Extension (C * -algebra)

In the mathematical theory of the extensions of C * -algebras one examines the structure of the class of all possible extensions of a pair of C * -algebras . The equivalence classes are examined more precisely with regard to a certain equivalence relation and in important cases an Abelian group is obtained . This assignment of an Abelian group to a pair of C * -algebras is an isomorphism invariant, which in the case of commutative C * -algebras defines a functor on the topological category of metrizable , compact spaces . Even if the theory for commutative C * algebras is historically earlier, we will start with the more general theory and later specialize in the commutative case.

Definitions

Let and be two C * -algebras, where be stable, that is, isomorphic to the tensor product with the C * -algebra of the compact operators on the Hilbert space of the square-summable sequences . An example that is important for many applications is . One now poses the question about all extensions of after , that is to say about the class of all short exact sequences ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle B\otimes K}$ ${\displaystyle K}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle B=K}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle {\mathcal {E}}(A,B)}$

${\displaystyle 0\rightarrow B\rightarrow C\rightarrow A\rightarrow 0}$

in the category of C * algebras. The quotient after a suitable equivalence relation is called later . Note the order in which the C * -algebras are named in this designation, which was chosen by analogy with the Ext functor of homological algebra . ${\displaystyle \mathrm {Ext} (A,B)}$

Two extensions are called isomorphic if there is an isomorphism that makes the following diagram commutative : ${\displaystyle 0\rightarrow B\rightarrow C_{i}\rightarrow A\rightarrow 0,\,i=1,2}$ ${\displaystyle \gamma :C_{1}\rightarrow C_{2}}$

${\displaystyle {\begin{array}{ccccccc}0\rightarrow &B&\rightarrow &C_{1}&\rightarrow &A&\rightarrow 0\\&\parallel &&\downarrow \gamma &&\parallel \\0\rightarrow &B&\rightarrow &C_{2}&\rightarrow &A&\rightarrow 0\\\end{array}}}$

This equivalence relation is very fine and it is clear that one does not want to regard two isomorphic extensions as "essentially different".

Two extensions are called unitary equivalent if there is an isomorphism and a unitary element from the multiplier algebra , so that the following diagram is commutative : ${\displaystyle 0\rightarrow B\rightarrow C_{i}\rightarrow A\rightarrow 0,\quad i=1,2}$ ${\displaystyle \gamma :C_{1}\rightarrow C_{2}}$ ${\displaystyle u}$ ${\displaystyle M(B)}$

${\displaystyle {\begin{array}{ccccccc}0\rightarrow &B&\rightarrow &C_{1}&\rightarrow &A&\rightarrow 0\\&\downarrow \alpha _{u}&&\downarrow \gamma &&\parallel \\0\rightarrow &B&\rightarrow &C_{2}&\rightarrow &A&\rightarrow 0\\\end{array}}}$

It is . ${\displaystyle \alpha _{u}:B\rightarrow B,\alpha _{u}(b)=u^{*}bu}$

It will later become clear with the introduction of addition, why this coarser relation of unitary equivalence, which we will denote in the following with , is the suitable equivalence relation and why we require stability. ${\displaystyle \sim _{u}}$ ${\displaystyle B}$

For later purposes we already define here that an extension

${\displaystyle 0\rightarrow B\rightarrow C{\xrightarrow {\sigma }}A\rightarrow 0}$

splitting means if there is a * homomorphism with . If there is only a completely positive mapping, the extension is called semi-divisive . ${\displaystyle \varphi :A\rightarrow C}$ ${\displaystyle \sigma \circ \varphi =\mathrm {id} _{A}}$ ${\displaystyle \varphi }$

Examples

${\displaystyle 0\rightarrow B{\xrightarrow {b\mapsto (0,b)}}A\oplus B{\xrightarrow {(a,b)\mapsto a}}A\rightarrow 0}$

is called trivial extension . The splitting extensions are exactly those for the trivial isomorphic extensions.

The Toeplitz algebra leads to an extension ${\displaystyle T}$

${\displaystyle 0\rightarrow K\rightarrow T\rightarrow C(S^{1})\rightarrow 0}$,

where is the C * -algebra of the continuous functions on the circular line . ${\displaystyle C(S^{1})}$ ${\displaystyle S^{1}\subset \mathbb {C} }$

The Busby invariant

We arrange each extension here

${\displaystyle E:0\rightarrow B{\xrightarrow {\iota }}C{\xrightarrow {\sigma }}A\rightarrow 0}$

to a * -homomorphism , where let be the outer algebra of . This homomorphism is called the Busby invariant , named after R. Busby . ${\displaystyle \tau _{E}:A\rightarrow Q(B)}$ ${\displaystyle Q(B)=M(B)/B}$ ${\displaystyle B}$

To construct this homomorphism we choose a prototype with respect to . Because a two-sided ideal in , this element defines a multiplier from . By transition to one makes oneself independent of the choice of the archetype and therefore one defines , where the quotient mapping is. The mapping defined in this way is a * homomorphism . ${\displaystyle a\in A}$ ${\displaystyle c\in C}$ ${\displaystyle \sigma }$ ${\displaystyle B\cong \iota (B)}$ ${\displaystyle C}$ ${\displaystyle m_{c}:B\rightarrow B,b\mapsto \iota ^{-1}(c\cdot \iota (b))}$ ${\displaystyle M(B)}$ ${\displaystyle m_{c}+B\in Q(B)}$ ${\displaystyle \tau _{E}(a):=q_{B}(m_{c})}$ ${\displaystyle q_{B}:M(B)\rightarrow Q(B)}$ ${\displaystyle A\rightarrow Q(B)}$

Conversely, every * homomorphism comes from an extension. Add to this ${\displaystyle \tau :A\rightarrow Q(B)}$

${\displaystyle C:=\{(a,m)\in A\times M(B)|\,\tau (a)=q_{B}(m)\}}$

and form the short exact sequence

${\displaystyle 0\rightarrow B{\xrightarrow {b\mapsto (0,b)}}C{\xrightarrow {(a,m)\mapsto a}}A\rightarrow 0}$.

The Busby invariant of this extension is then the given . ${\displaystyle \tau }$

Two Busby invariants are called unitary equivalent , in signs if there is a unitary element with for all . ${\displaystyle \tau _{i}:A\rightarrow Q(B)}$ ${\displaystyle \tau _{1}\sim _{u}\tau _{2}}$ ${\displaystyle u\in M(B)}$ ${\displaystyle \tau _{1}(a)=q_{B}(u)^{*}\tau _{2}(a)q_{B}(u)}$ ${\displaystyle a\in A}$

• Two extensions are isomorphic if and only if they have the same Busby invariant.

This statement looks amazing at first glance, since the C * algebra does not appear in the Busby invariant. But the Busby invariant by construction also contains information about the operation of pre-images of elements as multipliers , and as shown above, this is sufficient for the construction of a too isomorphic C * -algebra. ${\displaystyle C}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle C}$

• Two extensions are unitarily equivalent if and only if the associated Busby invariants are unitarily equivalent.

• An extension is isomorphic to the trivial extension if and only if the Busby invariant is 0.

• The Busby invariant of an extension is injective if and only if is an essential ideal . This is why the associated extension and its Busby invariant are also called essential . ${\displaystyle 0\rightarrow B{\xrightarrow {\iota }}C\rightarrow A\rightarrow 0}$ ${\displaystyle \iota (B)\subset C}$

We already see here that the isomorphism classes can form into a very large set. If special is and , then the Busby invariant to an extension is a * homomorphism and every such homomorphism is already uniquely determined by its image of 1. Hence there are as many isomorphism classes in as there are projections in Calkin's algebra . This consideration alone motivates the search for coarser equivalence relations. ${\displaystyle {\mathcal {E}}(A,B)}$ ${\displaystyle A=\mathbb {C} }$ ${\displaystyle B=K}$ ${\displaystyle \mathbb {C} \rightarrow Q(K)}$ ${\displaystyle {\mathcal {E}}(\mathbb {C} ,K)}$

In the following it is much more convenient to talk about Busby invariants than about extensions.

Ext (A, B)

We will now define a semigroup . To do this, we first create an addition and then a suitable sub-group, which we make a neutral element. Finally, we investigate the question of when this semigroup is even a group by examining the invertible elements. ${\displaystyle \mathrm {Ext} (A,B)}$

addition

With the addition to be presented here, it becomes clear why we assume stable and why we will use the equivalence relation of the unitary equivalence. Da , is , where is the algebra of the complex matrices and accordingly the algebra of the with components . This isomorphism continues , and continues. The last isomorphism, which goes back to the required stability and which we designate with , allows the direct sum of Busby invariants to be formed: ${\displaystyle B}$ ${\displaystyle \ell ^{2}\cong \ell ^{2}\oplus \ell ^{2}}$ ${\displaystyle K\cong K\otimes M_{2}\cong M_{2}(K)}$ ${\displaystyle M_{2}}$ ${\displaystyle 2\times 2}$ ${\displaystyle M_{2}(K)}$ ${\displaystyle 2\times 2}$ ${\displaystyle K}$ ${\displaystyle M_{2}(B)\cong B}$ ${\displaystyle M_{2}(M(B))\cong M(B)}$ ${\displaystyle M_{2}(Q(B))\cong Q(B)}$ ${\displaystyle \theta _{B}}$

${\displaystyle \tau _{1}\oplus \tau _{2}:=\theta _{B}\circ {\begin{pmatrix}\tau _{1}&0\\0&\tau _{2}\end{pmatrix}}:A\rightarrow Q(B)}$.

It is now obvious to consider this addition . However, the above definition depends on and therefore on the selected isomorphism and two different decompositions lead to a unitary element that mediates a unitary equivalence between the Busby invariants of the sums. In order to be independent of this choice, one goes over to the equivalence classes of Busby invariants with respect to unitary equivalence and shows that ${\displaystyle {\mathcal {E}}(A,B)}$ ${\displaystyle \theta _{B}}$ ${\displaystyle \ell ^{2}\cong \ell ^{2}\oplus \ell ^{2}}$ ${\displaystyle u\in M(B)}$ ${\displaystyle [\tau ]}$

${\displaystyle [\tau _{1}]+[\tau _{2}]:=[\tau _{1}\oplus \tau _{2}]}$

is well-defined and makes it into an Abelian semigroup, i.e. the addition defined above is associative and commutative, for which the unitary equivalence is also required. ${\displaystyle {\overline {\mathcal {E}}}(A,B):={\mathcal {E}}(A,B)/\!\sim _{u}}$

Degenerate extensions

Here we turn to the question of a neutral element in the semigroup defined above. A Busby invariant or an associated extension is called degenerate if there is a * homomorphism with . The class of the degenerate Busby invariants is closed under addition and unitary equivalence and we define . Obviously the trivial Busby invariant 0 is degenerate. ${\displaystyle \tau :A\rightarrow Q(B)}$ ${\displaystyle {\hat {\tau }}:A\rightarrow M(B)}$ ${\displaystyle \tau =q_{B}\circ {\hat {\tau }}}$ ${\displaystyle {\mathcal {D}}(A,B)}$ ${\displaystyle {\overline {\mathcal {D}}}(A,B):={\mathcal {D}}(A,B)/\!\sim _{u}}$

If separable and σ-unital , there is always an essential, degenerate extension. According to one theorem by Voiculescu, two essential, degenerate Busby invariants that map the one element to the one element are unitarily equivalent. ${\displaystyle A}$ ${\displaystyle B}$

Definition of Ext (A, B)

After these preparations we now define

${\displaystyle \mathrm {Ext} (A,B):={\overline {\mathcal {E}}}(A,B)/{\overline {\mathcal {D}}}(A,B)}$

and thus get a commutative semigroup with a neutral element ${\displaystyle 0:={\overline {\mathcal {D}}}(A,B)}$

The definition of addition requires that it is stable in order to be able to form the direct sum of the Busby invariants. If not stable, then we stabilize, that is, we set ${\displaystyle B}$ ${\displaystyle B}$

${\displaystyle \mathrm {Ext} (A,B):=\mathrm {Ext} (A,B\otimes K)}$,

which leads to the same definition for something that is already stable . ${\displaystyle B}$

We also define

${\displaystyle \mathrm {Ext} ^{-1}(A,B)}$ : = Group of invertible elements in ${\displaystyle \mathrm {Ext} (A,B)}$

If a Busby invariant is used, the associated element is also designated with and often only with . ${\displaystyle \tau }$ ${\displaystyle \mathrm {Ext} (A,B)}$ ${\displaystyle [\tau ]}$ ${\displaystyle \tau }$

Finally , one sometimes omits the mention of and is defined ${\displaystyle B=K}$ ${\displaystyle K}$

${\displaystyle \mathrm {Ext} (A):=\mathrm {Ext} (A,K)=\mathrm {Ext} (A,\mathbb {C} )}$.

Invertible elements

The question now arises as to when there is even a group, that is, when it applies. For this we need the following characterization of the invertible elements: ${\displaystyle \mathrm {Ext} (A,B)}$ ${\displaystyle \mathrm {Ext} ^{-1}(A,B)=\mathrm {Ext} (A,B)}$

The following statements are equivalent for: ${\displaystyle \tau \in \mathrm {Ext} (A,B)}$

• ${\displaystyle \tau }$ is invertible.
• The associated extension is semi-divisive.
• There is a completely positive picture with and ${\displaystyle \psi :A\rightarrow M(B)}$ ${\displaystyle \|\psi \|\leq 1}$ ${\displaystyle \tau =q_{B}\circ \psi }$
• There is a * homomorphism with ${\displaystyle \pi :A\rightarrow M_{2}(M(B))}$

${\displaystyle {\begin{pmatrix}\tau (\cdot )&0\\0&0\end{pmatrix}}=(q_{B}\otimes \mathrm {id} _{M_{2}})({\begin{pmatrix}1&0\\0&0\end{pmatrix}}\pi (\cdot ){\begin{pmatrix}1&0\\0&0\end{pmatrix}})}$

The essential tool for proving this theorem is Stinespring's Theorem , in order to get from a completely positive map to the * -homomorphism of the last statement.

According to a lifting theorem by Choi and Effros , every complete positive mapping can be lifted on a separable, nuclear C * -algebra and we therefore get for separable, nuclear C * -algebras, or in other words: ${\displaystyle \mathrm {Ext} ^{-1}(A,B)=\mathrm {Ext} (A,B)}$

• If separable and nuclear is a group. ${\displaystyle A}$ ${\displaystyle \mathrm {Ext} (A,B)}$

${\displaystyle \mathrm {Ext} (A)=\mathrm {Ext} (A,K)}$ is not always a group.

properties

Functoriality in the first component

If there is a * -homomorphism between C * -algebras, then it is a homomorphism . In this way a contravariant functor is obtained from the category of C * -algebras to the category of Abelian semigroups. If one restricts this consideration to the full sub-category of the separable, nuclear C * -algebras, one obtains a functor with values ​​in the category of the Abelian groups. ${\displaystyle \varphi :A_{1}\rightarrow A_{2}}$ ${\displaystyle [\tau ]\mapsto [\tau \circ \varphi ]}$ ${\displaystyle \varphi ^{*}:\mathrm {Ext} (A_{2},B)\rightarrow \mathrm {Ext} (A_{1},B)}$ ${\displaystyle B}$ ${\displaystyle \mathrm {Ext} (-,B)}$

Functoriality in the second component

If the C * -algebra is fixed , one obtains a covariant functor from the category of the C * -algebras to the category of the Abelian semigroups (or groups, if it is separable and nuclear). A homomorphism is obtained for each . This construction is a bit more complex and should not be reproduced here, see ${\displaystyle A}$ ${\displaystyle \mathrm {Ext} (A,-)}$ ${\displaystyle A}$ ${\displaystyle \varphi :B_{1}\rightarrow B_{2}}$ ${\displaystyle \varphi _{*}:\mathrm {Ext} (A,B_{1})\rightarrow \mathrm {Ext} (A,B_{2})}$

Relationship to the KK theory

From the last statement of the above characterization of invertible elements one can construct a * -homomorphism and a projection for invertible elements , so that . Then the pair is a KK ¹ -cycle and one has a total of one element out if σ-unital . In this way an isomorphism is obtained ${\displaystyle \tau }$ ${\displaystyle \pi :A\rightarrow M(B)}$ ${\displaystyle p\in M(B)}$ ${\displaystyle \tau (\cdot )=q_{B}(p\pi (\cdot ))}$ ${\displaystyle (p,\pi )}$ ${\displaystyle KK^{1}(A,B)}$ ${\displaystyle B}$

${\displaystyle \mathrm {Ext} ^{-1}(A,B)\cong KK^{1}(A,B)}$.

If it is also nuclear, so is ${\displaystyle A}$

${\displaystyle \mathrm {Ext} (A,B)\cong KK^{1}(A,B)}$.

Homotopy invariance

Two homomorphisms are called homotopic if there are further homomorphisms such that all mappings are continuous, with the elements of passing through. One can now ask whether the induced homomorphisms and are the same for homotopic mappings , that would be the homotopy invariance in the first component. Similarly, one can ask about the homotopy invariance in the second component. There are only partial results here. GG Kasparow has shown by means of the above relation to the KK theory that for homotopy invariance there is both components if is separable and σ-unital. ${\displaystyle \varphi _{0},\varphi _{1}:A\rightarrow B}$ ${\displaystyle \varphi _{t}:A\rightarrow B,\,0<t<1}$ ${\displaystyle [0,1]\ni t\mapsto \varphi _{t}(a)\in B}$ ${\displaystyle a}$ ${\displaystyle A}$ ${\displaystyle \varphi _{0},\varphi _{1}:A_{1}\rightarrow A_{2}}$ ${\displaystyle \varphi _{1}^{*}}$ ${\displaystyle \varphi _{2}^{*}}$ ${\displaystyle \mathrm {Ext} ^{-1}(A,B)}$ ${\displaystyle A}$ ${\displaystyle B}$

Commutative C * algebras, BDF theory

If a metrizable , compact space , then the algebra of continuous complex-valued functions is a commutative C * -algebra. In this case it is ${\displaystyle X}$ ${\displaystyle C(X)}$

${\displaystyle \mathrm {Ext} (X):=\mathrm {Ext} (C(X))=\mathrm {Ext} (C(X),K)}$

of interest. For such is separable and, as commutative C * -algebra, also nuclear. Therefore the above theory can be applied: ${\displaystyle X}$ ${\displaystyle C(X)}$

${\displaystyle X\mapsto \mathrm {Ext} (X)}$ is a homotopy-invariant functor from the category of metrizable, compact spaces to the category of Abelian groups.

Historically, the commutative case has been dealt with earlier, particularly in the work of LG Brown , RG Douglas, and PA Fillmore , published in closed form in 1980. This commutative theory is also called the BDF theory after the first letters of the mathematicians involved . Since is separable and nuclear, there is a simpler situation compared to the theory presented above, in particular there is always an Abelian group. Some proofs are also a bit easier for commutative C * -algebras. ${\displaystyle C(X)}$ ${\displaystyle \mathrm {Ext} (X)}$

One of the motivations was the investigation of essentially normal operators, because this leads to the investigation of , where is the essential spectrum of the operator in question, that is, the spectrum of the image in Calkin algebra. If such an essentially normal operator is the image in Calkin algebra and the spectrum of , then the C * -algebra generated by and the unit element is isomorphic to and one has an extension ${\displaystyle \mathrm {Ext} (X)}$ ${\displaystyle X\subset \mathbb {C} }$ ${\displaystyle T}$ ${\displaystyle t=T+K\in Q(K)}$ ${\displaystyle X=\sigma _{e}(T)=\sigma (t)}$ ${\displaystyle t}$ ${\displaystyle t}$ ${\displaystyle C^{*}(\{t,1\})}$ ${\displaystyle C(X)}$

${\displaystyle 0\rightarrow K\rightarrow C^{*}(\{T,1\}\cup K)\rightarrow C^{*}(\{t,1\})\cong C(X)\rightarrow 0}$.

Hence one is led to the investigation of . Here you can find the following result ${\displaystyle \mathrm {Ext} (X)}$

For is the group of homotopy classes of continuous functions with a compact carrier . Hence is a Cartesian product with a factor for each bounded connected component of . The isomorphism has the form ${\displaystyle X\subset \mathbb {C} }$ ${\displaystyle \mathrm {Ext} (X)=[\mathbb {C} \setminus X,\mathbb {Z} ]_{0}}$ ${\displaystyle \mathbb {C} \setminus X\rightarrow \mathbb {Z} }$ ${\displaystyle \mathrm {Ext} (X)}$ ${\displaystyle \prod \mathbb {Z} }$ ${\displaystyle \mathbb {C} \setminus X}$

${\displaystyle \textstyle [\tau ]\mapsto \prod _{n}\mathrm {index} (T-\lambda _{n})}$,

where is a substantially normal operator with a substantial spectrum , so that is the * -homomorphism, which maps to and the constant one function to the unit element in Calkin algebra, which are an element of every bounded connected component and where is the Fredholm index . ${\displaystyle T}$ ${\displaystyle X}$ ${\displaystyle \tau :C(X)\rightarrow Q(K)}$ ${\displaystyle \mathrm {id} _{X}}$ ${\displaystyle \mapsto T+K\in Q(K)}$ ${\displaystyle \lambda _{n}}$ ${\displaystyle \mathrm {index} }$

Since it has exactly one restricted connected component, and the extension to Toeplitz's algebra listed above under the examples is a generating element. ${\displaystyle \mathbb {C} \setminus S^{1}}$ ${\displaystyle \mathrm {Ext} (S^{1})\cong \mathbb {Z} }$

Individual evidence

1. ↑ R. Busby: Double centrilizers and extensions of C * -algebras , Transactions Amer. Math. Soc. (1968) Vol. 132, pp. 79-99
2. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , Lemma 3.2.3
3. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 15.12.3
4. ^ J. Anderson: AC * -algebra A for which Ext (A) is not a group , Annals of Math. (1978), Volume 107, pages 455-458
5. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , September 15
6. ^ R. Douglas: C * -Algebra Extensions and K-Homology , Princeton University Press (1980), Annals of Mathematical Studies, Volume 95
7. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 16.2.1.