Hilbert C*-module

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

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

that satisfies the following properties:

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

\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

\mathbb {C}

-linear in its second argument).

For all $x$ , $y$ in $E$ , and $a$ in $A$ :

\langle x,ya\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 ^{\frac {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_{\lambda }\rightarrow a\Rightarrow xa_{\lambda }\rightarrow xa.

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

xe_{\lambda }\rightarrow x.

Whence it follows that $EA$ is dense in $E$ , and $x1_{A}=x$ 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 $\mathbb {C}$ are a C*-algebra with an involution given by complex conjugation, a complex Hilbert space ${\mathcal {H}}$ is a Hilbert $\mathbb {C}$ -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 $X$ . ^{[citation needed]}

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

can be made into a Hilbert $A$ -module by defining

\langle (a_{i}),(b_{i})\rangle _{A}=\sum _{i=1}^{n}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)={\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 $A^{n}$ ), the resulting Hilbert $A$ -module is called the standard Hilbert module over $A$ .

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 $E$ is a countably generated Hilbert $A$ -module, there is an isometric isomorphism $E\oplus \ell ^{2}(A)\cong \ell ^{2}(A).$ ^[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 ${\mathcal {L}}(E,F)$ , normed by the operator norm.

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

In the special case where $A$ is $\mathbb {C}$ 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 linear and 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 $\mathbb {B} (E,F)$ of ${\mathcal {L}}(E,F)$ , which is complete in the operator norm.

In the case $F=E$ , the space $\mathbb {B} (E,E)$ of adjointable operators from $E$ to itself is denoted $\mathbb {B} (E)$ , 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 $\mathbb {K} (E,F)$ are defined to be the closed span of

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

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

As with the bounded operators, $\mathbb {K} (E,E)$ is denoted $\mathbb {K} (E)$ . This is a (closed, two-sided) ideal of $\mathbb {B} (E)$ .^[13]

C*-correspondences

If $A$ and $B$ are C*-algebras, an $(A,B)$ C*-correspondence is 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 $E\otimes _{B}F$ . The left- and right-actions of $A$ 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 ${\mathcal {T}}(E)$ 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.^[18]

In particular, graph algebras , crossed products by $\mathbb {Z}$ , 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(ae)=\phi (a)S(e)

and

S(ea)=S(e)\phi (a)

for

e\in E

and

a\in A

.

Toeplitz algebra

The Toeplitz algebra ${\mathcal {T}}(E)$ is the universal Toeplitz representation. That is, there is a Toeplitz representation $(T,\iota )$ of $E$ in ${\mathcal {T}}(E)$ such that if $(S,\phi )$ is any Toeplitz representation of $E$ (in an arbitrary algebra $D$ ) there is a unique *-homomorphism $\Phi \colon {\mathcal {T}}(E)\to D$ 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 $\mathbb {C} ^{n}$ , endowed with the natural $(\mathbb {C} ,\mathbb {C} )$ -bimodule structure, the corresponding Toeplitz algebra is the universal algebra generated by $n$ isometries with mutually orthogonal range projections.^[21]

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

Notes

^ Kaplansky, I. (1953). "Modules over operator algebras". American Journal of Mathematics. 75 (4): 839–853. doi:10.2307/2372552. JSTOR 2372552.
^ 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.
^ Rieffel, M. A. (1974). "Induced representations of C*-algebras". Advances in Mathematics. 13 (2): 176–257. doi:10.1016/0001-8708(74)90068-1.
^ Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. Theta Foundation: 133–150.
^ Rieffel, M. A. (1982). "Morita equivalence for operator algebras". Proceedings of Symposia in Pure Mathematics. 38. American Mathematical Society: 176–257.
^ 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.
^ 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.
^ Arveson, William (1976). An Invitation to C*-Algebras. Springer-Verlag. p. 35.
^ 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 $A$ .
^ 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.
^ Kasparov, G. G. (1980). "Hilbert C*-modules: Theorems of Stinespring and Voiculescu". Journal of Operator Theory. 4. ThetaFoundation: 133–150.
^ Wegge-Olsen 1993, pp. 240-241.
^ Wegge-Olsen 1993, pp. 242-243.
^ Brown, Ozawa 2008, section 4.6.
^ Buss, Meyer, Zhu, 2013, section 2.2.
^ Brown, Ozawa 2008, pp. 138-139.
^ Buss, Meyer, Zhu 2013, section 2.2.
^ Brown, Ozawa, 2008, section 4.6.
^ Fowler, Raeburn, 1999, section 1.
^ Fowler, Raeburn, 1999, Proposition 1.3.
^ 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

[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 $A$ 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. (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.

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Hilbert C*-module

Definitions

Inner-product C*-modules

Hilbert C*-modules

Examples

Hilbert spaces

Vector bundles

C*-algebras

The standard Hilbert module

Maps between Hilbert modules

Adjointable maps

Compact adjointable maps

C*-correspondences

Tensor products and the bicategory of correspondences

Toeplitz algebra of a correspondence

Toeplitz representations

Toeplitz algebra

Examples

See also

Notes

References

External links