KK theory

The KK theory is a mathematical theory from the field of functional analysis . The name comes from the fact that it is a K-theory with two variables, which generalizes the classical K-theory for C * -algebras and the theory of the extensions of C * -algebras . The KK theory goes back to GG Kasparow .

Constructions

The groups of the KK theory to be defined below are formed by equivalence classes of Hilbert C * modules with an additional structure and depend on two C * algebras and . The C * -algebras have a - graduation and for some sometimes very technical constructions, all C * -algebras that appear as the first variable should be separable and all C * -algebras that appear as the second variable should be σ-unital , which for the sake of simplicity is always assumed below. For C * -algebras without graduation one can assume the trivial graduation, which is generated by the identity as graduation automorphism; then one obtains statements for non-graduated C * -algebras. ${\displaystyle KK(A,B)}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \mathbb {Z} /2}$

Kasparov modules

A Kasparow-AB module is a triple consisting of ${\displaystyle (E,\varphi ,F)}$

• a countably generated , graduated Hilbert law module , ${\displaystyle B}$ ${\displaystyle E}$
• a graduated * homomorphism , where the C * -algebra of the -linear operators is on with the graduation resulting from (see Hilbert-C * module ), ${\displaystyle \varphi :A\rightarrow L_{B}(E)}$ ${\displaystyle L_{B}(E)}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle E}$
• an operator with the following properties ${\displaystyle F\in L_{B}(E)}$
  • ${\displaystyle \partial F=1}$, that is, is homogeneous of degree 1, ${\displaystyle F}$
  • ${\displaystyle [F,\varphi (a)]\in K_{B}(E)}$for all , ${\displaystyle a\in A}$
  • ${\displaystyle (F^{2}-1)\varphi (a)\in K_{B}(E)}$for all , ${\displaystyle a\in A}$
  • ${\displaystyle (F^{*}-F)\varphi (a)\in K_{B}(E)}$for everyone . ${\displaystyle a\in A}$

Here referred to the graduated commutator and the two-sided ideal of the "compact" Operators in . The letter for the operator should be reminiscent of the Fredholm operator . Note that through becomes an AB bimodule. ${\displaystyle [\cdot ,\cdot ]}$ ${\displaystyle K_{B}(E)}$ ${\displaystyle L_{B}(E)}$ ${\displaystyle F}$ ${\displaystyle E}$ ${\displaystyle \varphi }$

If one considers the elements from as "small" compared to the more general ones , just as one sometimes regards a real compact operator as a "small" perturbation of a Hilbert space operator, the conditions state that the quantities should be "small". If these quantities are even 0, the Kasparov module is called degenerate . The trivial module is a degenerate Kasparov AB module; note that very different objects are marked with 0. ${\displaystyle K_{B}(E)}$ ${\displaystyle L_{B}(E)}$ ${\displaystyle F}$ ${\displaystyle [F,\varphi (a)],(F^{2}-1)\varphi (a),(F^{*}-F)\varphi (a)}$ ${\displaystyle (0,0,0)}$

The following is the class of Kasparov AB modules. We will define an addition and a suitable equivalence relation on which makes the set of equivalence classes a group. That will be the KK groups. ${\displaystyle \mathrm {E} (A,B)}$ ${\displaystyle \mathrm {E} (A,B)}$

Direct sum

The addition in the KK groups will be defined by means of the direct sum, so we briefly describe the direct sum of finitely many Kasparov modules. Let be Kasparov AB modules. We define the direct sum ${\displaystyle {\mathcal {E}}_{i}=(E_{i},\varphi _{i},F_{i}),\,i=1,\ldots ,n}$

${\displaystyle (E,\varphi ,F)={\mathcal {E}}_{1}\oplus \ldots \oplus {\mathcal {E}}_{n}}$

by

${\displaystyle E=E_{1}\oplus \ldots \oplus E_{n}}$with graduation , whereby the graduations are on the . ${\displaystyle S_{E}(x_{1},\ldots ,x_{n})=(S_{E_{1}}(x_{1}),\ldots ,(S_{E_{n}}(x_{n}))}$ ${\displaystyle S_{E_{i}}}$ ${\displaystyle E_{i}}$

${\displaystyle \varphi (a):={\begin{pmatrix}\varphi _{1}(a)&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &\varphi _{n}(a)\end{pmatrix}}\in L_{B}(E_{1}\oplus \ldots \oplus E_{n})}$

${\displaystyle F:={\begin{pmatrix}F_{1}(a)&\cdots &0\\\vdots &\ddots &\vdots \\0&\cdots &F_{n}(a)\end{pmatrix}}\in L_{B}(E_{1}\oplus \ldots \oplus E_{n})}$

We check that the triple is again a Kasparov- AB module, which we call the direct sum of . ${\displaystyle (E,\varphi ,F)}$ ${\displaystyle {\mathcal {E}}_{i}}$

Pushout

For the announced definition of the equivalence relation on we need the following construction called pushout . Let it be a Kasparow AB module and a surjective * homomorphism. Then show you that at any operator it exactly an operator is using , wherein the push out the Hilbert B modulus with respect to , and the quotient map . Then defines a * homomorphism and ${\displaystyle \mathrm {E} (A,B)}$ ${\displaystyle {\mathcal {E}}=(E,\varphi ,F)}$ ${\displaystyle f:B\rightarrow C}$ ${\displaystyle T\in L_{B}(E)}$ ${\displaystyle T_{f}\in L_{C}(E_{f})}$ ${\displaystyle T_{f}(q(x))=q(T(x))}$ ${\displaystyle E_{f}}$ ${\displaystyle E}$ ${\displaystyle f}$ ${\displaystyle q:E\rightarrow E_{f}}$ ${\displaystyle \varphi _{f}(a):=\varphi (a)_{f}}$ ${\displaystyle \varphi _{f}:A\rightarrow L_{C}(E_{f})}$

${\displaystyle {\mathcal {E}}_{f}:=(E_{f},\varphi _{f},F_{f})}$

is a Kasparov AC module called the pushout of regarding . ${\displaystyle {\mathcal {E}}}$ ${\displaystyle f}$

Equivalence relations

Similar to the K-theory, in which the direct sum defines a link to suitable equivalence classes, we will proceed here. It is common to define several equivalence relations, of which it is then shown that they coincide under suitable countability assumptions on the C * -algebras, which we generally assume as mentioned above. We will only address one of these equivalence relations, the so-called homotopy relation, and only mention at this point that parts of the theory are developed in parallel for several equivalence relations in order to be able to show equality later. Since we only want to present results, we will not trace this path here.

To define the homotopy relation we first need the finer unitary equivalence. Two Kasparov- AB modules and are called unitary equivalent , in signs , if there is an operator that conveys a unitary equivalence of the graduated Hilbert C * modules such that ${\displaystyle {\mathcal {E}}_{0}=(E_{0},\varphi _{0},F_{0})}$ ${\displaystyle {\mathcal {E}}_{1}=(E_{1},\varphi _{1},F_{1})}$ ${\displaystyle {\mathcal {E}}_{0}\cong _{u}{\mathcal {E}}_{1}}$ ${\displaystyle U\in L_{B}(E_{0},E_{1})}$

• ${\displaystyle \varphi _{0}(a)=U^{*}\varphi _{1}(a)U}$ for all ${\displaystyle a\in A}$
• ${\displaystyle F_{0}=U^{*}F_{1}U}$

For a graduated C * -algebra, let the C * -algebra of the continuous functions on the unit interval with values ​​in . With the graduation induced by , this is again a graduated C * -algebra. The evaluations are surjective * homomorphisms with which pushouts can therefore be formed. Now write , if there is a Kasparov AC ([0,1], B) module with and . ${\displaystyle B}$ ${\displaystyle C([0,1],B)}$ ${\displaystyle [0,1]}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle f_{t}:C([0,1],B)\rightarrow B,f_{t}(a):=a(t),\,t\in [0,1]}$ ${\displaystyle {\mathcal {E}}_{0}\sim {\mathcal {E}}_{1}}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle {\mathcal {E}}_{0}\cong _{u}{\mathcal {E}}_{f_{0}}}$ ${\displaystyle {\mathcal {E}}_{1}\cong _{u}{\mathcal {E}}_{f_{1}}}$

Finally, one calls and homotop , in signs , if there is a finite number of Kasparov AB modules with ${\displaystyle {\mathcal {E}}_{0}}$ ${\displaystyle {\mathcal {E}}_{1}}$ ${\displaystyle {\mathcal {E}}_{0}\sim _{h}{\mathcal {E}}_{1}}$ ${\displaystyle {\tilde {\mathcal {E}}}_{i},\,i=0,\ldots ,n}$

${\displaystyle {\mathcal {E}}_{0}\sim {\tilde {\mathcal {E}}}_{0}\sim {\tilde {\mathcal {E}}}_{1}\sim \ldots \sim {\tilde {\mathcal {E}}}_{n}\sim {\mathcal {E}}_{1}}$.

One shows that there is an equivalence relation on the class of the Kasparov- AB modules. The equivalence class of a Kasparov module is denoted by. ${\displaystyle \sim _{h}}$ ${\displaystyle \mathrm {E} (A,B)}$ ${\displaystyle {\mathcal {E}}}$ ${\displaystyle [{\mathcal {E}}]}$

The KK groups

definition

Let and C * algebras. Then be ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle KK(A,B):=\mathrm {E} (A,B)/\!\sim _{h}}$

the set of equivalence classes of the Kasparov AB modules. Since the countably generated Hilbert B-modules according to the stabilization theorem of Kasparow can be understood as sub -modules of up to unitary equivalence , there is actually a lot here. One can show that the direct sum is an addition ${\displaystyle {\hat {H}}_{B}}$

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

which makes an Abelian group. ${\displaystyle KK(A,B)}$

The zero element is the class of the degenerate Kasparov modules. We briefly describe the inverse formation in this group. It is

${\displaystyle -[(E,\varphi ,F)]=[(E^{op},{\tilde {\varphi }},-F)]}$.

It is the Hilbert module with the opposite graduation, that is, the modules have the same homogeneous elements, only the degrees of these homogeneous elements of a are compared with the degrees of homogeneous elements of the other by 1 modulo shifted. Next is , the homogeneous elements of degree of were. ${\displaystyle E^{op}}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle \mathbb {Z} /2}$ ${\displaystyle {\tilde {\varphi }}(a^{(0)}+a^{(1)})=\varphi (a^{(0)}-a^{(1)})}$ ${\displaystyle a^{(i)}}$ ${\displaystyle i}$ ${\displaystyle A}$

Finally be

${\displaystyle KK^{1}(A,B):=KK(A,B{\hat {\otimes }}\mathbb {C} _{1})}$.

It is the tensor product of graduated and , said two-dimensional C * algebra by wearing defined graduation. For a better understanding of this tensor product, let the graduation automorphism be and be the homomorphism that maps 0 to identity and 1 to . Then is a C * -dynamic system and the graduated tensor product is isomorphic to the cross product of this C * -dynamic system. ${\displaystyle B{\hat {\otimes }}\mathbb {C} _{1}}$ ${\displaystyle B}$ ${\displaystyle \mathbb {C} _{1}:=\mathbb {C} \oplus \mathbb {C} }$ ${\displaystyle \alpha (z,w):=(w,z)}$ ${\displaystyle \beta _{B}}$ ${\displaystyle B}$ ${\displaystyle \beta :\mathbb {Z} /2\rightarrow \mathrm {Aut} (B)}$ ${\displaystyle \beta _{B}}$ ${\displaystyle (B,\mathbb {Z} /2,\beta )}$ ${\displaystyle B{\hat {\otimes }}\mathbb {C} _{1}}$ ${\displaystyle B\ltimes _{\beta }\mathbb {Z} /2}$

Tensor products with C * algebras

Let a Kasparaow AB module be another, separable, graduated C * algebra. Then there is a graduated Hilbert D module and the Hilbert B⊗D module , the outer, graduated tensor product of the two Hilbert C * modules, can be formed. is then a * -homomorphism and an operator that makes a Kasparov- A⊗DB⊗D -module. In this way we get homomorphism ${\displaystyle (E,\varphi ,F)}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle E\otimes D}$ ${\displaystyle \varphi \otimes \mathrm {id} _{D}}$ ${\displaystyle A\otimes D\rightarrow L_{B\otimes D}(E\otimes D)}$ ${\displaystyle F\otimes \mathrm {id} _{D}}$ ${\displaystyle (E\otimes D,\varphi \otimes \mathrm {id} _{D},F\otimes \mathrm {id} _{D})}$

${\displaystyle \tau _{D}:KK(A,B)\rightarrow KK(A\otimes D,B\otimes D),\quad [(E,\varphi ,F)]\mapsto [(E\otimes D,\varphi \otimes \mathrm {id} _{D},F\otimes \mathrm {id} _{D})]}$.

Bott periodicity

The group is also written as . By iterating the formation of the graduated tensor product with , one could also define higher KK groups: ${\displaystyle KK(A,B)}$ ${\displaystyle KK^{0}(A,B)}$ ${\displaystyle \mathbb {C} _{1}}$

${\displaystyle KK^{n+1}(A,B):=KK^{n}(A,B{\hat {\otimes }}\mathbb {C} _{1}),\quad n=0,1,2,\ldots }$

but that turns out to be unnecessary, because you can show that

${\displaystyle KK^{0}(A,B)\cong KK^{1}(A{\hat {\otimes }}\mathbb {C} _{1},B)\cong KK^{1}(A,B{\hat {\otimes }}\mathbb {C} _{1})\cong KK^{0}(A{\hat {\otimes }}\mathbb {C} _{1},B{\hat {\otimes }}\mathbb {C} _{1})}$.

This is called the formal Bott periodicity because it behaves similarly to the Bott periodicity of the K-theory. The formal Bott periodicity can essentially be traced back to the relationship and is therefore much simpler than the real Bott periodicity, which uses hangings . But this real Bott periodicity can also be proven in the KK theory. ${\displaystyle \mathbb {C} _{1}{\hat {\otimes }}\mathbb {C} _{1}\cong M_{2}(\mathbb {C} )}$

If a C * -algebra, then denote the suspension of , that is, the C * -algebra of all continuous functions that vanish at infinity . Then applies ${\displaystyle A}$ ${\displaystyle SA}$ ${\displaystyle A}$ ${\displaystyle \mathbb {R} \rightarrow A}$

• ${\displaystyle KK^{1}(A,B)\cong KK^{0}(A,SB)\cong KK(SA,B)}$,
• ${\displaystyle KK^{0}(A,B)\cong KK^{1}(A,SB)\cong KK^{1}(SA,B)}$

${\displaystyle \cong KK^{0}(SSA,B)\cong KK^{0}(A,SSB)\cong KK^{0}(SA,SB)}$.

Alternative description of CC ¹ (A, B)

Here we define so-called KK ¹ -cycles and show how such cycles can be used for an alternative description of trivially graduated C * -algebras by means of a suitable equivalence relation . ${\displaystyle KK^{1}(A,B)}$

A KK ¹ cycle is a pair consisting of an element and a * homomorphism such that ${\displaystyle (v,\lambda )}$ ${\displaystyle v\in M(B)}$ ${\displaystyle \lambda :A\rightarrow M(B)}$

${\displaystyle v\lambda (a)-\lambda (a)v,(v^{*}-v)\lambda (a),(v^{2}-v)\lambda (a)\in B}$ for all ${\displaystyle a\in A}$

Let it be the set of such KK ¹ -cycles. Two KK ¹ -cycles and are called homomtop , if there are with ${\displaystyle E^{1}(A,B)}$ ${\displaystyle (v_{0},\lambda _{0})}$ ${\displaystyle (v_{1},\lambda _{1})}$ ${\displaystyle (v,\lambda )\in E^{1}(A,C([0,1],B))}$

${\displaystyle v_{0}={\overline {\pi }}_{0}(v),v_{1}={\overline {\pi }}_{1}(v),\lambda _{0}={\overline {\pi }}_{0}\circ \lambda ,\lambda _{1}={\overline {\pi }}_{1}\circ \lambda }$,

where the evaluation mapping is in the point and its unambiguous, strictly continuous continuation to the multiplier algebras. This relation marked with is an equivalence relation, the equivalence classes of which we write in square brackets. ${\displaystyle \pi _{t}:C([0,1],B)\rightarrow B}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle {\overline {\pi _{t}}}}$ ${\displaystyle \sim _{h}}$

To you by ${\displaystyle E^{1}(A,B)/\!\sim _{h}}$

${\displaystyle [(v_{1},\lambda _{1})]+[(v_{2},\lambda _{2})]:=[(\theta _{B}{\begin{pmatrix}v&0\\0&v\end{pmatrix}},\theta _{B}\circ {\begin{pmatrix}\lambda (\cdot )&0\\0&\lambda (\cdot )\end{pmatrix}})]}$

defined addition, where denotes a unitary isomorphism . ${\displaystyle \theta _{B}}$ ${\displaystyle M_{2}(M(B))\cong M(B)}$

We now describe an isomorphism from to . In addition, let and the isomorphism defined by . For a KK ¹ cycle, set ${\displaystyle E^{1}(A,B)/\!\sim _{h}}$ ${\displaystyle KK^{1}(A,B)}$ ${\displaystyle B_{1}=B\otimes \mathbb {C} _{1}\cong B\oplus B}$ ${\displaystyle \varphi :M(B)\oplus M(B)\rightarrow M(B_{1})}$ ${\displaystyle \varphi (m_{1},m_{2})(b_{1}\oplus b_{2}):=m_{1}b_{1}\oplus m_{2}b_{2}}$ ${\displaystyle (v,\lambda )\in E^{1}(A,B)}$

${\displaystyle {\mathcal {E}}_{v,\lambda }=(B_{1},\varphi \circ (\lambda \oplus \lambda ),\varphi ((2v-1)\oplus (1-2v)))}$.

Then is and ${\displaystyle {\mathcal {E}}_{v,\lambda }\in KK(A,B_{1})=KK^{1}(A,B)}$

${\displaystyle E^{1}(A,B)/\!\sim _{h}\rightarrow KK^{1}(A,B),\quad (v,\lambda )\mapsto {\mathcal {E}}_{v,\lambda }}$

is an isomorphism. It is thus described independently of the formal Bott periodicity. ${\displaystyle KK^{1}(A,B)}$

Examples

Homomorphisms

If there is a graduated * -homomorphism between C * -algebras, then it is a Kasparov- AB -module. Please note the above agreed requirement that -unital; thus, as a Hilbert module (with trivial graduation), it is actually generated countable. The equivalence class is often just referred to with . The elements will take on the role of identities in the Kasparov product to be discussed below. ${\displaystyle f:A\rightarrow B}$ ${\displaystyle (B,f,0)}$ ${\displaystyle B}$ ${\displaystyle \sigma }$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle [(B,f,0)]\in KK(A,B)}$ ${\displaystyle [f]}$ ${\displaystyle [\mathrm {id} _{A}]\in KK(A,A)}$

KK (ℂ, ℂ)

Let it be with graduation and the * homomorphism with . Next is the quotient mapping in the Calkin algebra . Let be an operator such that is unitary. Then ${\displaystyle E={\hat {H}}_{\mathbb {C} }=\ell ^{2}\oplus \ell ^{2}}$ ${\displaystyle S_{E}(\xi \oplus \eta ):=\xi \oplus -\eta }$ ${\displaystyle \varphi :\mathbb {C} \rightarrow L_{\mathbb {C} }(E)}$ ${\displaystyle \varphi (1)=\mathrm {id} _{E}}$ ${\displaystyle \pi :L(\ell ^{2})\rightarrow L(\ell ^{2})/K(\ell ^{2})}$ ${\displaystyle T\in L(\ell ^{2})}$ ${\displaystyle \pi (T)}$

${\displaystyle [(E,\varphi ,{\begin{pmatrix}0&T^{*}\\T&0\end{pmatrix}})]\in KK(\mathbb {C} ,\mathbb {C} )}$.

Since is unitary, is a Fredholm operator , and it can be shown that ${\displaystyle \pi (T)}$ ${\displaystyle T}$

${\displaystyle KK(\mathbb {C} ,\mathbb {C} )\rightarrow \mathbb {Z} ,\quad [(E,\varphi ,{\begin{pmatrix}0&T^{*}\\T&0\end{pmatrix}})]\mapsto \mathrm {index} (T)}$

is a group isomorphism, where index denotes the Fredholm index .

K groups

We show here how the K groups of a C * algebra (with trivial graduation) reappear in the KK theory. ${\displaystyle B}$

Let it be a unitary element of the outer multiplier algebra of the C * -algebra , i.e. from the quotient of the multiplier algebra of the tensor product and the C * -algebra of the compact operators over a separable Hilbert space according to this tensor product. Be a lift from , that is . Then ${\displaystyle u\in Q^{s}(B):=M(B\otimes K)/(B\otimes K)}$ ${\displaystyle B}$ ${\displaystyle B}$ ${\displaystyle K}$ ${\displaystyle T_{u}\in M^{s}(B):=M(B\otimes K)}$ ${\displaystyle u}$ ${\displaystyle u=T_{u}+B\otimes K\in Q^{s}(B)}$

${\displaystyle [({\hat {H}}_{B},\varphi ,{\begin{pmatrix}0&T_{u}^{*}\\T_{u}&0\end{pmatrix}})]\in KK(\mathbb {C} ,B)}$.

Note that and therefore the third component of the specified element is actually off and obviously has degree 1. is the * homomorphism with . Then you can show the assignment ${\displaystyle L({\hat {H}}_{B})\cong L(H_{B}\oplus H_{B}))\cong M_{2}(L(H_{B}))\cong M_{2}(M^{s}(B))}$ ${\displaystyle L_{B}({\hat {H}}_{B})}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi (1)=\mathrm {id} _{{\hat {H}}_{B}}}$

${\displaystyle K_{1}(Q^{s}(B))\rightarrow KK(\mathbb {C} ,B),\quad [u]\mapsto [({\hat {H}}_{B},\varphi ,{\begin{pmatrix}0&T_{u}^{*}\\T_{u}&0\end{pmatrix}})]}$

is a group isomorphism. As stated in the article on multiplier algebras, you also have a natural isomorphism , so you get an isomorphism overall . ${\displaystyle K_{0}(B)\cong K_{1}(Q^{s}(B))}$ ${\displaystyle K_{0}(B)\cong KK(\mathbb {C} ,B)}$

Either through similar considerations or using the Bott periodicity presented above, one also comes to an isomorphism , so that overall one arrives at the following easily memorable formula: ${\displaystyle K_{1}(B)\cong KK^{1}(\mathbb {C} ,B)}$

${\displaystyle K_{i}(B)\cong KK^{i}(\mathbb {C} ,B),\,i=0,1}$.

Extensions

Using the alternative description of KK ¹ cycles presented above , an isomorphism can be constructed, the former denoting the group of invertible elements in Ext (A, B) . As the article on extensions of C * -algebras executed belong to Busby-invariant one invertible element a homomorphism and a projection with . Then there is a KK ¹ -cycle and we get a group isomorphism ${\displaystyle KK^{1}(A,B)}$ ${\displaystyle \mathrm {Ext} ^{-1}(A,B)\cong KK^{1}(A,B)}$ ${\displaystyle \tau :A\rightarrow Q(B)}$ ${\displaystyle \pi :A\rightarrow M(B)}$ ${\displaystyle p\in M(B)}$ ${\displaystyle \tau (\cdot )=p\pi (\cdot )+B}$ ${\displaystyle (p,\pi )}$

${\displaystyle \mathrm {Ext} ^{-1}(A,B)\rightarrow KK^{1}(A,B),\quad [\tau ]\mapsto [{\mathcal {E}}_{p,\pi }]}$.

Functoriality

The assignment of two C * -algebras to their KK-group can be expanded to a functor if one fixes one C * -algebra for each. These functors even turn out to be homotopy-invariant.

Functoriality in the first component

If there is a Kasparov AB module and a graduated * homomorphism , then there is a Kasparov CB module and a group homomorphism is obtained ${\displaystyle (E,\varphi ,F)}$ ${\displaystyle \psi :C\rightarrow A}$ ${\displaystyle (E,\varphi \circ \psi ,F)}$

${\displaystyle \psi ^{*}:KK(A,B)\rightarrow KK(C,B),\quad [(E,\varphi ,F)]\mapsto [(E,\varphi \circ \psi ,F)]}$.

As a result, for a fixed to a contravariant functor on the category of separable, graduated C * algebras in the category of abelian groups. If one considers the trivial graduation on every C * -algebra, we get a contravariant functor from the category of the separable C * -algebras into the category of the Abelian groups. ${\displaystyle KK(-,B)}$ ${\displaystyle B}$

This functor is homotopy-invariant, i.e. there are * -homomorphisms for , so that the mappings are continuous for all , so is . ${\displaystyle \psi _{t}:A\rightarrow B}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle t\mapsto \psi _{t}(a)}$ ${\displaystyle a\in A}$ ${\displaystyle {\psi _{0}}^{*}={\psi _{1}}^{*}}$

Functoriality in the second component

If there is a Kasparov AB module and a graduated * homomorphism, then form the inner tensor product . This is a Hilbert module and is a * homomorphism . This definition gives a group homomorphism ${\displaystyle (E,\varphi ,F)}$ ${\displaystyle \psi :B\rightarrow C}$ ${\displaystyle E\otimes _{\varphi }B}$ ${\displaystyle D}$ ${\displaystyle {\tilde {\varphi }}(a):=\varphi (a)\otimes \mathrm {id} _{D}}$ ${\displaystyle A\rightarrow L_{D}(E\otimes _{\varphi }B)}$

${\displaystyle \psi _{*}:KK(A,B)\rightarrow KK(A,D),\quad [(E,\varphi ,F)]\mapsto [(E\otimes _{\varphi }B,{\tilde {\varphi }},F\otimes \mathrm {id} _{D})]}$.

As a result, for a fixed to a covariant functor on the category of -unitalen, C * algebras graduated in the category of abelian groups. If one considers the trivial graduation on every C * -algebra, we get a covariant functor from the category of the -unital C * -algebras to the category of the Abelian groups. ${\displaystyle KK(A,-)}$ ${\displaystyle A}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

This functor is homotopy-invariant, i.e. there are * -homomorphisms for , so that the mappings are continuous for all , so is . ${\displaystyle \psi _{t}:A\rightarrow B}$ ${\displaystyle t\in [0,1]}$ ${\displaystyle t\mapsto \psi _{t}(a)}$ ${\displaystyle a\in A}$ ${\displaystyle {\psi _{0}}_{*}={\psi _{1}}_{*}}$

The Kasparov product

Construction and properties

The Kasparov product is an illustration

${\displaystyle KK(A,D)\times KK(D,B)\mapsto KK(A,B)}$

which is a powerful tool in applications. Both the construction, which can only be indicated below, and the verification of the properties listed below require a high level of technical effort.

For construction are and . Then the graduated inner tensor product is a Hilbert module and is a * homomorphism . With great technical effort can be a suitable operator to construct and so an element define which one the Kasparaow-product of the two elements and calls, and the following properties: ${\displaystyle [(E_{1},\varphi _{1},F_{1})]\in KK(A,D)}$ ${\displaystyle [(E_{2},\varphi _{2},F_{2})]\in KK(D,B)}$ ${\displaystyle E_{1}\otimes _{\varphi _{2}}E_{2}}$ ${\displaystyle B}$ ${\displaystyle \varphi :=\varphi _{1}\otimes \mathrm {id} _{E_{2}}}$ ${\displaystyle A\rightarrow L_{B}(E)}$ ${\displaystyle F\in L_{B}(E)}$ ${\displaystyle [(E,\varphi ,F)]\in KK(A,B)}$ ${\displaystyle KK(A,D)}$ ${\displaystyle KK(D,B)}$

${\displaystyle (x\cdot y)\cdot z=x\cdot (y\cdot z)}$	For ${\displaystyle x\in KK(A,B),y\in KK(B,C),z\in KK(C,D)}$
${\displaystyle f^{*}(x\cdot y)=f^{*}(x)\cdot y}$	for * homomorphism ${\displaystyle x\in KK(A,B),y\in KK(B,C),f:{\tilde {A}}\rightarrow A}$
${\displaystyle f^{*}(x)\cdot y=x\cdot f^{*}(y)}$	for * homomorphism ${\displaystyle x\in KK(A,B),y\in KK(B,C),f:B\rightarrow {\tilde {B}}}$
${\displaystyle f_{*}(x\cdot y)=x\cdot f_{*}(y)}$	for * homomorphism ${\displaystyle x\in KK(A,B),y\in KK(B,C),f:C\rightarrow {\tilde {C}}}$
${\displaystyle [\mathrm {id} _{A}]\cdot x=x\cdot [\mathrm {id} _{B}]=x}$	For ${\displaystyle x\in KK(A,B)}$
${\displaystyle x\cdot (y+z)=x\cdot y+x\cdot z}$	For ${\displaystyle x\in KK(A,B),y,z\in KK(B,C)}$
${\displaystyle (x+y)\cdot z=x\cdot z+y\cdot z}$	For ${\displaystyle x,y\in KK(A,B),z\in KK(B,C),}$
${\displaystyle [f]\cdot x=f^{*}(x)}$	for * homomorphism ${\displaystyle x\in KK(A,B),f:C\rightarrow A}$
${\displaystyle x\cdot [f]=f_{*}(x)}$	for * homomorphism. ${\displaystyle x\in KK(A,B),f:B\rightarrow C}$

In particular, for every separable C * -algebra there is a ring with a unity . The group isomorphism presented above turns out to be a ring isomorphism. If an AF-C * algebra is isomorphic to the endomorphism ring of the group . ${\displaystyle KK(A,A)}$ ${\displaystyle [\mathrm {id} _{A}]}$ ${\displaystyle KK(\mathbb {C} ,\mathbb {C} )\cong \mathbb {Z} }$ ${\displaystyle A}$ ${\displaystyle KK(A,A)}$ ${\displaystyle K_{0}(A)}$

Extended Kasparov product

The Kasparov product can become a product as follows

${\displaystyle KK(A_{1},B_{1}\otimes D)\times KK(D\otimes A_{2},B_{2})\rightarrow KK(A_{1}\otimes A_{2},B_{1}\otimes B_{2})}$

be generalized, whereby the occurring C * -algebras should fulfill the countability conditions agreed above and the tensor product is always the graduated tensor product. For and are ${\displaystyle x\in KK(A_{1},B_{1}\otimes D)}$ ${\displaystyle y\in KK(D\otimes A_{2},B_{2})}$

${\displaystyle \tau _{A_{2}}(x)\in KK(A_{1}\otimes A_{2},B_{1}\otimes D\otimes A_{2})}$ and

${\displaystyle \tau _{B_{1}}(y)\in KK(D\otimes A_{2}\otimes B_{1},B_{2}\otimes B_{1})\cong KK(B_{1}\otimes D\otimes A_{2},B_{1}\otimes B_{2})}$,

so that one can form the Kasparov product , an element in . This product is labeled again with and confirms that it does not conflict with the Kasparov product already defined, which is essentially due to the fact that the identity is. Overall, we get the announced bilinear mapping ${\displaystyle \tau _{A_{2}}(x)\cdot \tau _{B_{1}}(y)}$ ${\displaystyle KK(A_{1}\otimes A_{2},B_{1}\otimes B_{2})}$ ${\displaystyle x\cdot y}$ ${\displaystyle \tau _{\mathbb {C} }}$

${\displaystyle \cdot :KK(A_{1},B_{1}\otimes D)\times KK(D\otimes A_{2},B_{2})\rightarrow KK(A_{1}\otimes A_{2},B_{1}\otimes B_{2}),\quad (x,y)\mapsto x\cdot y}$.

For one gets back the already known Kasparov product, because tensing with leads to isomorphic C * -algebras and is the identity. In this sense, the above product represents a generalization of the Kasparov product introduced earlier. ${\displaystyle A_{2}=B_{1}=\mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \tau _{\mathbb {C} }}$

As an important special case we want to deal with tensorization , because according to the above definition this leads to KK ¹ groups. If special is and , then when the tensor product occurs, one can move over to the KK ¹ group and omit the tensor . A bilinear mapping is therefore obtained from the above ${\displaystyle \mathbb {C} _{1}}$ ${\displaystyle B_{1}=\mathbb {C} _{1}}$ ${\displaystyle A_{2}=\mathbb {C} }$ ${\displaystyle B_{1}}$ ${\displaystyle A_{2}}$

${\displaystyle KK^{1}(A_{1},D)\times KK(D,B_{2})\rightarrow KK^{1}(A_{1},B_{1})}$.

So you can multiply elements from the left with elements from and get one element from . ${\displaystyle KK(D,B_{2})}$ ${\displaystyle KK^{1}(A_{1},D)}$ ${\displaystyle KK^{1}(A_{1},B_{1})}$

By setting analog and respectively , you get bilinear maps ${\displaystyle B_{1}=\mathbb {C} }$ ${\displaystyle A_{2}=\mathbb {C} _{1}}$ ${\displaystyle B_{1}=A_{2}=\mathbb {C} _{1}}$

${\displaystyle KK(A_{1},D)\times KK^{1}(D,B_{2})\rightarrow KK^{1}(A_{1},B_{2})}$

and

${\displaystyle KK^{1}(A_{1},D)\times KK^{1}(D,B_{2})\rightarrow KK(A_{1},B_{2})}$,

The formal Bott periodicity was used for the last figure.

Cyclic, 6-part, exact sequences

The cyclic, 6-part, exact sequences known from the K-theory can also be proven in the KK-theory. We assume a short, exact sequence

${\displaystyle 0\rightarrow J{\xrightarrow {\iota }}A{\xrightarrow {\sigma }}A/J\rightarrow 0}$

which arises from a closed, two-sided ideal in a C * -algebra with one element and of which we want to assume that it is semi-splitting . Then this sequence is an invertible extension and therefore determines an element according to the above . The multiplication with defined by the Kasparov product is defined for another C * -algebra homomorphisms ${\displaystyle J\subset A}$ ${\displaystyle e\in \mathrm {Ext} ^{-1}(A/J,J)\cong KK^{1}(A/J,J)}$ ${\displaystyle e}$ ${\displaystyle D}$

${\displaystyle KK(D,A/J)\rightarrow KK^{1}(D,J),\quad x\mapsto x\cdot e}$,

${\displaystyle KK^{1}(D,A/J)\rightarrow KK(D,J),\quad y\mapsto y\cdot e}$,

${\displaystyle KK(J,D)\rightarrow KK^{1}(A/J,D),\quad x\mapsto e\cdot x}$,

${\displaystyle KK^{1}(J,D)\rightarrow KK(A/J,D),\quad y\mapsto e\cdot y}$,

which are all designated with. Then there are the following cyclical, exact sequences: ${\displaystyle \delta }$

${\displaystyle {\begin{array}{ccccc}KK(D,J)&{\xrightarrow {\iota _{*}}}&KK(D,A)&{\xrightarrow {\sigma _{*}}}&KK(D,A/J)\\\uparrow \delta &&&&\downarrow \delta \\KK^{1}(D,A/J)&{\xleftarrow {\sigma _{*}}}&KK^{1}(D,A)&{\xleftarrow {\iota _{*}}}&KK^{1}(D,J)\\\end{array}}}$

and

${\displaystyle {\begin{array}{ccccc}KK(J,D)&{\xleftarrow {\iota ^{*}}}&KK(A,D)&{\xleftarrow {\sigma ^{*}}}&KK(A/J,D)\\\downarrow \delta &&&&\uparrow \delta \\KK^{1}(A/J,D)&{\xrightarrow {\sigma ^{*}}}&KK^{1}(A,D)&{\xrightarrow {\iota ^{*}}}&KK^{1}(J,D)\\\end{array}}}$

Such sequences are often helpful when calculating KK groups, especially if one or two members of such a sequence are 0, because then some images can be detected as injective, surjective or even bijective by means of accuracy.

Web links

• KK theory (nLab)

Individual evidence

1. ^ GG Kasparow: The operator K-functor and extensions of C * -algebras , Izv. Akad. Nauk. SSSR Ser. Mat. (1980) Vol. 44, pp. 571-636
2. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , definition 17.1.1
3. ^ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , definition 2.1.1.
4. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhauser-Verlag (1991), ISBN 0-8176-3496-7 , Section 2.1.5 .: pushout
5. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , definition 2.1.9 + Lemma 2.1.12
6. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 17.3.3
7. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Corollary 17.8.9
8. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Corollary 19.2.2
9. ^ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , sentence 3.3.6
10. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 17.5.5
11. ^ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , Corollary 3.3.11
12. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , August 17: Functoriality , September 17: Homotopy Invariance
13. ↑ KK Jensen, K. Thomsen: Elements of KK-Theory , Birkhäuser-Verlag (1991), ISBN 0-8176-3496-7 , Chapter 2.2: The Kasparov Product
14. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , sentences 18.4.4, 18.6.1, 18.7.1, 18.7.2
15. ^ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 18.9.1
16. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 19.5.7