nLab Hilbert bimodule

Idea

The notion of Hilbert $C^\ast$-bimodule adapts the notion of bimodules over associative algebras to operator algebra/C-star-algebra theory.

Definition

Definition

For $A,B \in$ C*Alg two C-star algebras, an $(A,B)$-Hilbert $C^\ast$-bimodule (or just Hilbert bimodule, for short) is

• a right $B$-Hilbert C*-module $(N, \langle -,-\rangle)$;

• equipped with a further left $A$-representation $A \to \mathcal{B}(N)$ by adjointable operators, hence such that $\langle a -,- \rangle = \langle -,a^\ast -\rangle$ for all $a \in A$.

A isomorphism between two $(A,B)$-bimodules $(N_1, \langle -,-\rangle) \to (N_2, \langle -,-\rangle_2)$ is a linear operator $N_1\to N_2$ which is unitary with respect to $\langle -,-\rangle_2$.

Definition

Given an $(A,B)$-Hilbert bimodule $(N_1, \langle -,-\rangle_1)$ and a $(B,C)$-Hilbert bimodule $(N_2, \langle -,-\rangle_2)$, the tensor product of Hilbert bimodules $N_1 \otimes_B N_2$ is the $(A,C)$-Hilbert bimodule obtained from the ordinary (algebraic) tensor product of modules over $\mathbb{C}$ $N_1 \otimes_{\mathbb{C}} N_2$ by

1. equipping it with the $C$-valued inner product defined by

   $$\left\langle \xi_1 \otimes \eta_1 , \xi_2 \otimes \eta_2\right\rangle \coloneqq \left\langle \eta_1, \left\langle \xi_1, \xi_2\right\rangle \cdot \eta_2 \right\rangle \,.$$

2. forming the quotient by the submodule of elements $v$ for which $\langle v,v\rangle = 0$;

3. forming the completion of this quotient with respect to the induced norm.

Remark

Def. really does yield a kind of tensor product over $B$: elements of the form

$$v \cdot b \otimes w - v \otimes b \cdot w$$

are in the submodule that it being divided out, because

$$\begin{aligned} \langle v \cdot b \otimes w - v \otimes b \cdot w \;,\; v \cdot b \otimes w - v \otimes b \cdot w \rangle & \coloneqq \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle \\ & + \langle b \cdot w, \langle v,v\rangle \cdot b \cdot w\rangle \\ & - \langle w, \langle v\cdot b, v \rangle \cdot b \cdot w\rangle \\ & - \langle b \cdot w, \langle v, v \cdot b \rangle \cdot w\rangle \\ & = (1+1-1-1) \langle w , \langle v \cdot b, v \cdot b\rangle \cdot w \rangle \\ & = 0 \end{aligned} \,,$$

where we use that by definition the left actions are required to have adjoints, so that for instance

$$\begin{aligned} \langle b \cdot w , \langle v,v\rangle \cdot b \cdot w\rangle & = \langle w , b^\ast \cdot \langle v,v\rangle \cdot b \cdot t\rangle \\ & = \langle w , \langle v \cdot b,v \cdot b\rangle \cdot w\rangle \end{aligned}$$

Proposition

There is a (2,1)-category $C^\ast Alg_b$ whose

• objects are C*-algebras;

• 1-morphisms are Hilbert bimodules;

• 2-morphisms are isomorphisms of Hilbert bimodules;

• vertical composition is the evident composition of intertwiners;

• horizontal composition is the tensor product of Hilbert bimodules, def. .

(Buss-Zhu-Meyer 09)

Examples

An $(\mathbb{C},B)$-Hilbert $C^\ast$-bimodule is equialently just an $B$-Hilbert C*-module.

An $(A, \mathbb{C})$-Hilbert $C^\ast$-bimodule is equivalently just as representation of a C-star-algebra.

Properties

(…)

Related concepts

• amplimorphism

• generalized elliptic operator

• KK-theory

• C*-correspondence

References

For instance

• Sergio Doplicher, Claudia Pinzari, Rita Zuccante, The $C^\ast$-algebra of a Hilbert Bimodule [arXiv:funct-an/9707006]

• Nik Weaver, Hilbert bimodules with involution, Canad. Math. Bull. 44 3 (2001) 355–369 [arXiv:math/9908119, pdf]

The tensor product of Hilbert bimodules and the induced 2-category structure is discussed in

• Alcides Buss, Chenchang Zhu, Ralf Meyer, A higher category approach to twisted actions on $C^\ast$-algebras (arXiv:0908.0455)