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Hilbert C*-modules are mathematical objects that generalise the notion of Hilbert spaces (which are themselves generalisations of Euclidean space), in that they endow a linear space with an "inner product" that takes values in a C*-algebra.

They were first introduced in the work of Irving Kaplansky in 1953, which developed the theory for commutative, unital algebras (though Kaplansky observed that the assumption of a unit element was not "vital").

In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras.

Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory, and provide the right framework to extend the notion of Morita equivalence to C*-algebras. They can be viewed as the generalization of vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory, and groupoid C*-algebras.

Definitions

Inner-product C*-modules

Let ${\displaystyle A}$ be a C*-algebra (not assumed to be commutative or unital), its involution denoted by ${\displaystyle {}^{*}}$. An inner-product ${\displaystyle A}$-module (or pre-Hilbert ${\displaystyle A}$-module) is a complex linear space ${\displaystyle E}$ equipped with a compatible right ${\displaystyle A}$-module structure, together with a map

${\displaystyle \langle \,\cdot \,,\,\cdot \,\rangle _{A}:E\times E\rightarrow A}$

that satisfies the following properties:

• For all ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$ in ${\displaystyle E}$, and ${\displaystyle \alpha }$, ${\displaystyle \beta }$ in ${\displaystyle \mathbb {C} }$:

${\displaystyle \langle x,y\alpha +z\beta \rangle _{A}=\langle x,y\rangle _{A}\alpha +\langle x,z\rangle _{A}\beta }$

(i.e. the inner product is ${\displaystyle \mathbb {C} }$-linear in its second argument).

• For all ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$, and ${\displaystyle a}$ in ${\displaystyle A}$:

${\displaystyle \langle x,ya\rangle _{A}=\langle x,y\rangle _{A}a}$

• For all ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$:

${\displaystyle \langle x,y\rangle _{A}=\langle y,x\rangle _{A}^{*},}$

from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).

• For all ${\displaystyle x}$ in ${\displaystyle E}$:

${\displaystyle \langle x,x\rangle _{A}\geq 0}$

in the sense of being a positive element of A, and

${\displaystyle \langle x,x\rangle _{A}=0\iff x=0.}$

(An element of a C*-algebra ${\displaystyle A}$ is said to be positive if it is self-adjoint with non-negative spectrum.)

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product ${\displaystyle A}$-module ${\displaystyle E}$:

${\displaystyle \langle x,y\rangle _{A}\langle y,x\rangle _{A}\leq \Vert \langle y,y\rangle _{A}\Vert \langle x,x\rangle _{A}}$

for ${\displaystyle x}$, ${\displaystyle y}$ in ${\displaystyle E}$.

On the pre-Hilbert module ${\displaystyle E}$, define a norm by

${\displaystyle \Vert x\Vert =\Vert \langle x,x\rangle _{A}\Vert ^{\frac {1}{2}}.}$

The norm-completion of ${\displaystyle E}$, still denoted by ${\displaystyle E}$, is said to be a Hilbert ${\displaystyle A}$-module or a Hilbert C*-module over the C*-algebra ${\displaystyle A}$. The Cauchy–Schwarz inequality implies the inner product is jointly continuous in norm and can therefore be extended to the completion.

The action of ${\displaystyle A}$ on ${\displaystyle E}$ is continuous: for all ${\displaystyle x}$ in ${\displaystyle E}$

${\displaystyle a_{\lambda }\rightarrow a\Rightarrow xa_{\lambda }\rightarrow xa.}$

Similarly, if ${\displaystyle (e_{\lambda })}$ is an approximate unit for ${\displaystyle A}$ (a net of self-adjoint elements of ${\displaystyle A}$ for which ${\displaystyle ae_{\lambda }}$ and ${\displaystyle e_{\lambda }a}$ tend to ${\displaystyle a}$ for each ${\displaystyle a}$ in ${\displaystyle A}$), then for ${\displaystyle x}$ in ${\displaystyle E}$

${\displaystyle xe_{\lambda }\rightarrow x.}$

Whence it follows that ${\displaystyle EA}$ is dense in ${\displaystyle E}$, and ${\displaystyle x1_{A}=x}$ when ${\displaystyle A}$ is unital.

Let

${\displaystyle \langle E,E\rangle _{A}=\operatorname {span} \{\langle x,y\rangle _{A}\mid x,y\in E\},}$

then the closure of ${\displaystyle \langle E,E\rangle _{A}}$ is a two-sided ideal in ${\displaystyle A}$. Two-sided ideals are C*-subalgebras and therefore possess approximate units. One can verify that ${\displaystyle E\langle E,E\rangle _{A}}$ is dense in ${\displaystyle E}$. In the case when ${\displaystyle \langle E,E\rangle _{A}}$ is dense in ${\displaystyle A}$, ${\displaystyle E}$ is said to be full. This does not generally hold.

Examples

Hilbert spaces

Since the complex numbers ${\displaystyle \mathbb {C} }$ are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space ${\displaystyle {\mathcal {H}}}$ is a Hilbert ${\displaystyle \mathbb {C} }$-module under scalar multipliation by complex numbers and its inner product.

Vector bundles

If ${\displaystyle X}$ is a locally compact Hausdorff space and ${\displaystyle E}$ a vector bundle over ${\displaystyle X}$ with projection ${\displaystyle \pi \colon E\to X}$ a Hermitian metric ${\displaystyle g}$, then the space of continuous sections of ${\displaystyle E}$ is a Hilbert ${\displaystyle C(X)}$-module. Given sections ${\displaystyle \sigma ,\rho }$ of ${\displaystyle E}$ and ${\displaystyle f\in C(X)}$ the right action is defined by

${\displaystyle \sigma f(x)=\sigma (x)f(\pi (x)),}$

and the inner product is given by

${\displaystyle \langle \sigma ,\rho \rangle _{C(X)}(x):=g(\sigma (x),\rho (x)).}$

The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra ${\displaystyle A=C(X)}$ is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over ${\displaystyle X}$.

C*-algebras

Any C*-algebra ${\displaystyle A}$ is a Hilbert ${\displaystyle A}$-module with the action given by right multiplication in ${\displaystyle A}$ and the inner product ${\displaystyle \langle a,b\rangle =a^{*}b}$. By the C*-identity, the Hilbert module norm coincides with C*-norm on ${\displaystyle A}$.

The (algebraic) direct sum of ${\displaystyle n}$ copies of ${\displaystyle A}$

${\displaystyle A^{n}=\bigoplus _{i=1}^{n}A}$

can be made into a Hilbert ${\displaystyle A}$-module by defining

${\displaystyle \langle (a_{i}),(b_{i})\rangle _{A}=\sum _{i=1}^{n}a_{i}^{*}b_{i}.}$

If ${\displaystyle p}$ is a projection in the C*-algebra ${\displaystyle M_{n}(A)}$, then ${\displaystyle pA^{n}}$ is also a Hilbert ${\displaystyle A}$-module with the same inner product as the direct sum.

The standard Hilbert module

One may also consider the following subspace of elements in the countable direct product of ${\displaystyle A}$

${\displaystyle \ell _{2}(A)={\mathcal {H}}_{A}={\Big \{}(a_{i})|\sum _{i=1}^{\infty }a_{i}^{*}a_{i}{\text{ converges in }}A{\Big \}}.}$

Endowed with the obvious inner product (analogous to that of ${\displaystyle A^{n}}$), the resulting Hilbert ${\displaystyle A}$-module is called the standard Hilbert module over ${\displaystyle A}$.

The fact that there is a unique separable Hilbert space has a generalization to Hilbert modules in the form of the Kasparov stabilization theorem, which states that if ${\displaystyle E}$ is a countably generated Hilbert ${\displaystyle A}$-module, there is an isometric isomorphism ${\displaystyle E\oplus \ell ^{2}(A)\cong \ell ^{2}(A).}$

Maps between Hilbert modules

Let ${\displaystyle E}$ and ${\displaystyle F}$ be two Hilbert modules over the same C*-algebra ${\displaystyle A}$. These are then Banach spaces, so it is possible to speak of the Banach space of bounded linear maps ${\displaystyle {\mathcal {L}}(E,F)}$, normed by the operator norm.

The adjointable and compact adjointable operators are subspaces of this Banach space defined using the inner product structures on ${\displaystyle E}$ and ${\displaystyle F}$.

In the special case where ${\displaystyle A}$ is ${\displaystyle \mathbb {C} }$ these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

A map (not necessarily linear) ${\displaystyle T\colon E\to F}$ is defined to be adjointable if there is another map ${\displaystyle T^{*}\colon F\to E}$, known as the adjoint of ${\displaystyle T}$, such that for every ${\displaystyle e\in E}$ and ${\displaystyle f\in F}$,

${\displaystyle \langle f,Te\rangle =\langle T^{*}f,e\rangle .}$

Both ${\displaystyle T}$ and ${\displaystyle T^{*}}$ are then automatically linear and also ${\displaystyle A}$-module maps. The closed graph theorem can be used to show that they are also bounded.

Analogously to the adjoint of operators on Hilbert spaces, ${\displaystyle T^{*}}$ is unique (if it exists) and itself adjointable with adjoint ${\displaystyle T}$. If ${\displaystyle S\colon F\to G}$ is a second adjointable map, ${\displaystyle ST}$ is adjointable with adjoint ${\displaystyle S^{*}T^{*}}$.

The adjointable operators ${\displaystyle E\to F}$ form a subspace ${\displaystyle \mathbb {B} (E,F)}$ of ${\displaystyle {\mathcal {L}}(E,F)}$, which is complete in the operator norm.

In the case ${\displaystyle F=E}$, the space ${\displaystyle \mathbb {B} (E,E)}$ of adjointable operators from ${\displaystyle E}$ to itself is denoted ${\displaystyle \mathbb {B} (E)}$, and is a C*-algebra.

Compact adjointable maps

Given ${\displaystyle e\in E}$ and ${\displaystyle f\in F}$, the map ${\displaystyle |f\rangle \langle e|\colon E\to F}$ is defined, analogously to the rank one operators of Hilbert spaces, to be

${\displaystyle g\mapsto f\langle e,g\rangle .}$

This is adjointable with adjoint ${\displaystyle |e\rangle \langle f|}$.

The compact adjointable operators ${\displaystyle \mathbb {K} (E,F)}$ are defined to be the closed span of

${\displaystyle \{|f\rangle \langle e|\mid e\in E,\;f\in F\}}$

in ${\displaystyle \mathbb {B} (E,F)}$.

As with the bounded operators, ${\displaystyle \mathbb {K} (E,E)}$ is denoted ${\displaystyle \mathbb {K} (E)}$. This is a (closed, two-sided) ideal of ${\displaystyle \mathbb {B} (E)}$.

C*-correspondences

If ${\displaystyle A}$ and ${\displaystyle B}$ are C*-algebras, an ${\displaystyle (A,B)}$ C*-correspondence is a Hilbert ${\displaystyle B}$-module equipped with a left action of ${\displaystyle A}$ by adjointable maps that is faithful. (NB: Some authors require the left action to be non-degenerate instead.) These objects are used in the formulation of Morita equivalence for C*-algebras, see applications in the construction of Toeplitz and Cuntz-Pimsner algebras, and can be employed to put the structure of a bicategory on the collection of C*-algebras.

Tensor products and the bicategory of correspondences

If ${\displaystyle E}$ is an ${\displaystyle (A,B)}$ and ${\displaystyle F}$ a ${\displaystyle (B,C)}$ correspondence, the algebraic tensor product ${\displaystyle E\odot F}$ of ${\displaystyle E}$ and ${\displaystyle F}$ as vector spaces inherits left and right ${\displaystyle A}$- and ${\displaystyle C}$-module structures respectively.

It can also be endowed with the ${\displaystyle C}$-valued sesquilinear form defined on pure tensors by

${\displaystyle \langle e\odot f,e'\odot f'\rangle _{C}:=\langle f,\langle e,e'\rangle _{B}f\rangle _{C}.}$

This is positive semidefinite, and the Hausdorff completion of ${\displaystyle E\odot F}$ in the resulting seminorm is denoted ${\displaystyle E\otimes _{B}F}$. The left- and right-actions of ${\displaystyle A}$ and ${\displaystyle C}$ extend to make this an ${\displaystyle (A,C)}$ correspondence.

The collection of C*-algebras can then be endowed with the structure of a bicategory, with C*-algebras as objects, ${\displaystyle (A,B)}$ correspondences as arrows ${\displaystyle B\to A}$, and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.

Toeplitz algebra of a correspondence

Given a C*-algebra ${\displaystyle A}$, and an ${\displaystyle (A,A)}$ correspondence ${\displaystyle E}$, its Toeplitz algebra ${\displaystyle {\mathcal {T}}(E)}$ is defined as the universal algebra for Toeplitz representations (defined below).

The classical Toeplitz algebra can be recovered as a special case, and the Cuntz-Pimsner algebras are defined as particular quotients of Toeplitz algebras.

In particular, graph algebras , crossed products by ${\displaystyle \mathbb {Z} }$ , and the Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations

A Toeplitz representation of ${\displaystyle E}$ in a C*-algebra ${\displaystyle D}$ is a pair ${\displaystyle (S,\phi )}$ of a linear map ${\displaystyle S\colon E\to D}$ and a homomorphism ${\displaystyle \phi \colon A\to D}$ such that

• ${\displaystyle S}$ is "isometric":

${\displaystyle S(e)^{*}S(f)=\phi (\langle e,f\rangle )}$ for all ${\displaystyle e,f\in E}$,

• ${\displaystyle S}$ resembles a bimodule map:

${\displaystyle S(ae)=\phi (a)S(e)}$ and ${\displaystyle S(ea)=S(e)\phi (a)}$ for ${\displaystyle e\in E}$ and ${\displaystyle a\in A}$.

Toeplitz algebra

The Toeplitz algebra ${\displaystyle {\mathcal {T}}(E)}$ is the universal Toeplitz representation. That is, there is a Toeplitz representation ${\displaystyle (T,\iota )}$ of ${\displaystyle E}$ in ${\displaystyle {\mathcal {T}}(E)}$ such that if ${\displaystyle (S,\phi )}$ is any Toeplitz representation of ${\displaystyle E}$ (in an arbitrary algebra ${\displaystyle D}$) there is a unique *-homomorphism ${\displaystyle \Phi \colon {\mathcal {T}}(E)\to D}$ such that ${\displaystyle S=\Phi \circ T}$ and ${\displaystyle \phi =\Phi \circ \iota }$.

Examples

If ${\displaystyle A}$ is taken to be the algebra of complex numbers, and ${\displaystyle E}$ the vector space ${\displaystyle \mathbb {C} ^{n}}$, endowed with the natural ${\displaystyle (\mathbb {C} ,\mathbb {C} )}$-bimodule structure, the corresponding Toeplitz algebra is the universal algebra generated by ${\displaystyle n}$ isometries with mutually orthogonal range projections.

In particular, ${\displaystyle {\mathcal {T}}(\mathbb {C} )}$ is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.

See also

• Operator algebra

Notes

1. Kaplansky, I. (1953). "Modules over operator algebras". American Journal of Mathematics. 75 (4): 839–853. doi:10.2307/2372552. JSTOR 2372552.
2. Paschke, W. L. (1973). "Inner product modules over B*-algebras". Transactions of the American Mathematical Society. 182: 443–468. doi:10.2307/1996542. JSTOR 1996542.
3. Rieffel, M. A. (1974). "Induced representations of C*-algebras". Advances in Mathematics. 13 (2): 176–257. doi:10.1016/0001-8708(74)90068-1.
4. Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. Theta Foundation: 133–150.
5. Rieffel, M. A. (1982). "Morita equivalence for operator algebras". Proceedings of Symposia in Pure Mathematics. 38. American Mathematical Society: 176–257.
6. Baaj, S.; Skandalis, G. (1993). "Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres". Annales Scientifiques de l'École Normale Supérieure. 26 (4): 425–488. doi:10.24033/asens.1677.
7. Woronowicz, S. L. (1991). "Unbounded elements affiliated with C*-algebras and non-compact quantum groups". Communications in Mathematical Physics. 136 (2): 399–432. Bibcode:1991CMaPh.136..399W. doi:10.1007/BF02100032. S2CID 118184597.
8. Arveson, William (1976). An Invitation to C*-Algebras. Springer-Verlag. p. 35.
9. In the case when ${\displaystyle A}$ is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to ${\displaystyle A}$.
10. This result in fact holds for semi-inner-product ${\displaystyle A}$-modules, which may have non-zero elements ${\displaystyle A}$ such that ${\displaystyle \langle x,x\rangle _{A}=0}$, as the proof does not rely on the nondegeneracy property.
11. Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. ThetaFoundation: 133–150.
12. Wegge-Olsen 1993, pp. 240-241.
13. Wegge-Olsen 1993, pp. 242-243.
14. Brown, Ozawa 2008, section 4.6.
15. Buss, Meyer, Zhu, 2013, section 2.2.
16. Brown, Ozawa 2008, pp. 138-139.
17. Buss, Meyer, Zhu 2013, section 2.2.
18. Brown, Ozawa, 2008, section 4.6.
19. Fowler, Raeburn, 1999, section 1.
20. Fowler, Raeburn, 1999, Proposition 1.3.
21. Brown, Ozawa, 2008, Example 4.6.10.

References

• Lance, E. Christopher (1995). Hilbert C*-modules: A toolkit for operator algebraists. London Mathematical Society Lecture Note Series. Cambridge, England: Cambridge University Press.
• Wegge-Olsen, N. E. (1993). K-Theory and C*-Algebras. Oxford University Press.
• Brown, Nathanial P.; Ozawa, Narutaka (2008). C*-Algebras and Finite-Dimensional Approximations. American Mathematical Society.
• Buss, Alcides; Meyer, Ralf; Zhu, Chenchang (2013). "A higher category approach to twisted actions on c* -algebras". Proceedings of the Edinburgh Mathematical Society. 56 (2): 387–426. arXiv:0908.0455. doi:10.1017/S0013091512000259.
• Fowler, Neal J.; Raeburn, Iain (1999). "The Toeplitz algebra of a Hilbert bimodule". Indiana University Mathematics Journal. 48 (1): 155–181. arXiv:math/9806093. doi:10.1512/iumj.1999.48.1639. JSTOR 24900141.

External links

• Weisstein, Eric W. "Hilbert C*-Module". MathWorld.
• Hilbert C*-Modules Home Page, a literature list

• C*-algebras
• Operator theory
• Theoretical physics