B)$ can be viewed as a subalgebra of $M(A)$. Assume that $u,v$ are invertible elements in $M(B)$ so that $E(vu\otimes 1)E=E$. This last condition is obviously fulfilled if $u$ and $v$ are each other inverses, but there are also other cases. Now modify $\Delta$ and define $\Delta'(a)=(u\otimes 1)\Delta(a)(v\otimes 1)$ for all $a \in A$. We show in this paper that $(A,\Delta')$ is again a regular weak multiplier Hopf algebra and we obtain formulas for the various data of $(A,\Delta')$ in terms of the data associated with the original pair $(A,\Delta)$. In the case of a finite-dimensional weak Hopf algebra, the above deformation is a special case of the twists as studied by Nikshych and Vainerman. It is known that any regular weak multiplier Hopf algebra gives rise in a natural way to a regular multiplier Hopf algebroid. This result applies to both the original weak multiplier Hopf algebra $(A,\Delta)$ and the modified version $(A,\Delta')$. However, the same method can be used to associate another regular multiplier Hopf algebroid to the triple $(A,\Delta,\Delta')$. This turns out to give an example of a regular multiplier Hopf algebroid that does not arise from a regular weak multiplier Hopf algebra although the base algebra is separable Frobenius.

It is well-known that any weak Hopf algebra gives rise to a Hopf algebroid. Moreover it is possible to characterize those Hopf algebroids that arise in this way. Recently, the notion of a weak Hopf algebra has been extended to the case of algebras without identity. This led to the theory of weak multiplier Hopf algebras. Similarly also the theory of Hopf algebroids was recently developed for algebras without identity. They are called multiplier Hopf algebroids. Then it is quite natural to investigate the expected link between weak multiplier Hopf algebras and multiplier Hopf algebroids. This relation has been considered already in the original paper on multiplier Hopf algebroids. In this note, we investigate the connection further. First we show that any regular weak multiplier Hopf algebra gives rise, in a natural way, to a regular multiplier Hopf algebroid. Secondly we give a characterization, mainly in terms of the base algebra, for a regular multiplier Hopf algebroid to have an underlying weak multiplier Hopf algebra. We illustrate this result with some examples. In particular, we give examples of multiplier Hopf algebroids that do not arise from a weak multiplier Hopf algebra.

The Larson-Sweedler theorem says that a finite-dimensional bialgebra with a faithful integral is a Hopf algebra. The result has been generalized to finite-dimensional weak Hopf algebras by Vecserny\'es. In this paper, we show that the result is still true for weak multiplier Hopf algebras. The notion of a weak multiplier bialgebra was introduced by B\"ohm, G\'omez-Torrecillas and L\'opez-Centella. In this note it is shown that a weak multiplier bialgebra with a regular and full coproduct is a regular weak multiplier Hopf algebra if there is a faithful set of integrals. This result is important for the development of the theory of locally compact quantum groupoids in the operator algebra setting. Our treatment of this material is motivated by the prospect of such a theory.

In this paper, we continue the study of weak multiplier Hopf algebras. We recall the notions of the source and target maps $\varepsilon_s$ and $\varepsilon_t$, as well as of the source and target algebras. Then we investigate these objects further. Among other things, we show that the canonical idempotent $E$ (which is eventually $\Delta(1)$) belongs to the multiplier algebra $M(B\otimes C)$ where $B=\varepsilon_s(A)$ and $C=\varepsilon_t(A)$ and that it is a separability idempotent. We also consider special cases and examples in this paper. In particular, we see how for any weak multiplier Hopf algebra, it is possible to make $C\otimes B$ (with $B$ and $C$ as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the 'Hopf algebra part' of the original weak multiplier Hopf algebra and only remembers the source and target algebras. It is in turn generalized to the case of any pair of non-degenerate algebras $B$ and $C$ with a separability idempotent $E\in M(B\otimes C)$. We get another example using this theory associated to any discrete quantum group (a multiplier Hopf algebra of discrete type with a normalized cointegral). Finally we also consider the well-known 'quantization' of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras).

Consider two non-degenerate algebras B and C over the complex numbers. We study a certain class of idempotent elements E in the multiplier algebra of the tensor product of B with C, called separability idempotents. The conditions include the existence of non-degenerate anti-homomorphisms from B to M(C) and from C to M(B), the multiplier algebras of C and B respectively. They are called the antipodal maps. There also exist what we call distinguished linear functionals. They are unique and faithful in the regular case. The notion is more restrictive than what is generally considered in the case of (finite-dimensional) algebras with identity. The separability idempotents we consider in this paper are of a Frobenius type. It seems to be quite natural to consider this type of separability idempotents in the case of non-degenerate algebras possibly without an identity. One example is coming from a discrete quantum group A. Here we take B=C=A and E is the image under the coproduct of h, where h is the normalized cointegral. The antipodal maps coincide with the original antipode and the distinguished linear functionals are the integrals. Another example is obtained from a weak multiplier Hopf algebra A. Now B and C are the images of the source and target maps. For E we take the canonical idempotent. Again the antipodal maps come from the antipode of the weak multiplier Hopf algebra. These two examples have motivated the study of separability idempotents as used in this paper. We use the separability idempotent to study modules over the base algebras B and C. In the regular case, we obtain a structure theorem for these algebras.

A weak multiplier Hopf algebra is a pair (A,\Delta) of a non-degenerate idempotent algebra A and a coproduct $\Delta$ on A. The coproduct is a coassociative homomorphism from A to the multiplier algebra M(A\otimes A) with some natural extra properties (like the existence of a counit). Further we impose extra but natural conditions on the ranges and the kernels of the canonical maps T_1 and T_2 defined from A\otimes A to M(A\otimes A) by T_1(a\otimes b)=\Delta(a)(1\otimes b) and T_2(a\ot b)=(a\otimes 1)\Delta(b). The first condition is about the ranges of these maps. It is assumed that there exists an idempotent element E\in M(A\otimes A) such that \Delta(A)(1\ot A)=E(A\ot A) and (A\otimes 1)\Delta(A)=(A\otimes A)E. The second condition determines the behavior of the coproduct on the legs of E. We require (\Delta\otimes \iota)(E)=(\iota\otimes\Delta)(E)=(1\otimes E)(E\ot 1)=(E\otimes 1)(1\otimes E) where $\iota$ is the identity map and where $\Delta\otimes \iota$ and $\iota\otimes\Delta$ are extensions to the multipier algebra M(A\otimes A). Finally, the last condition determines the kernels of the canonical maps T_1 and T_2 in terms of this idempotent E by a very specific relation. From these conditions we develop the theory. In particular, we construct a unique antipode satisfying the expected properties and various other data. Special attention is given to the regular case (that is when the antipode is bijective) and the case of a *-algebra (where regularity is automatic). Weak Hopf algebras are special cases of such weak multiplier Hopf algebras. Conversely, if the underlying algebra of a (regular) weak multiplier Hopf algebra has an identity, it is a weak Hopf algebra. Also any groupoid, finite or not, yields two weak multiplier Hopf algebras in duality.

Let $G$ be a {\it finite group}. Consider the algebra $A$ of all complex functions on G (with pointwise product). Define a coproduct $\Delta$ on A by $\Delta(f)(p,q)=f(pq)$ where $f\in A$ and $p,q\in G$. Then $(A,\Delta)$ is a Hopf algebra. If $G$ is only a {\it groupoid}, so that the product of two elements is not always defined, one still can consider $A$ and define $\Delta(f)(p,q)$ as above when $pq$ is defined. If we let $\Delta(f)(p,q)=0$ otherwise, we still get a coproduct on $A$, but $\Delta(1)$ will no longer be the identity in $A\ot A$. The pair $(A,\Delta)$ is not a Hopf algebra but a weak Hopf algebra. If $G$ is a {\it group}, but {\it no longer finite}, one takes for $A$ the algebra of functions with finite support. Then $A$ has no identity and $(A,\Delta)$ is not a Hopf algebra but a multiplier Hopf algebra. Finally, if $G$ is a {\it groupoid}, but {\it not necessarily finite}, the standard construction above, will give, what we call in this paper, a weak multiplier Hopf algebra. Indeed, this paper is devoted to the development of this 'missing link': {\it weak multiplier Hopf algebras}. We spend a great part of this paper to the motivation of our notion and to explain where the various assumptions come from. The goal is to obtain a good definition of a weak multiplier Hopf algebra. Throughout the paper, we consider the basic examples and use them, as far as this is possible, to illustrate what we do. In particular, we think of the finite-dimensional weak Hopf algebras. On the other hand however, we are also inspired by the far more complicated existing analytical theory. In forthcoming papers on the subject, we develop the theory further.

Let $A$ and $B$ be two algebraic quantum groups (i.e. multiplier Hopf algebras with integrals). Assume that $B$ is a right $A$-module algebra and that $A$ is a left $B$-comodule coalgebra. If the action and coaction are matched, it is possible to define a coproduct $\Delta_#$ on the smash product $A # B$ making the pair $(A # B,\Delta_#)$ into an algebraic quantum group. In this paper, we continue the study of these objects. First, we study the various data of the bicrossproduct $A # B$, such as the modular automorphisms, the modular elements, ... and obtain formulas in terms of the data of the components $A$ and $B$. Secondly, we look at the dual of $A # B$ (in the sense of algebraic quantum groups) and we show it is itself a bicrossproduct (of the second type) of the duals $\hatA$ and $\hatB$. The result is immediate for finite-dimensional Hopf algebras and therefore it is expected also for algebraic quantum groups. However, it turns out that some aspects involve a careful argument, mainly due to the fact that coproducts and coactions have ranges in the multiplier algebras of the tensor products and not in the tensor product itself. Finally, we also treat some examples in this paper. We have included some of the examples that are known for finite-dimensional Hopf algebras and show how they can also be obtained for more general algebraic quantum groups. We also give some examples that are more typical for algebraic quantum groups. In particular, we focus on the extra structure, provided by the integrals and associated objects. It should be mentioned that with examples of bicrossproducts of algebraic quantum groups, we do get examples that are essentially different from those commonly known in Hopf algebra theory.

In this paper, we generalize Majid's bicrossproduct construction. We start with a pair (A,B) of two regular multiplier Hopf algebras. We assume that B is a right A-module algebra and that A is a left B-comodule coalgebra. We recall and discuss the two notions in the first sections of the paper. The right action of A on B gives rise to the smash product A # B. The left coaction of B on A gives a possible coproduct on A # B. We will discuss in detail the necessary compatibility conditions between the action and the coaction for this to be a proper coproduct on A # B. The result is again a regular multiplier Hopf algebra. Majid's construction is obtained when we have Hopf algebras. We also look at the dual case, constructed from a pair (C,D) of regular multiplier Hopf algebras where now C is a left D-module algebra while D is a right C-comodule coalgebra. We will show that indeed, these two constructions are dual to each other in the sense that a natural pairing of A with C and of B with D will yield a duality between A # B and the smash product C # D. We show that the bicrossproduct of algebraic quantum groups is again an algebraic quantum group (i.e. a regular multiplier Hopf algebra with integrals). The *-algebra case will also be considered. Some special cases will be treated and they will be related with other constructions available in the literature. Finally, the basic example, coming from a (not necessarily finite) group G with two subgroups H and K such that G=KH and the intersection of H and K is trivial will be used throughout the paper for motivation and illustration of the different notions and results. The cases where either H or K is a normal subgroup will get special attention.

Let $(A,\Delta)$ be a multiplier Hopf algebra. In general, the underlying algebra $A$ need not have an identity and the coproduct $\Delta$ does not map $A$ into $A\otimes A$ but rather into its multiplier algebra $M(A\otimes A)$. In this paper, we study {\it some tools} that are frequently used when dealing with such multiplier Hopf algebras and that are typical for working with algebras without identity in this context. The {\it basic ingredient} is a unital left $A$-module $X$. And the basic construction is that of extending the module by looking at linear maps $\rho:A\to X$ satisfying $\rho(aa')=a\rho(a')$ where $a,a'\in A$. We write the module action as multiplication. Of course, when $x\in X$, and when $\rho(a)=ax$, we get such a linear map. And if $A$ has an identity, all linear maps $\rho$ have this form for $x=\rho(1)$. However, the point is that in the case of a non-unital algebra, the space of such maps is in general strictly bigger than $X$ itself. We get an {\it extended module}, denoted by $\bar X$ (for reasons that will be explained in the paper). We study all sorts of more complicated situations where such extended modules occur and we illustrate all of this with {\it several examples}, from very simple ones to more complex ones where iterated extensions come into play. We refer to cases that appear in the literature. We use this basic idea of extending modules to explain, in a more rigorous way, the so-called {\it covering technique}, which is needed when using {\it Sweedler notations} for coproducts and coactions. Again, we give many examples and refer to the existing literature where this technique is applied.