Before we can even begin to properly define a locally compact quantum group, we first need to define a number of preliminary concepts and also state a few theorems.

Definition (weight). Let ${\displaystyle A}$  be a C*-algebra, and let ${\displaystyle A_{\geq 0}}$  denote the set of positive elements of ${\displaystyle A}$ . A weight on ${\displaystyle A}$  is a function ${\displaystyle \phi :A_{\geq 0}\to [0,\infty ]}$  such that

• ${\displaystyle \phi (a_{1}+a_{2})=\phi (a_{1})+\phi (a_{2})}$  for all ${\displaystyle a_{1},a_{2}\in A_{\geq 0}}$ , and
• ${\displaystyle \phi (r\cdot a)=r\cdot \phi (a)}$  for all ${\displaystyle r\in [0,\infty )}$  and ${\displaystyle a\in A_{\geq 0}}$ .

Some notation for weights. Let ${\displaystyle \phi }$  be a weight on a C*-algebra ${\displaystyle A}$ . We use the following notation:

• ${\displaystyle {\mathcal {M}}_{\phi }^{+}:=\{a\in A_{\geq 0}\mid \phi (a)<\infty \}}$ , which is called the set of all positive ${\displaystyle \phi }$ -integrable elements of ${\displaystyle A}$ .
• ${\displaystyle {\mathcal {N}}_{\phi }:=\{a\in A\mid \phi (a^{*}a)<\infty \}}$ , which is called the set of all ${\displaystyle \phi }$ -square-integrable elements of ${\displaystyle A}$ .
• ${\displaystyle {\mathcal {M}}_{\phi }:={\text{Span}}~{\mathcal {M}}_{\phi }^{+}={\text{Span}}~{\mathcal {N}}_{\phi }^{*}{\mathcal {N}}_{\phi }}$ , which is called the set of all ${\displaystyle \phi }$ -integrable elements of ${\displaystyle A}$ .

Types of weights. Let ${\displaystyle \phi }$  be a weight on a C*-algebra ${\displaystyle A}$ .

• We say that ${\displaystyle \phi }$  is faithful if and only if ${\displaystyle \phi (a)\neq 0}$  for each non-zero ${\displaystyle a\in A_{\geq 0}}$ .
• We say that ${\displaystyle \phi }$  is lower semi-continuous if and only if the set ${\displaystyle \{a\in A_{\geq 0}\mid \phi (a)\leq \lambda \}}$  is a closed subset of ${\displaystyle A}$  for every ${\displaystyle \lambda \in [0,\infty ]}$ .
• We say that ${\displaystyle \phi }$  is densely defined if and only if ${\displaystyle {\mathcal {M}}_{\phi }^{+}}$  is a dense subset of ${\displaystyle A_{\geq 0}}$ , or equivalently, if and only if either ${\displaystyle {\mathcal {N}}_{\phi }}$  or ${\displaystyle {\mathcal {M}}_{\phi }}$  is a dense subset of ${\displaystyle A}$ .
• We say that ${\displaystyle \phi }$  is proper if and only if it is non-zero, lower semi-continuous and densely defined.

Definition (one-parameter group). Let ${\displaystyle A}$  be a C*-algebra. A one-parameter group on ${\displaystyle A}$  is a family ${\displaystyle \alpha =(\alpha _{t})_{t\in \mathbb {R} }}$  of *-automorphisms of ${\displaystyle A}$  that satisfies ${\displaystyle \alpha _{s}\circ \alpha _{t}=\alpha _{s+t}}$  for all ${\displaystyle s,t\in \mathbb {R} }$ . We say that ${\displaystyle \alpha }$  is norm-continuous if and only if for every ${\displaystyle a\in A}$ , the mapping ${\displaystyle \mathbb {R} \to A}$  defined by ${\displaystyle t\mapsto {\alpha _{t}}(a)}$  is continuous (surely this should be called strongly continuous?).

Definition (analytic extension of a one-parameter group). Given a norm-continuous one-parameter group ${\displaystyle \alpha }$  on a C*-algebra ${\displaystyle A}$ , we are going to define an analytic extension of ${\displaystyle \alpha }$ . For each ${\displaystyle z\in \mathbb {C} }$ , let

${\displaystyle I(z):=\{y\in \mathbb {C} \mid |\Im (y)|\leq |\Im (z)|\}}$ ,

which is a horizontal strip in the complex plane. We call a function ${\displaystyle f:I(z)\to A}$  norm-regular if and only if the following conditions hold:

• It is analytic on the interior of ${\displaystyle I(z)}$ , i.e., for each ${\displaystyle y_{0}}$  in the interior of ${\displaystyle I(z)}$ , the limit ${\displaystyle \displaystyle \lim _{y\to y_{0}}{\frac {f(y)-f(y_{0})}{y-y_{0}}}}$  exists with respect to the norm topology on ${\displaystyle A}$ .
• It is norm-bounded on ${\displaystyle I(z)}$ .
• It is norm-continuous on ${\displaystyle I(z)}$ .

Suppose now that ${\displaystyle z\in \mathbb {C} \setminus \mathbb {R} }$ , and let

${\displaystyle D_{z}:=\{a\in A\mid {\text{There exists a norm-regular}}~f:I(z)\to A~{\text{such that}}~f(t)={\alpha _{t}}(a)~{\text{for all}}~t\in \mathbb {R} \}.}$ 

Define ${\displaystyle \alpha _{z}:D_{z}\to A}$  by ${\displaystyle {\alpha _{z}}(a):=f(z)}$ . The function ${\displaystyle f}$  is uniquely determined (by the theory of complex-analytic functions), so ${\displaystyle \alpha _{z}}$  is well-defined indeed. The family ${\displaystyle (\alpha _{z})_{z\in \mathbb {C} }}$  is then called the analytic extension of ${\displaystyle \alpha }$ .

Theorem 1. The set ${\displaystyle \cap _{z\in \mathbb {C} }D_{z}}$ , called the set of analytic elements of ${\displaystyle A}$ , is a dense subset of ${\displaystyle A}$ .

Definition (K.M.S. weight). Let ${\displaystyle A}$  be a C*-algebra and ${\displaystyle \phi :A_{\geq 0}\to [0,\infty ]}$  a weight on ${\displaystyle A}$ . We say that ${\displaystyle \phi }$  is a K.M.S. weight ('K.M.S.' stands for 'Kubo-Martin-Schwinger') on ${\displaystyle A}$  if and only if ${\displaystyle \phi }$  is a proper weight on ${\displaystyle A}$  and there exists a norm-continuous one-parameter group ${\displaystyle (\sigma _{t})_{t\in \mathbb {R} }}$  on ${\displaystyle A}$  such that

• ${\displaystyle \phi }$  is invariant under ${\displaystyle \sigma }$ , i.e., ${\displaystyle \phi \circ \sigma _{t}=\phi }$  for all ${\displaystyle t\in \mathbb {R} }$ , and
• for every ${\displaystyle a\in {\text{Dom}}(\sigma _{i/2})}$ , we have ${\displaystyle \phi (a^{*}a)=\phi (\sigma _{i/2}(a)[\sigma _{i/2}(a)]^{*})}$ .

We denote by ${\displaystyle M(A)}$  the multiplier algebra of ${\displaystyle A}$ .

Theorem 2. If ${\displaystyle A}$  and ${\displaystyle B}$  are C*-algebras and ${\displaystyle \pi :A\to M(B)}$  is a non-degenerate *-homomorphism (i.e., ${\displaystyle \pi [A]B}$  is a dense subset of ${\displaystyle B}$ ), then we can uniquely extend ${\displaystyle \pi }$  to a *-homomorphism ${\displaystyle {\overline {\pi }}:M(A)\to M(B)}$ .

Theorem 3. If ${\displaystyle \omega :A\to \mathbb {C} }$  is a state (i.e., a positive linear functional of norm ${\displaystyle 1}$ ) on ${\displaystyle A}$ , then we can uniquely extend ${\displaystyle \omega }$  to a state ${\displaystyle {\overline {\omega }}:M(A)\to \mathbb {C} }$  on ${\displaystyle M(A)}$ .

Definition (Locally compact quantum group). A (C*-algebraic) locally compact quantum group is an ordered pair ${\displaystyle {\mathcal {G}}=(A,\Delta )}$ , where ${\displaystyle A}$  is a C*-algebra and ${\displaystyle \Delta :A\to M(A\otimes A)}$  is a non-degenerate *-homomorphism called the co-multiplication, that satisfies the following four conditions:

• The co-multiplication is co-associative, i.e., ${\displaystyle {\overline {\Delta \otimes \iota }}\circ \Delta ={\overline {\iota \otimes \Delta }}\circ \Delta }$ .
• The sets ${\displaystyle \left\{{\overline {\omega \otimes {\text{id}}}}(\Delta (a))~{\big |}~\omega \in A^{*},~a\in A\right\}}$  and ${\displaystyle \left\{{\overline {{\text{id}}\otimes \omega }}(\Delta (a))~{\big |}~\omega \in A^{*},~a\in A\right\}}$  are linearly dense subsets of ${\displaystyle A}$ .
• There exists a faithful K.M.S. weight ${\displaystyle \phi }$  on ${\displaystyle A}$  that is left-invariant, i.e., ${\displaystyle \phi \!\left({\overline {\omega \otimes {\text{id}}}}(\Delta (a))\right)={\overline {\omega }}(1_{M(A)})\cdot \phi (a)}$  for all ${\displaystyle \omega \in A^{*}}$  and ${\displaystyle a\in {\mathcal {M}}_{\phi }^{+}}$ .
• There exists a K.M.S. weight ${\displaystyle \psi }$  on ${\displaystyle A}$  that is right-invariant, i.e., ${\displaystyle \psi \!\left({\overline {{\text{id}}\otimes \omega }}(\Delta (a))\right)={\overline {\omega }}(1_{M(A)})\cdot \psi (a)}$  for all ${\displaystyle \omega \in A^{*}}$  and ${\displaystyle a\in {\mathcal {M}}_{\phi }^{+}}$ .

From the definition of a locally compact quantum group, it can be shown that the right-invariant K.M.S. weight ${\displaystyle \psi }$  is automatically faithful. Therefore, the faithfulness of ${\displaystyle \psi }$  is a redundant condition and does not need to be postulated.

The category of locally compact quantum groups allows for a dual construction with which one can prove that the bi-dual of a locally compact quantum group is isomorphic to the original one. This result gives a far-reaching generalization of Pontryagin duality for locally compact Hausdorff abelian groups.

The theory has an equivalent formulation in terms of von Neumann algebras.

• Locally compact space
• Locally compact field
• Locally compact group

• Johan Kustermans & Stefaan Vaes. "Locally Compact Quantum Groups." Annales Scientifiques de l’École Normale Supérieure. Vol. 33, No. 6 (2000), pp. 837–934.
• Thomas Timmermann. "An Invitation to Quantum Groups and Duality – From Hopf Algebras to Multiplicative Unitaries and Beyond." EMS Textbooks in Mathematics, European Mathematical Society (2008).