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Hilbert C*-module

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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").[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 notion of Morita equivalence to C*-algebras.[5] 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,[6][7] and groupoid C*-algebras.


Definitions

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Inner-product C*-modules

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

that satisfies the following properties:

  • For all , , in , and , in :
(i.e. the inner product is -linear in its second argument).
  • For all , in , and in :
  • For all , in :
from which it follows that the inner product is conjugate linear in its first argument (i.e. it is a sesquilinear form).
  • For all in :
in the sense of being a positive element of A, and
(An element of a C*-algebra is said to be positive if it is self-adjoint with non-negative spectrum.)[8][9]

Hilbert C*-modules

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An analogue to the Cauchy–Schwarz inequality holds for an inner-product -module :[10]

for , in .

On the pre-Hilbert module , define a norm by

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

The action of on is continuous: for all in

Similarly, if is an approximate unit for (a net of self-adjoint elements of for which and tend to for each in ), then for in

Whence it follows that is dense in , and when is unital.

Let

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

Examples

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Hilbert spaces

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

Vector bundles

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If is a locally compact Hausdorff space and a vector bundle over with projection a Hermitian metric , then the space of continuous sections of is a Hilbert -module. Given sections of and the right action is defined by

and the inner product is given by

The converse holds as well: Every countably generated Hilbert C*-module over a commutative unital C*-algebra is isomorphic to the space of sections vanishing at infinity of a continuous field of Hilbert spaces over . [citation needed]

C*-algebras

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Any C*-algebra is a Hilbert -module with the action given by right multiplication in and the inner product . By the C*-identity, the Hilbert module norm coincides with C*-norm on .

The (algebraic) direct sum of copies of

can be made into a Hilbert -module by defining

If is a projection in the C*-algebra , then is also a Hilbert -module with the same inner product as the direct sum.

The standard Hilbert module

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One may also consider the following subspace of elements in the countable direct product of

Endowed with the obvious inner product (analogous to that of ), the resulting Hilbert -module is called the standard Hilbert module over .

The standard Hilbert module plays an important role in the proof of the Kasparov stabilization theorem which states that for any countably generated Hilbert -module there is an isometric isomorphism [11]

Maps between Hilbert modules

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Let and be two Hilbert modules over the same C*-algebra . These are then Banach spaces, so it is possible to speak of the Banach space of bounded linear maps , normed by the operator norm.

The adjointable and compact adjointable operators are subspaces of this Banach space defined using the inner product structures on and .

In the special case where is these reduce to bounded and compact operators on Hilbert spaces respectively.

Adjointable maps

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A map (not necessarily linear) is defined to be adjointable if there is another map , known as the adjoint of , such that for every and ,

Both and are then automatically linear and also -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, is unique (if it exists) and itself adjointable with adjoint . If is a second adjointable map, is adjointable with adjoint .

The adjointable operators form a subspace of , which is complete in the operator norm.

In the case , the space of adjointable operators from to itself is denoted , and is a C*-algebra.[12]

Compact adjointable maps

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Given and , the map is defined, analogously to the rank one operators of Hilbert spaces, to be

This is adjointable with adjoint .

The compact adjointable operators are defined to be the closed span of

in .

As with the bounded operators, is denoted . This is a (closed, two-sided) ideal of .[13]

C*-correspondences

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If and are C*-algebras, an C*-correspondence is a Hilbert -module equipped with a left action of 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

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If is an and a correspondence, the algebraic tensor product of and as vector spaces inherits left and right - and -module structures respectively.

It can also be endowed with the -valued sesquilinear form defined on pure tensors by

This is positive semidefinite, and the Hausdorff completion of in the resulting seminorm is denoted . The left- and right-actions of and extend to make this an correspondence.[16]

The collection of C*-algebras can then be endowed with the structure of a bicategory, with C*-algebras as objects, correspondences as arrows , and isomorphisms of correspondences (bijective module maps that preserve inner products) as 2-arrows.[17]

Toeplitz algebra of a correspondence

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Given a C*-algebra , and an correspondence , its Toeplitz algebra 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.[18]

In particular, graph algebras , crossed products by , and the Cuntz algebras are all quotients of specific Toeplitz algebras.

Toeplitz representations

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A Toeplitz representation[19] of in a C*-algebra is a pair of a linear map and a homomorphism such that

  • is "isometric":
for all ,
  • resembles a bimodule map:
and for and .

Toeplitz algebra

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The Toeplitz algebra is the universal Toeplitz representation. That is, there is a Toeplitz representation of in such that if is any Toeplitz representation of (in an arbitrary algebra ) there is a unique *-homomorphism such that and .[20]

Examples

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If is taken to be the algebra of complex numbers, and the vector space , endowed with the natural -bimodule structure, the corresponding Toeplitz algebra is the universal algebra generated by isometries with mutually orthogonal range projections.[21]

In particular, is the universal algebra generated by a single isometry, which is the classical Toeplitz algebra.

See also

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Notes

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  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 is non-unital, the spectrum of an element is calculated in the C*-algebra generated by adjoining a unit to .
  10. ^ This result in fact holds for semi-inner-product -modules, which may have non-zero elements such that , 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

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