Hilbert C*-module explained

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 aC*-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").^[1]

In the 1970s the theory was extended to non-commutative C*-algebras independently by William Lindall Paschke^[2] and Marc Rieffel, the latter in a paper that used Hilbert C*-modules to construct a theory of induced representations of C*-algebras.^[3]

Hilbert C*-modules are crucial to Kasparov's formulation of KK-theory,^[4] and provide the right framework to extend the notionof Morita equivalence to C*-algebras.^[5] They can be viewed as the generalizationof vector bundles to noncommutative C*-algebras and as such play an important role in noncommutative geometry, notably in C*-algebraic quantum group theory,^{[6] [7]} and groupoid C*-algebras.

Definitions

Inner-product C*-modules

Let

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

that satisfies the following properties:

• For all

x , y , z in E , and \alpha , \beta in C :\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

C -linear in its second argument).

• For all

x , y in E , and a in A :\langle x, y a \rangle _{A} = \langle x, y \rangle _{A} a

• For all

x , y in E :\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

x in E :\langle x, x \rangle _{A} \geq 0

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

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

(An element of a C*-algebra

A is said to be positive if it is self-adjoint with non-negative spectrum.)^{[8] [9]}

Hilbert C*-modules

An analogue to the Cauchy–Schwarz inequality holds for an inner-product

A -module E :^[10] \langle x, y \rangle _{A} \langle y, x \rangle _{A} \leq \Vert \langle y, y \rangle _{A} \Vert \langle x, x \rangle _{A}

for

x , y in E .

On the pre-Hilbert module

E , define a norm by\Vert x \Vert = \Vert \langle x, x \rangle _{A} \Vert ^{(1)/(2)} .

The norm-completion of

E , still denoted by E , is said to be a Hilbert A -module or a Hilbert C*-module over the C*-algebra 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

A on E is continuous: for all x in E a _{λ} → a ⇒ xa _{λ} → xa.

Similarly, if

(e _{λ)} is an approximate unit for A (a net of self-adjoint elements of A for which a e _{λ} and e _{λ} a tend to a for each a in A ), then for x in E xe _{λ} → x.

Whence it follows that

EA is dense in E , and x 1 _{A} = x when A is unital. Let\langle E, E \rangle _{A} = \operatorname{span} \{ \langle x, y \rangle _{A} \mid x, y \in E \},

then the closure of

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

Examples

Hilbert spaces

Since the complex numbers

C are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space l{H} is a Hilbert C -module under scalar multipliation by complex numbers and its inner product.

Vector bundles

If

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

A = C(X) is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over X .

C*-algebras

Any C*-algebra

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

The (algebraic) direct sum of

n copies of A A ^{n} = o plus ^{n} _{i =1} A

can be made into a Hilbert

A -module by defining\langle (a _{i),} (b _{i)} \rangle _{A} = \sum ^{n} _{i =1} a ^{*} _{i} b _{i.}

If

p is a projection in the C*-algebra M _{n(A)} , then pA ^{n} is also a Hilbert 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

A \ell _{2(A)=} l{H} _{A} = \ { (a _{i)} | \sum ^{inf ty} _{i =1} a ^{*} _{i} a _{iconvergesinA} \ }.

Endowed with the obvious inner product (analogous to that of

A ^{n} ), the resulting Hilbert A -module is called the standard Hilbert module over A .

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert

A -module E there is an isometric isomorphism E ⊕ \ell ^{2(A)} \cong \ell ^{2(A).} ^[11]

Maps between Hilbert modules

Let

E and F be two Hilbert modules over the same C*-algebra A . These are then Banach spaces, so it is possible to speak of the Banach space of bounded linear maps l{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

E and F .

In the special case where

A is C these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

A map (not necessarily linear)

T \colon E \to F is defined to be adjointable if there is another map T ^{*} \colon F \to E , known as the adjoint of T , such that for every e \in E and f \in F ,\langle f, Te \rangle = \langle T ^{*} f, e \rangle.

Both

T and T ^{*} are then automatically linearand also 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,

T ^{*} is unique (if it exists) and itself adjointable with adjoint T . If S \colon F \to G is a second adjointable map, ST is adjointable with adjoint S ^{*} T ^{*} .

The adjointable operators

E \to F form a subspace B (E,F) of l{L}(E,F) , which is complete in the operator norm.

In the case

F = E , the space B (E,E) of adjointable operators from E to itself is denoted B (E) , and is a C*-algebra.^[12]

Compact adjointable maps

Given

e \in E and f \in F , the map | f \rangle \langle e | \colon E \to F is defined, analogously to the rank one operators of Hilbert spaces, to be g \mapsto f \langle e, g \rangle.

This is adjointable with adjoint

| e \rangle \langle f | .

The compact adjointable operators

K (E,F) are defined to be the closed span of \{ | f \rangle \langle e | \mid e \in E, f \in F \}

in

B (E,F) .

As with the bounded operators,

K (E,E) is denoted K (E) . This is a (closed, two-sided) ideal of B (E) .^[13]

C*-correspondences

If

A and B are C*-algebras, an (A,B) C*-correspondenceis a Hilbert B -module equipped with a left action of 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,^[14] and can be employed to put the structure of a bicategory on the collection of C*-algebras.^[15]

Tensor products and the bicategory of correspondences

If

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

It can also be endowed with the

C -valued sesquilinear form defined on pure tensors by \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

E \odot F in the resulting seminorm is denoted E ⊗ _{B} F . The left- and right-actions ofA and C extend to make this an (A,C) correspondence.^[16]

The collection of C*-algebras can then be endowed with the structure of a bicategory, with C*-algebras as objects,

(A,B) correspondences as arrows B \to A , and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.^[17]

Toeplitz algebra of a correspondence

Given a C*-algebra

A , and an (A,A) correspondence E ,its Toeplitz algebra l{T}(E) is defined as the universal algebrafor Toeplitz representations (defined below).

The classical Toeplitz algebra can be recovered as a special case, and the Cuntz-Pimsner algebrasare defined as particular quotients of Toeplitz algebras.^[18]

In particular, graph algebras, crossed products by

Z , and the Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations

A Toeplitz representation^[19] of

E in a C*-algebra D is a pair (S,\phi) of a linear map S \colon E \to D and a homomorphism \phi \colon A \to D such that S is "isometric":S(e) ^{*} S(f) = \phi(\langle e, f \rangle) for all e,f \in E ,S resembles a bimodule map:S(a e) = \phi(a) S(e) and S(ea) = S(e) \phi(a) for e \in E and a \in A .

Toeplitz algebra

The Toeplitz algebra

l{T}(E) is the universal Toeplitz representation.That is, there is a Toeplitz representation (T, \iota) of E in l{T}(E) such that if (S,\phi) is any Toeplitz representationof E (in an arbitrary algebra D ) there is a unique *-homomorphism \Phi \colon l{T}(E) \to D such that S = \Phi \circ T and \phi = \Phi \circ \iota .^[20]

Examples

If

A is taken to be the algebra of complex numbers, and E the vector space C ^{n} , endowed with the natural ( C , C ) -bimodule structure, the corresponding Toeplitz algebrais the universal algebra generated by n isometries with mutually orthogonal range projections.^[21]

In particular,

l{T}( C ) is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.

See also

• Operator algebra

References

• Book: Lance, E. Christopher. Hilbert C*-modules: A toolkit for operator algebraists . London Mathematical Society Lecture Note Series. 1995 . Cambridge University Press . Cambridge, England.
• Book: Wegge-Olsen, N. E.. K-Theory and C*-Algebras . Oxford University Press . 1993.
• Book: 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 . 10.1017/S0013091512000259. 0908.0455 .
• Fowler . Neal J. . Raeburn . Iain . 1999 . The Toeplitz algebra of a Hilbert bimodule . Indiana University Mathematics Journal . 48 . 1 . 155–181. 10.1512/iumj.1999.48.1639 . 24900141 . math/9806093 .

External links

• • Hilbert C*-Modules Home Page, a literature list

Notes and References

1. Kaplansky. I.. Irving Kaplansky . Modules over operator algebras. American Journal of Mathematics. 75. 4. 839–853. 1953. 10.2307/2372552. 2372552.
2. Paschke. W. L.. Inner product modules over B*-algebras. Transactions of the American Mathematical Society. 182. 443–468. 1973. 10.2307/1996542. 1996542.
3. Rieffel. M. A.. Induced representations of C*-algebras. Advances in Mathematics. 13. 176–257. 1974. 10.1016/0001-8708(74)90068-1. free . 2.
4. Kasparov. G. G.. Hilbert C*-modules: Theorems of Stinespring and Voiculescu. Journal of Operator Theory. 4. 133–150. Theta Foundation. 1980.
5. Rieffel. M. A.. Morita equivalence for operator algebras. Proceedings of Symposia in Pure Mathematics. 38. 176–257. American Mathematical Society. 1982.
6. Baaj. S.. Skandalis, G.. Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres. Annales Scientifiques de l'École Normale Supérieure. 26. 4. 425–488. 1993. 10.24033/asens.1677. free.
7. Woronowicz. S. L.. S. L. Woronowicz. Unbounded elements affiliated with C*-algebras and non-compact quantum groups. Communications in Mathematical Physics. 136. 399–432. 1991. 10.1007/BF02100032. 1991CMaPh.136..399W. 2 . 118184597.
8. Book: Arveson, William. William Arveson

   . William Arveson. An Invitation to C*-Algebras. Springer-Verlag. 1976. 35.

9. In the case when A is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to A .
10. This result in fact holds for semi-inner-product A -modules, which may have non-zero elements A such that \langle x , x \rangle _{A} = 0 , as the proof does not rely on the nondegeneracy property.
11. Kasparov. G. G.. Hilbert C*-modules: Theorems of Stinespring and Voiculescu. Journal of Operator Theory. 4. 133–150. ThetaFoundation. 1980.
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.