A KK-like picture for E-theory of C*-algebras

TL;DR

This paper introduces a new way to represent elements of E-theory for C*-algebras using pairs of maps from the algebra itself, broadening the understanding of asymptotic homomorphisms.

Contribution

It provides a novel representation of E-theory elements via pairs of maps from the algebra, extending the classical asymptotic homomorphism approach.

Findings

E(A,B) can be represented by pairs of maps with the same deficiency from being homomorphisms

Pairs of maps can be viewed as asymptotic homomorphisms from a surjecting C*-algebra C

Examples of full surjections C→A show all classes in E(A,B) can be obtained from such pairs

Abstract

Let $A$, $B$ be separable C*-algebras, $B$ stable. Elements of the E-theory group $E(A,B)$ are represented by asymptotic homomorphisms from the second suspension of $A$ to $B$. Our aim is to represent these elements by (families of) maps from $A$ itself to $B$. We have to pay for that by allowing these maps to be even further from $*$-homomorphisms. We prove that $E(A,B)$ can be represented by pairs $(\varphi^+,\varphi^-)$ of maps from $A$ to $B$ that are not necessarily asymptotic homomorphisms, but have the same deficiency from being ones. Not surprisingly, such pairs of maps can be viewed as pairs of asymptotic homomorphisms from some C*-algebra $C$ that surjects onto $A$, and the two maps in a pair should agree on the kernel of this surjection. We give examples of full surjections $C\to A$, i.e. those, for which all classes in $E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$.

Equations

$$M^{+}(a,b){\phi }^{+}(c_1)\cdots {\phi }^{+}(c_n)=M^{-}(a,b){\phi }^{-}(c_1)\cdots {\phi }^{-}(c_n)$$

$$M^{\pm }(a,b){\Phi }^{+}(c_1)\cdots {\Phi }^{+}(c_n)({\Phi }^{+}(d)-{\Phi }^{-}(d))\in C_0({\mathbb{R}}_{+};B),$$

$$\underset{t\to \infty }{\lim }{M}_t^{\pm }(a,b){\Phi }_t^{+}(c_1)\cdots {\Phi }_t^{+}(c_n)({\Phi }_t^{+}(d)-{\Phi }_t^{-}(d))=0,$$

$$M^{+}(a,b)\sim M^{-}(a,b)(\text{mbox}thisis(a3)forn=0)$$

$$M^{+}(a,b){\Phi }^{+}(c)\sim M^{-}(a,b){\Phi }^{-}(c)(\text{mbox}thisis(a3)forn=1)$$

$$M^{+}(a,b){\Phi }^{+}(c_1)\cdots {\Phi }^{+}(c_n)\sim M^{-}(a,b){\Phi }^{-}(c_1)\cdots {\Phi }^{-}(c_n)$$

$$M^{+}(a,b){\Phi }^{+}(c_1)\cdots {\Phi }^{+}(c_{n+1})\sim M^{-}(a,b){\Phi }^{-}(c_1)\cdots {\Phi }^{-}(c_{n+1})$$

$${\Phi }^{+}(a){\Phi }^{+}(b)-{\Phi }^{-}(a){\Phi }^{-}(b)=({\Phi }^{+}(a)-{\Phi }^{-}(a)){\Phi }^{+}(b)+{\Phi }^{-}(a)({\Phi }^{+}(b)-{\Phi }^{-}(b))\in D,$$

$$M^{\pm }(a,b){\Phi }^{+}(c_1){\Phi }^{+}(c_2)({\Phi }^{+}(d)-{\Phi }^{-}(d))$$

$$=M^{\pm }(a,b)M^{+}(c_1,c_2)({\Phi }^{+}(d)-{\Phi }^{-}(d))+M^{\pm }(a,b){\Phi }^{+}(c_1{c}_2)({\Phi }^{+}(d)-{\Phi }^{-}(d)).$$

$$[({\Phi }^{+},{\Phi }^{-})]+[({\Psi }^{+},{\Psi }^{-})]=[({\tilde{s}}_1{\Phi }^{+}{\tilde{s}}_1^{\ast }+{\tilde{s}}_2{\Psi }^{+}{\tilde{s}}_2^{\ast },{\tilde{s}}_1{\Phi }^{-}{\tilde{s}}_1^{\ast }+{\tilde{s}}_2{\Psi }^{-}{\tilde{s}}_2^{\ast })].$$

$${\Psi }^{\pm }(a)(\tau )={\tilde{s}}_1(\tau ){\Psi }^{\pm }(a){\tilde{s}}_1(\tau {)}^{\ast }$$

$$[({\Phi }^{+},{\Phi }^{-})]+[({\Psi }^{+},{\Psi }^{-})]=[({\Psi }^{+},{\Psi }^{-})]+[({\Phi }^{+},{\Phi }^{-})]$$

$$[({\Phi }^{+},{\Phi }^{-})]+[({\Phi }^{-},{\Phi }^{+})]=0$$

$$\underset{r\to \infty }{\lim }\underset{t\in {\mathbb{R}}_{+}}{\sup }\Vert v_r(t)x_t-x_t\Vert =0$$

$$({\Psi }^{+}(ab)-{\Psi }^{+}(a){\Psi }^{+}(b))({\Psi }^{+}(c)-{\Psi }^{-}(c))$$

$$=v({\Phi }^{+}(ab)-{\Phi }^{+}(a){\Phi }^{+}(b))v({\Phi }^{+}(c)-{\Phi }^{-}(c))$$

$$+v{\Phi }^{+}(a)(1-v){\Phi }^{+}(b)v({\Phi }^{+}(c)-{\Phi }^{-}(c)).$$

$$M^{\pm }(a,b){\Psi }^{\pm }(c_1)\cdots {\Psi }^{\pm }(c_n)({\Psi }^{+}(d)-{\Psi }^{-}(d))\in C_0({\mathbb{R}}_{+};B)$$

$$p:EL(A,B)\to EM(A,B).$$

$$V=\left(\begin{array}{c}{w}^{1/2}\\ (1-w{)}^{1/2}\end{array}\right).$$

$$V{\Phi }^{\pm }(a)V^{\ast }=A^{\pm }=\left(\begin{array}{cc}{a}_{11}^{\pm }& a_{12}^{\pm }\\ a_{21}^{\pm }& a_{22}^{\pm }\end{array}\right).$$

$$(a_{ij}^{+}-a_{ij}^{-})(1-w{)}^{1/2}\sim 0\text{mbox}if(ij)\ne (11),$$

$$c_{11}^{+}-a_{11}^{+}{b}_{11}^{+}\sim c_{11}^{-}-a_{11}^{-}{b}_{11}^{-}.$$

$$C^{+}(\tau )-A^{+}(\tau )B^{+}(\tau )=\left(\begin{array}{cc}{c}_{11}^{+}-a_{11}^{+}{b}_{11}^{+}-{\tau }^2{a}_{12}^{+}{b}_{21}^{+}& \tau (c_{12}^{+}-a_{11}^{+}{b}_{12}^{+})-{\tau }^3{a}_{12}^{+}{b}_{22}^{+}\\ \tau (c_{21}^{+}-a_{21}^{+}{b}_{11}^{+})-{\tau }^3{a}_{22}^{+}{b}_{12}^{+}& {\tau }^2(c_{22}^{+}-a_{21}^{+}{b}_{12}^{+})-{\tau }^4{a}_{22}^{+}{b}_{22}^{+}\end{array}\right).$$

$$\Phi (a)=\sum _{i\ge 0}{\Phi }^{+}(a)e_{ii}+\sum _{i<0}{\Phi }^{-}(a)e_{ii},T=\sum _i{e}_{i,i+1}.$$

$$\underset{t\ge T_n}{\sup }\Vert M_t(a,b)v_r(t)\Vert <\frac{1}{n}$$

$$\alpha ({\Phi }^{+},{\Phi }^{-}{)}_t(f\otimes h_n\otimes a)=f(v(t))T^n{\Phi }_t(a).$$

$$\underset{t\to \infty }{\lim }\Vert \alpha ({\Phi }^{+},{\Phi }^{-}{)}_t(fg\otimes h_{n+m}\otimes ab)-\alpha ({\Phi }^{+},{\Phi }^{-}{)}_t(f\otimes h_n\otimes a)\alpha ({\Phi }^{+},{\Phi }^{-}{)}_t(g\otimes h_m\otimes b)\Vert =0$$

$$p_{+}=p_{+}(r,\varphi )=\left(\begin{array}{cc}f(r)& g(r)e^{i\varphi }\\ g(r)e^{-i\varphi }& 1-f(r)\end{array}\right);p_{-}=p_{-}(r,\varphi )=\left(\begin{array}{cc}f(r)& g(r)\\ g(r)& 1-f(r)\end{array}\right),$$

Full text

Introduction

There are several pictures for Kasparov’s $KK$-theory of $C^{*}$-algebras [7], [4], [2], [11], [13] each emphasizing one of the faces of $KK$-cycles, i.e. of representatives of $KK$-classes. One of the reasons behind this diversity is lack of homomorphisms between $C^{*}$-algebras. Any $*$-homomorphism $\varphi:A\to B$ represents an element in $KK(A,B)$, but if one wants to represent arbitrary element of $KK(A,B)$ then one has to use generalized morphisms. For example, the Cuntz picture of $KK$-theory uses the pairs $(\varphi^{+},\varphi^{-})$ of $*$-homomorphisms from $A$ to the multiplier algebra $MB$ of $B$ such that $\varphi^{+}(a)-\varphi^{-}(a)\in B$ for any $a\in A$ [4].

$E$-theory of Connes and Higson [3] shares the same problem: one has to use asymptotic homomorphisms in place of $*$-homomorphisms. Moreover, the asymptotic homomorphisms that represent elements in $E(A,B)$ have domain not in $A$ but in the suspension $SA$ over $A$ (or, even in the second suspension), and take values (if considered as $*$-homomorphisms) in $C_{b}(\mathbb{R}_{+};B)/C_{0}(\mathbb{R}_{+};B)$, where $\mathbb{R}_{+}=[0,\infty)$, and $C_{b}(\mathbb{R}_{+};B)$ denotes the algebra of bounded $B$-valued functions on $\mathbb{R}_{+}$.

Our aim is to represent elements of $E(A,B)$ by pairs of maps from $A$ itself to $C_{b}(\mathbb{R}_{+};B)/C_{0}(\mathbb{R}_{+};B)$ similarly to the Cuntz picture for $KK$-theory. We have to pay for that by allowing these maps to be even further from $*$-homomorphisms. We shall work with pairs of maps, which should be continuous, but may be not linear or not asymptotically linear, and we require that the two maps in a pair should have the same deficiency from being an asymptotic homomorphism. Namely, we show that elements of $E(A,B)$ can be represented by pairs $(\varphi^{+},\varphi^{-})$ of maps from $A$ to $C_{b}(\mathbb{R}_{+};B)/C_{0}(\mathbb{R}_{+};B)$ that satisfy

[TABLE]

for any $n=0,1,2,\ldots$ and any $a,b,c_{0},c_{1},\ldots,c_{n}\in A$, where $M^{\pm}(a,b)$ stands for either $\varphi^{\pm}(ab)-\varphi^{\pm}(a)\varphi^{\pm}(b)$ or $\varphi^{\pm}(a+b)-\varphi^{\pm}(a)-\varphi^{\pm}(b)$. Isomorphism between $E(A,B)$ and the group of homotopy classes of such pairs is proved using the explicit form of the Bott isomorphism obtained in [8].

Not surprisingly, such pairs of maps can be viewed as pairs of asymptotic homomorphisms from some $C^{*}$-algebra $C$ that surjects onto $A$, and the two maps in a pair should agree on the kernel of this surjection. We give examples of full surjections $C\to A$, i.e. those, for which all classes in $E(A,B)$ can be obtained from pairs of asymptotic homomorphisms from $C$.

Note that in topological applications, $A$ is often a group $C^{*}$-algebra of a discrete group $G$, or its augmentation ideal. Our interest in representing $E$-theory classes by maps from $A$ instead of suspensions over $A$ is caused by our hope that group actions may give more direct constructions of these classes.

Special cases $A=\mathbb{C}$ or $B=\mathbb{K}$ (the $C^{*}$-algebra of compact operators) were considered earlier in [9] and in [10] respectively.

Similar approach can be used in $KK$-theory. We shall describe it in a separate paper.

1. Asymptotic $KK$-cycles

Let $A$, $B$ be $C^{*}$-algebras, $A$ separable, $B$ stable and $\sigma$-unital, $MB$ the multiplier algebra for $B$. Let $C_{b}(\mathbb{R}_{+};MB)$ (resp. $C_{b,s}(\mathbb{R}_{+};MB)$) denote the algebras of $MB$-valued functions on $\mathbb{R}_{+}=[0,\infty)$ continuous with respect to the norm (resp. to the strict) topology. For $t\in\mathbb{R}_{+}$, $\operatorname{ev}_{t}$ denotes the evaluation at $t$.

Let $C$ be a $C^{*}$-algebra (usually $C$ is either $C_{b,s}(\mathbb{R}_{+};MB)$ or $C_{b}(\mathbb{R}_{+};B)$). Let $(\Phi^{+},\Phi^{-})$ be a pair of (not necessarily linear) continuous maps $\Phi^{\pm}:A\to C$. We denote by $E=E(\Phi^{+},\Phi^{-})=C^{*}(\Phi^{+}(A),\Phi^{-}(A))\subset C$ the $C^{*}$-algebra generated by all $\Phi^{\pm}(a)$, $a\in A$, and by $D=D(\Phi^{+},\Phi^{-})\subset E$ the ideal, in $E$, generated by $\Phi^{+}(a)-\Phi^{-}(a)$, $a\in A$.

For $x,y\in C_{b,s}(\mathbb{R}_{+};MB)$ we write $x\sim y$ if $x-y\in C_{0}(\mathbb{R}_{+};MB)$, i.e. if $\lim_{t\to\infty}x_{t}-y_{t}=0$.

Definition 1.1**.**

A pair $(\Phi^{+},\Phi^{-})$ of continuous maps $\Phi^{\pm}:A\to C_{b}(\mathbb{R}_{+};B)$ is a compact asymptotic $KK$-cycle from $A$ to $B$ if

• (a1)

$\Phi^{\pm}$ are $\mathbb{C}$-homogeneous and asymptotically involutive, i.e. $\Phi^{\pm}(\lambda a)=\lambda\Phi^{\pm}(a)$ for any $\lambda\in\mathbb{C}$ and for any $a\in A$, and $\Phi^{\pm}(a^{*})-\Phi^{\pm}(a)^{*}\in C_{0}(\mathbb{R}_{+};B)$ for any $a\in A$;

• (a2)

$(\Phi^{\pm}(a+b)-\Phi^{\pm}(a)-\Phi^{\pm}(b))x\in C_{0}(\mathbb{R}_{+};B)$ and $(\Phi^{\pm}(ab)-\Phi^{\pm}(a)\Phi^{\pm}(b))x\in C_{0}(\mathbb{R}_{+};B)$ for any $a,b\in A$ and for any $x\in D$.

Let $M^{\pm}(a,b)$ denote either $\Phi^{\pm}(a+b)-\Phi^{\pm}(a)-\Phi^{\pm}(b)$ or $\Phi^{\pm}(ab)-\Phi^{\pm}(a)\Phi^{\pm}(b)$. Note that (a2) is equivalent to

[TABLE]

or, equivalently,

[TABLE]

for any $a,b,c_{1},\ldots,c_{n},d\in A$ and any $n=0,1,2,\ldots$.

Definition 1.2**.**

A pair $(\Phi^{+},\Phi^{-})$ of continuous maps $\Phi^{\pm}:A\to C_{b}(\mathbb{R}_{+};B)$ has the same deficiency from being a homomorphism if

• (a1)

$\Phi^{\pm}$ are $\mathbb{C}$-homogeneous and asymptotically involutive;

• (a3)

$M^{+}(a,b)\Phi^{+}(c_{1})\cdots\Phi^{+}(c_{n})-M^{-}(a,b)\Phi^{-}(c_{1})\cdots\Phi^{-}(c_{n})\in C_{0}(\mathbb{R}_{+};B)$

for any $n=0,1,2,\ldots$ and any $a,b,c_{1},\ldots,c_{n}\in A$.

Proposition 1.3**.**

Definitions 1.1 and 1.2 are equivalent.

Proof.

We write $x\sim y$ if $x-y\in C_{0}(\mathbb{R}_{+};B)$. It follows from

[TABLE]

and

[TABLE]

that $M^{\pm}(a,b)(\Phi^{+}(c)-\Phi^{-}(c))\sim 0$. Similarly,

[TABLE]

and

[TABLE]

imply $M^{\pm}(a,b)\Phi^{\pm}(c_{1})\cdots\Phi^{\pm}(c_{n})(\Phi^{+}(c_{n+1})-\Phi^{-}(c_{n+1}))\sim 0$. Any $x\in D$ can be approximated by finite sums of products $\Phi^{+}(c_{1})\cdots\Phi^{+}(c_{n})(\Phi^{+}(c_{n+1})-\Phi^{-}(c_{n+1}))e$, with $e\in E$, hence (a2) follows from (a3).

To prove the opposite, note that

[TABLE]

hence $M^{+}(a,b)-M^{-}(a,b)\in D$. Passing to adjoints, $M^{+}(a,b)^{*}-M^{-}(a,b)^{*}\in D$. For shortness’ sake set $m_{\pm}=M^{\pm}(a,b)$. Since $m_{\pm}x\sim 0$ for $x=m_{+}^{*}-m_{-}^{*}\in D$, we have $(m_{+}-m_{-})(m_{+}-m_{-})^{*}\sim 0$. Then, by the $C^{*}$-property of the norm, we conclude that $m_{+}\sim m_{-}$, i.e. that $M^{+}(a,b)\sim M^{-}(a,b)$. Now, it is easy to check that (a3) holds for any $n$.

∎

Remark 1.4*.*

It suffices to check (a3) in Definition 1.2 only for $n=0$ and $n=1$:

[TABLE]

[TABLE]

Definition 1.5**.**

A pair $(\Phi^{+},\Phi^{-})$ of continuous maps $\Phi^{\pm}:A\to C_{b,s}(\mathbb{R}_{+};MB)$ is an asymptotic $KK$-cycle from $A$ to $B$ if

• (a1)

$\Phi^{\pm}$ are $\mathbb{C}$-homogeneous and asymptotically involutive;

• (b2)

$D\subset C_{b}(\mathbb{R}_{+};B)$;

• (b3)

$(\Phi(a+b)-\Phi(a)-\Phi(b))x\in C_{0}(\mathbb{R}_{+};B)$ and $(\Phi(ab)-\Phi(a)\Phi(b))x\in C_{0}(\mathbb{R}_{+};B)$ for any $a,b\in A$ and for any $x\in D$.

Two (compact) asymptotic $KK$-cycles $(\Phi^{+}_{0},\Phi^{-}_{0})$ and $(\Phi^{+}_{1},\Phi^{-}_{1})$ from $A$ to $B$ are homotopic if there is a (compact) asymptotic $KK$-cycle $(\Psi^{+},\Psi^{-})$ from $A$ to $IB=C([0,1];B)$ such that the obvious evaluation maps at 0 and at 1 give $(\Phi^{+}_{0},\Phi^{-}_{0})$ and $(\Phi^{+}_{1},\Phi^{-}_{1})$.

Lemma 1.6**.**

Let $\Psi^{\pm}_{t,\tau}:A\to MB$, $(t,\tau)\in\mathbb{R}_{+}\times[0,1]$, be a family of continuous maps. The following properties are equivalent:

1. (1)

The pair of maps $\Psi^{\pm}:A\times\mathbb{R}_{+}\times[0,1]\to MB$ is an asymptotic $KK$-cycle from $A$ to $IB$; 2. (2)

The maps $\Psi^{\pm}_{t,\tau}$ satisfy:

• (h1)

$\Psi^{\pm}_{t,\tau}$* are homogeneous and $\Psi^{\pm}_{t,\tau}(a^{*})-\Psi^{\pm}_{t,\tau}(a)\in C_{0}(\mathbb{R}_{+};IB)=C_{0}(\mathbb{R}_{+}\times[0,1];B)$ for any $a\in A$;*

• (h2)

$\Psi^{\pm}_{t,\tau}(a)\in C_{b,s}(\mathbb{R}_{+};M(IB))$* for any $a\in A$;*

• (h3)

$\Psi^{+}_{t,\tau}(a)-\Psi^{-}_{t,\tau}(a)\in C_{b}(\mathbb{R}_{+};IB)$* for any $a\in A$.*

• (h4)

$M^{\pm}_{t,\tau}(a,b)\Phi^{+}_{t,\tau}(c_{1})\cdots\Phi^{+}_{t,\tau}(c_{n})(\Phi^{+}_{t,\tau}(d)-\Phi^{-}_{t,\tau}(d))\in C_{0}(\mathbb{R}_{+};IB)=C_{0}(\mathbb{R}_{+}\times[0,1];B)$* for any $a,b,c_{1},\ldots,c_{n},d\in A$;*

∎

Denote the homotopy class of a (compact) asymptotic $KK$-cycle $(\Phi^{+},\Phi^{-})$ by $[(\Phi^{+},\Phi^{-})]$, and denote the set of all homotopy classes of (compact) asymptotic $KK$-cycles from $A$ to $B$ by $EL(A,B)$ (resp. by $EM(A,B)$).

Stability of $B$ allows to define addition on $EL(A,B)$ and on $EM(A,B)$: let $s_{1},s_{2}\in MB$ be isometries with $s_{1}s_{1}^{*}+s_{2}s_{2}^{*}=1$, and let $\tilde{s}_{1},\tilde{s}_{2}\in C_{b,s}(\mathbb{R}_{+};MB)$ be the constant functions with values $s_{1}$ and $s_{2}$ respectively. Let $(\Phi^{+},\Phi^{-})$, $(\Psi^{+},\Psi^{-})$ be two (compact) asymptotic $KK$-cycles. Set

[TABLE]

It is easy to see that the pair in the right-hand side is an asymptotic $KK$-cycle, and that the addition is well defined.

Lemma 1.7**.**

An asymptotic $KK$-cycle $(\Phi^{+},\Phi^{-})$ with $\Phi^{+}=\Phi^{-}$ represents the zero element in $EL(A,B)$.

Proof.

Let $\Phi=\Phi^{+}=\Phi^{-}$. Note that the ideal $D$ generated by $\Phi^{+}(a)-\Phi^{-}(a)$, $a\in A$, is zero, hence $(\tau\Phi,\tau\Phi)$ is an asymptotic $KK$-cycle for any $\tau\in[0,1]$, so $[(\Phi,\Phi)]=[(0,0)]$.

Let $(\Psi^{+},\Psi^{-})$ be an asymptotic $KK$-cycle. There are strictly continuous paths $s_{1}(\tau)$, $s_{2}(\tau)$, $\tau\in[0,1]$, such that, for $\tau\in(0,1]$, $s_{1}(\tau)$ and $s_{2}(\tau)$ are isometries with $s_{1}(\tau)s_{1}(\tau)^{*}+s_{2}(\tau)s_{2}(\tau)^{*}=1$, and $s_{1}(0)=1$, $s_{2}(0)=0$. Set

[TABLE]

This is an asymptotic $KK$-cycle from $A$ to $IB$ ((h1)-(h4) trivially hold), hence $[(\Psi^{+},\Psi^{-})]=[(\Psi^{+},\Psi^{-})]+[(0,0)]$.

∎

The standard rotation argument shows that

[TABLE]

and that

[TABLE]

in $EL(A,B)$, hence $EL(A,B)$ is an abelian group.

Similar results hold for compact asymptotic $KK$-cycles.

2. Making asymptotic $KK$-cycles compact

There is a canonical map $j:EM(A,B)\to EL(A,B)$ induced by the inclusion $B\subset MB$. Here we construct a map in the opposite direction and show that these two maps are inverse to each other.

Let $(\Phi^{+},\Phi^{-})$ be an asymptotic $KK$-cycle, $[(\Phi^{+},\Phi^{-})]\in EL(A,B)$. Recall that $D\subset C_{b}(\mathbb{R}_{+};B)$ denotes the ideal generated by $\Phi^{+}(a)-\Phi^{-}(a)$, $a\in A$, in the $C^{*}$-algebra $E$ generated by $\Phi^{\pm}(A)$. Let $(v_{r})_{r\in\mathbb{R}_{+}}$ be an approximate unit in $D$, quasicentral in $E$.

Lemma 2.1**.**

Let $\varphi:A\to E$ be a continuous map. There exists an order-preserving homeomorphism $r$ of $[0,\infty)$ (a reparametrization) such that

• (r1)

$\lim_{t\to\infty}[v_{r(t)},\varphi_{t}(a)]=0$* for any $a\in A$;*

• (r2)

$\lim_{t\to\infty}v_{r(t)}x_{t}-x_{t}=0$* for any $x\in D$.*

Proof.

As $v_{r}$ is an approximate unit for $D$, we have

[TABLE]

for any $x\in D$. Therefore, any function $r=r(t)$ satisfies (r2) if $\lim_{t\to\infty}r(t)=\infty$.

Consider a dense sequence $a_{1},a_{2},\ldots\in A$ and positive numbers $\varepsilon_{1},\varepsilon_{2},\ldots$ such that $\lim_{n\to\infty}\varepsilon_{n}=0$. Inductively, for $r=n\in\mathbb{N}$ find $t_{n}$ such that $t_{n}\geq t_{n-1}+1$ and $\|[v_{n}(t),\varphi_{t}(a_{i})]\|<\varepsilon_{n}$ for $i=1,2,\ldots,n$ when $t\geq t_{n}$. Extend the map $n\mapsto t_{n}$ to a homeomorphism of $[0,\infty)$ and take its inverse as $r$.

∎

As a bonus, we have $[v,\varphi(a)]\in C_{0}(\mathbb{R}_{+};B)$ (i.e. $[v,\varphi(a)]$ is norm-continuous) for any $a\in A$.

Set $v(t)=v_{r(t)}(t)$, $v\in C_{b}(\mathbb{R}_{+};B)$, and $\Psi^{\pm}(a)=v\Phi^{\pm}(a)\in C_{b}(\mathbb{R}_{+};B)$. Note that (a1) holds, i.e. that $(v\Phi^{\pm}(a))^{*}-v\Phi^{\pm}(a^{*})\sim 0$.

Then

[TABLE]

[TABLE]

[TABLE]

Properties of $v$ imply that the both terms here lie in $C_{0}(\mathbb{R}_{+};B)$.

Similarly, one can show that

[TABLE]

for any $a,b,c,d\in A$, where $M^{\pm}(a,b)$ is either $\Psi^{+}(a+b)-\Psi^{\pm}(a)-\Psi^{\pm}(b)$ or $\Psi^{+}(ab)-\Psi^{\pm}(a)\Psi^{\pm}(b)$, hence $(\Psi^{+},\Psi^{-})$ is a compact asymptotic $KK$-cycle.

Two different choices of an approximate unit $v$ and of a reparametrization give rise to two homotopic compact asymptotic $KK$-cycles. Also, if $(\Phi^{+},\Phi^{-})$ and $(\Xi^{+},\Xi^{-})$ are two homotopic asymptotic $KK$-cycles then one can find an approximate unit $v$ and a reparametrization related to this homotopy, so that $v^{0}=\operatorname{ev}_{0}v$, $v^{1}=\operatorname{ev}_{1}v$ are approximate units for the given two asymptotic $KK$-cycles, and $(v^{0}\Phi^{+},v^{0}\Phi^{-})$ and $(v^{1}\Xi^{+},v^{1}\Xi^{-})$ are homotopic. Thus this construction defines a map

[TABLE]

Theorem 2.2**.**

The map $p$ is an isomorphism.

Claim 1. $p\circ j$ is the identity map on $EM(A,B)$.

Let $(\Psi^{+},\Psi^{-})$ be a compact asymptotic $KK$-cycle. Consider it as an asymptotic $KK$-cycle, and find an appropriate approximate unit $v$ such that $[(v\Psi^{+},v\Psi^{-})]=p([(\Psi^{+},\Psi^{-})])$. Since $\Psi^{\pm}(A)\subset C_{b}(\mathbb{R}_{+};B)$, there exists $w\in C_{b}(\mathbb{R}_{+};B)$ such that $0\leq w\leq 1$ and $(1-w)\Psi^{\pm}(a)\sim 0$ for any $a\in A$. Then $[(w\Psi^{+},w\Psi^{-})]=[(\Psi^{+},\Psi^{-})]$. Connecting $v$ and $w$ by the linear homotopy, we conclude that $p([(\Psi^{+},\Psi^{-})])=[(\Psi^{+},\Psi^{-})]$. ∎

Claim 2. $j\circ p$ is the identity map on $EL(A,B)$.

Let $(\Phi^{+},\Phi^{-})$ be an asymptotic $KK$-cycle. Consider the $C^{*}$-algebra $\mathcal{E}$ generated by $\Phi^{\pm}(A)$ and by constant $B$-valued functions on $\mathbb{R}_{+}$, and let $\mathcal{D}\subset\mathcal{E}$ be the ideal generated by $\Phi^{+}(a)-\Phi^{-}(a)$, $a\in A$, and by constant $B$-valued functions on $\mathbb{R}_{+}$. Let $(w_{r})_{r\in\mathbb{R}_{+}}$ be an approximate unit in $\mathcal{D}$, quasicentral in $\mathcal{E}$. Set $w(t)=w_{r(t)}(t)$, $t\in\mathbb{R}_{+}$, $w\in C_{b}(\mathbb{R}_{+};B)$, where $r$ satisfies $[w_{r(t)},\Phi^{\pm}_{t}(a)]\sim 0$ for any $a\in A$. Then the linear homotopy connecting $v$ and $w$ shows that the asymptotic $KK$-cycles $(v\Phi^{+},v\Phi^{-})$ and $(w\Phi^{+},w\Phi^{-})$ are homotopic. Note that $w\Phi^{\pm}(a)\sim w^{1/2}\Phi^{\pm}(a)w^{1/2}$.

Lemma 2.3**.**

Let $(\Psi^{+}_{t},\Psi^{-}_{t})$ be an asymptotic $KK$-cycle from $A$ to $B$, $U_{t}\in M(B)$ a norm-continuous family of unitaries, $t\in\mathbb{R}_{+}$. Then $(U_{t}\Psi^{+}_{t}U^{*}_{t},U_{t}\Psi^{-}_{t}U^{*}_{t})$ is an asymptotic $KK$-cocycle homotopic to $(\Psi^{+}_{t},\Psi^{-}_{t})$.

Proof.

Note that $(U_{\tau t}\Psi^{+}_{t}U^{*}_{\tau t},U_{\tau t}\Psi^{-}_{t}U^{*}_{\tau t})$ is an asymptotic $KK$-cycle from $A$ to $IB$ as $U_{\tau t}\in C_{b}(\mathbb{R}_{+};M(IB))$. Then use contractibility of the unitary group of $M(B)$.

∎

Set

[TABLE]

$V$ is an isometry, and there is a norm-continuous family $U_{t}$ of unitaries such that $V\Phi^{\pm}V^{*}\oplus 0=U(\Phi^{\pm}\oplus 0)U^{*}$, hence $[(\Phi^{+},\Phi^{-})]=[(V\Phi^{+}V^{*},V\Phi^{-}V^{*})]$. Write $V\Phi^{\pm}V^{*}$ as a matrix:

[TABLE]

Note that

[TABLE]

as $wx\sim x$ for any $x\in D$.

For $\tau\in[0,1]$ set $A^{\pm}(\tau)=\left(\begin{smallmatrix}1&0\\ 0&\tau\end{smallmatrix}\right)V\Phi^{\pm}(a)V^{*}\left(\begin{smallmatrix}1&0\\ 0&\tau\end{smallmatrix}\right)$.

Lemma 2.4**.**

This is a homotopy connecting $(V\Phi^{+}V^{*},V\Phi^{-}V^{*})$ with $(w^{1/2}\Phi^{+}w^{1/2},w^{1/2}\Phi^{-}w^{1/2})$.

Proof.

Let $a,b\in A$, $c=ab$, and let $B^{\pm}=V\Phi^{\pm}(b)V^{*}$, $C^{\pm}=V\Phi^{\pm}(c)V^{*}$. As $(\Phi^{+},\Phi^{-})$ is an asymptotic $KK$-cycle, we have $C^{+}-A^{+}B^{+}\sim C^{-}-A^{-}B^{-}$, which is equivalent to $C^{+}-C^{-}=(A^{+}-A^{-})B^{+}+A^{-}(B^{+}-B^{-})$. Looking at the upper left corner and taking into account (2.1), we obtain $c^{+}_{11}-c^{-}_{11}\sim(a^{+}_{11}-a^{-}_{11})b^{+}_{11}+a^{-}_{11}(b^{+}_{11}-b^{-}_{11})$, whence

[TABLE]

Replacing here $A,B,C$ by $A(\tau),B(\tau),C(\tau)$, we see that (2.2) does not depend on $\tau$.

Let $d\in A$, $D^{\pm}=V\Phi^{\pm}(d)V^{*}$. We know that $(C^{+}(\tau)-A^{+}(\tau)B^{+}(\tau))(D^{+}(\tau)-D^{-}(\tau))\sim 0$ when $\tau=1$. Note that $D^{+}(\tau)-D^{-}(\tau)\sim\left(\begin{matrix}d^{+}_{11}-d^{-}_{11}&0\\ 0&0\end{matrix}\right)$,

[TABLE]

As $(1-w)^{1/2}(d^{+}_{11}-d^{-}_{11})\sim 0$, so $a^{+}_{ij}b^{+}_{kl}(d^{+}_{11}-d^{-}_{11})\sim 0$ for all $(ijkl)\neq(1111)$, therefore, $(D^{+}(\tau)-D^{-}(\tau))(C^{+}(\tau)-A^{+}(\tau)B^{+}(\tau))\sim 0$ for any $\tau$, and uniformly in $\tau$. This proves (h4). (h2) and (h3) are obvious.

∎

Remark 2.5*.*

We did not use in Lemma 2.4 that $w\in C_{b}(\mathbb{R}_{+};B)$. It suffices to require that $w\in C_{b}(\mathbb{R}_{+};MB))$, $w\Phi^{\pm}(a)\in C_{b}(\mathbb{R}_{+};B)$, $[w,\Phi^{\pm}(a)]\sim 0$ and $w(\Phi^{+}(a)-\Phi^{-}(a))\sim\Phi^{+}(a)-\Phi^{-}(a)$ for any $a\in A$.

3. A map $EL(A,B)\to E(A,B)$

Let $e_{ij}$, $i,j\in\mathbb{Z}$, denote the standard matrix units for $M(B\otimes\mathbb{K})$. For an asymptotic $KK$-cycle $(\Phi^{+},\Phi^{-})$ set

[TABLE]

Let $\widetilde{E}$ be the $C^{*}$-algebra generated by $\{\Phi(a):a\in A\}$ and by $T$, $\widetilde{D}=D\otimes\mathbb{K}$, where $D$ is the ideal in $E$ generated by $\Phi^{+}(a)-\Phi^{-}(a)$, $a\in A$, and $E$ is the $C^{*}$-algebra generated by $\Phi^{\pm}(A)$. Then $\widetilde{D}\subset\widetilde{E}\subset C_{b,s}(\mathbb{R}_{+};MB)$, and $\widetilde{D}$ is an ideal in $\widetilde{E}$.

Let $(v_{r})_{r\in\mathbb{R}_{+}}$ be an approximate unit in $\widetilde{D}$, quasicentral in $\widetilde{E}$.

Let $\Phi_{t}=\operatorname{ev}_{t}\circ\Phi$, $v_{r,t}=\operatorname{ev}_{t}(v_{r})$, $M_{t}(a,b)=\operatorname{ev}_{t}(M(a,b))$, where $M(a,b)$ is either $\Phi(a+b)-\Phi(a)-\Phi(b)$ or $\Phi(ab)-\Phi(a)\Phi(b)$.

Lemma 3.1**.**

There exists a reparametrization $r$ such that

• (v1)

$v\in C_{b}(\mathbb{R}_{+};B\otimes\mathbb{K})$;

• (v2)

$f(v)x-x,xf(v)-x\in C_{0}(\mathbb{R}_{+};B\otimes\mathbb{K})$* for any $x\in\widetilde{D}$ and for any $f\in C_{0}(0,1]$;*

• (v3)

$f(v)y-yf(v)\in C_{0}(\mathbb{R}_{+};B\otimes\mathbb{K})$* for any $y\in\widetilde{E}$ and for any $f\in C[0,1]$;*

• (v4)

$f(v)M(a,b)\in C_{0}(\mathbb{R}_{+};B\otimes\mathbb{K})$* for any $a,b\in A$ and any $f\in C_{0}(0,1]$.*

Proof.

Items (v1)-(v2) hold for any reparametrization. We have already shown how to satisfy (v3) in Lemma 2.1, where a sequence $t_{1},t_{2},\ldots$ was constructed. To satisfy (v4), note that $M(a,b)v_{r}\sim 0$ for any $a,b\in A$ and any $r\in\mathbb{R}_{+}$. As $A$ is separable, fix a dense sequence $(c_{n})_{n\in\mathbb{N}}\subset A$. For each $n\in\mathbb{N}$ find $T_{n}\in\mathbb{R}_{+}$ such that $T_{n}\geq t_{n}$ and

[TABLE]

for $a,b\in\{c_{1},\ldots,c_{n}\}$ and for any $r\in[0,n]$. Then assigning $T_{n}$ to $n$, we get a reparametrization $r=r(t)$ such that $r(T_{n})=n$.

∎

For $a\in A$, $f\in C_{0}(0,1)$, $h_{n}(s)=e^{2\pi ins}$, set

[TABLE]

It follows from (v2)-(v4) that

[TABLE]

for any $f,g\in C_{0}(0,1)$, any $a,b\in A$ and any $n,m\in\mathbb{N}$, hence (3.1) determines an asymptotic homomorphism $C_{0}(0,1)\otimes C(\mathbb{S}^{1})\otimes A\to C_{b}(\mathbb{R}_{+};B\otimes\mathbb{K})$, which can be restricted to $C_{0}((0,1)^{2})\otimes A$. We denote this asymptotic homomorphism by $\alpha(\Phi^{+},\Phi^{-})$, $\alpha(\Phi^{+},\Phi^{-})_{t}:S^{2}A\to C_{b}(\mathbb{R}_{+};B\otimes\mathbb{K})$.

It is clear that the map $\alpha:EL(A,B)\to E(A,B\otimes\mathbb{K})$ doesn’t depend on the choices of the approximate unit and of the approximation, as any two such choices can be connected by a linear homotopy.

4. A map $E(A,B)\to EL(A,B)$

Let $\varphi:S^{2}A\to C_{b}(\mathbb{R}_{+};B)$ be a continuous map such that $(\varphi_{t})_{t\in\mathbb{R}_{+}}:S^{2}A\to B$ is an asymptotic homomorphism, where $\varphi_{t}=\operatorname{ev}\circ\varphi$. We identify here $S^{2}A$ with $C_{0}(\mathbb{D};A)$, where $\mathbb{D}$ is the open unit ball on the complex plane.

Let $(\alpha_{i})_{i=0,1,2,\ldots}$ be a family of functions in $C_{0}[0,1)$ such that

• ($\alpha$1)

$\operatorname{supp}\alpha_{0}=[0,\frac{1}{2}]$ and $\alpha_{0}(0)=1$;

• ($\alpha$2)

$\operatorname{supp}\alpha_{i}=[\frac{i-1}{i},\frac{i+1}{i+2}]$, $i=1,2,\ldots$;

• ($\alpha$3)

$\alpha_{i}\geq 0$ for any $i=0,1,2,\ldots$;

• ($\alpha$4)

$\sum_{i=0}^{\infty}\alpha_{i}^{2}=1$ (convergence here is pointwise).

Let $f:[0,1]\to[0,1]$ satisfy $f(1-t)=1-f(t)$ and $f(t)=0$ for $t\in[0,\frac{1}{10}]$, and let $g=\sqrt{f-f^{2}}$. Set

[TABLE]

where $(r,\phi)$ are the polar coordinates on $\mathbb{D}$. Note that $p_{\pm}\alpha_{i}\in M_{2}(C_{0}(\mathbb{D}))$ for any $i=0,1,2,\ldots$.

Set

[TABLE]

where $\epsilon_{ij}$, $i,j=0,1,2,\ldots$, are the matrix units in $M(B\otimes\mathbb{K})$ (later on, we shall use two copies of $\mathbb{K}$, so we have to use two different sets of matrix units — $e_{ij}$ and $\epsilon_{ij}$).

Then the maps $\Phi^{\pm}_{t}$ are well-defined (the sum is strictly convergent), but are not necessarily asymptotic homomorphisms (because algebraic properties of $(\varphi_{t})$ hold asymptotically in a non-uniform way).

Lemma 4.1**.**

$(\Phi^{+},\Phi^{-})$* is an asymptotic $KK$-cycle.*

Proof.

(a1) is trivial. As $p_{+}\alpha_{i}=p_{-}\alpha_{i}$ for $i\geq 10$, so (b2) holds. To check (b3), let $P_{n}=\sum_{i=0}^{n}e_{ii}$. Then, as $p_{\pm}$ are projections, so, for any $n\in\mathbb{N}$ and for any $a,b\in A$, one has

[TABLE]

where $M^{\pm}_{t}(a,b)$ is either $\Phi^{\pm}_{t}(a+b)-\Phi^{\pm}_{t}(a)-\Phi^{\pm}_{t}(b)$ or $\Phi^{\pm}_{t}(ab)-\Phi^{\pm}_{t}(a)\Phi^{\pm}_{t}(b)$. As $p_{+}(r,\phi)=p_{-}(r,\phi)$ for $r\geq\frac{9}{10}$, so

[TABLE]

for any $a\in A$. Note that $\alpha_{i}\alpha_{j}=0$ when $|i-j|\geq 2$, so $\Phi^{\pm}(a)$ (4.1) has tri-diagonal form, therefore, combining (4.2) and (4.3), we get

[TABLE]

for any $a,b,c_{1},\ldots,c_{n},d\in A$.

∎

This construction doesn’t depend, up to homotopy, on the choice of the functions $\alpha_{i}$, $i=0,1,2,\ldots$, $f$, and on the choice of $\varphi$ within its class $[\varphi]\in E(A,B)$, so this gives us a map $\beta(\varphi)=(\Phi^{+},\Phi^{-})$, $\beta:E(A,B)\to EL(A,B\otimes\mathbb{K})$.

5. Isomorphism $EL(A,B)\cong E(A,B)$

Theorem 5.1**.**

The maps $\alpha$ and $\beta$ are isomorphisms between $E(A,B)$ and $EL(A,B)$.

Proof.

Claim 1. $\beta\circ\alpha:EL(A,B)\to EL(A,B\otimes\mathbb{K}\otimes\mathbb{K}\otimes M_{2})$ coincides with the isomorphism induced by an isomorphism $B\cong B\otimes\mathbb{K}\otimes\mathbb{K}\otimes M_{2}$.

Set

[TABLE]

First step of homotopy.

Let $w_{r(t)}=\sum_{i=0}^{\infty}\lambda_{i}(r(t))\epsilon_{ii}$, where $r$ is a reparametrization (i.e. an order-preserving homeomorphism of $[0,\infty)$), and $\lambda_{i}$, $i=0,1,\ldots$, are continuous functions such that

• ($\lambda$1)

$0\leq\ldots\leq\lambda_{i}(t)\leq\lambda_{i+1}(t)\leq\ldots\leq 1$ for any $t\in[0,\infty)$ and any $i=0,1,\ldots$;

• ($\lambda$2)

for any $t\in[0,\infty)$ there is $N\in\mathbb{N}$ such that $\lambda_{i}(t)=0$ for any $i\geq N$;

• ($\lambda$3)

for any $i=0,1,\ldots$ $\lim_{t\to\infty}\lambda_{i}(t)=1$;

• ($\lambda$4)

$\lim_{t\to\infty}\lambda_{i+1}(t)-\lambda_{i}(t)=0$ for any $i=0,1,\ldots$;

Note that tri-diagonal structure of $\Xi^{\pm}$ and ($\lambda$4) imply that

[TABLE]

for any $a\in A$ for an appropriately chosen reparametrization. In fact, any sufficiently slow reparametrization would do. We shall fix it later.

By ($\lambda$2), $w_{r}(t)\Xi^{\pm}_{t}(a)$ is norm-continuous in $t$ ($\Xi^{\pm}$ is only strictly continuous).

Tri-diagonality of $\Xi^{\pm}$ and ($\lambda$4) imply that

[TABLE]

for any $a\in A$, and ($\lambda$3) implies that

[TABLE]

for any $a\in A$. All these hold for any reparametrization $r$. Therefore, there exists a homotopy connecting $(\Xi^{+},\Xi^{-})$ with $(w\Xi^{+},w\Xi^{-})$ (and with $(w\Xi^{+}w,w\Xi^{-}w)$) as in Theorem 2.2 (see Lemma 2.4 and Remark after it).

Set $u=1-v$, $u=(u_{t})_{t\in\mathbb{R}_{+}}$, where an approximate unit $(v_{r})_{r\in\mathbb{R}_{+}}$ and a reparametrization $r$ are as in Lemma 3.1; set

[TABLE]

and set

[TABLE]

Note that

[TABLE]

for any $i,j\in\mathbb{N}$ and for any $a\in A$, and

[TABLE]

if $|i-j|>1$, so we have

[TABLE]

and the asymptotic $KK$-cycles $(\Xi^{+},\Xi^{-})$ and $(\Psi^{+},\Psi^{-})$ are homotopic.

Second step of homotopy.

Set $\gamma(\tau)=\sqrt{(1-\tau)+\tau f(u)}$, $\delta(\tau)=\sqrt{1-\gamma^{2}(\tau)}$, $\tau\in[0,1]$; $s=\gamma(1)$, $c=\delta(1)$. Note that $\gamma(0)=1$, $\delta(0)=0$. Set

[TABLE]

[TABLE]

Now, let us fix a reparametrization.

Lemma 5.2**.**

Let $\varphi:A\to C_{b,s}(\mathbb{R}_{+};MB)$ be a continuous map, let $v\in C_{b}(\mathbb{R}_{+};MB)$ satisfy $0\leq v\leq 1$ and

[TABLE]

*for any $a\in A$. Let $\beta_{i}$, $i=0,1,\ldots$, be the functions defined by $\beta_{i}(x)=\alpha_{i}(1-x)$, $x\in[0,1]$.

1. There exists a reparametrization $r$ such that*

[TABLE]

for any $a\in A$, uniformly in $\tau$; 2) For any reparametrization $r$ and for any $i=0,1,\ldots$ one has

[TABLE]

Proof.

1. Let $a_{1},a_{2},\ldots$ be a dense sequence in $A$, $\varepsilon_{1}>\varepsilon_{2}>\ldots>0$, $\lim_{j\to\infty}\varepsilon_{j}=0$. Set $g_{\rho}(x)=\sum_{i=0}^{\infty}\beta_{i}^{2}(x)\lambda_{i}^{2}(\rho)$, $x\in[0,1]$, $\rho\in[0,\infty)$. We may think of $\varphi(a)$ and the function $\tilde{v}$ given by $\tau\mapsto v(\tau)=\tau v+(1-\tau)1$ as elements of $C_{b}(\mathbb{R}_{+};MB)\otimes C[0,1]$. Let $q:C_{b}(\mathbb{R}_{+};MB)\otimes C[0,1]\to C_{b}(\mathbb{R}_{+};MB)/C_{0}(\mathbb{R}_{+};MB)\otimes C[0,1]$ be the quotient map. Then (5.4) implies that $q(\varphi(a))$ and $q(\tilde{v})$ commute. Then $q(\varphi(a))$ commutes with $q(g(\tilde{v}))=g(q(\tilde{v}))$ for any $g\in C[0,1]$, hence $\lim_{t\to\infty}[\varphi_{t}(a),g(v_{t}(\tau))]=0$ for any $a\in A$ uniformly in $\tau$.

Let $k_{1},k_{2},\ldots$ be a sequence of numbers, $k_{i}\geq 1$, $i\in\mathbb{N}$. Start with $\rho=1$ and find $t_{1}\in[0,\infty)$ such that $[g_{1}(v_{t}(\tau)),\varphi_{t}(a_{1})]<\varepsilon_{1}$ for any $t\geq t_{1}$ and any $\tau\in[0,1]$. Then take $\rho=2$ and find $t_{2}\in[0,\infty)$ such that $t_{2}\geq t_{1}+k_{1}$ and $[g_{2}(v_{t}(\tau)),\varphi_{t}(a_{i})]<\varepsilon_{2}$ for $i=1,2$, for any $t\geq t_{2}$ and for any $\tau\in[0,1]$. Proceeding infinitely, we obtain a map $\rho\mapsto t_{\rho}$ when $\rho\in\mathbb{N}$. Extend this map to a homeomorphism on $[0,\infty)$, and take its inverse as $r$.

2. $\beta_{i}(v_{t})\sum_{i=0}^{\infty}\beta_{i}^{2}(v_{t})\lambda_{i}^{2}(r(t))=\beta_{i}(v_{t})g_{r(t)}(v_{t})$, so the claim follows from the fact that $g_{r}$ is an approximate unit in $C_{0}(0,1]$.

∎

When finding a reparametrization, we should keep in mind that it should be sufficiently slow to satisfy also (5.2). This can be done by adjusting the numbers $k_{i}$, $i\in\mathbb{N}$.

Note that

[TABLE]

and

[TABLE]

uniformly in $\tau$, as $[T,\gamma(\tau)]\sim 0$, $[T,\delta(\tau)]\sim 0$ uniformly in $\tau\in[0,1]$ by (v3).

Lemma 5.3**.**

(5.3) is a homotopy of asymptotic $KK$-cycles.

Proof.

We have to check (h1)-(h4) from Lemma 1.6. (h1) is trivial. (h2) and (h3) follow from (5.5). To check (h4), set $M^{\tau}_{\pm}(a,b)=\Psi^{\pm}(ab)(\tau)-\Psi^{\pm}(a)(\tau)\Psi^{\pm}(b)(\tau)$, $M_{\pm}(a,b)=\Phi_{\pm}(ab)-\Phi_{\pm}(a)\Phi_{\pm}(b)$, $a,b\in A$. Let $c\in A$. By (5.6) and Lemma 5.2, using that $\lim_{t\to\infty}[\alpha_{i}(u_{t}),U_{\pm}(\tau)]=0$ uniformly in $\tau$ for any $i=0,1,\ldots$, $\lim_{t\to\infty}[\alpha_{i}(u_{t}),\Phi^{\pm}_{t}(a)]=0$, and $\lim_{t\to\infty}[p_{\pm}(u_{t},T),\Phi_{t}(a)]=0$ we have

[TABLE]

[TABLE]

Let us write the second multiplicand (5.7) as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Then it would suffice to show that $M_{+}(a,b)(p_{+}(u,T)-p_{-}(u,T))$, $M_{+}(a,b)p_{-}(u,T)\Phi(c)(U_{+}^{*}(\tau)-U_{-}^{*}(\tau))$ and $M_{+}(a,b)U_{+}^{*}(\tau)(U_{+}(\tau)-U_{-}(\tau))$ vanish as $t\to\infty$ uniformly in $\tau$.

Note that $p_{+}(u,T)-p_{-}(u,T)=\left(\begin{matrix}0&g(u)(T-1)\\ (T^{*}-1)g(u)&0\end{matrix}\right)$, where $g(0)=g(1)=0$, hence, by (v4) $M_{+}(a,b)g(u)\in C_{0}(\mathbb{R}_{+};B)$.

To deal with the two remaining terms, note that

[TABLE]

and that $\delta(\tau)=\sqrt{1-\gamma^{2}(\tau)}=\sqrt{\tau(1-f(u))}=\sqrt{\tau f(v)}$, so $M_{+}(a,b)\delta(\tau)\sim 0$ by (v4) uniformly in $\tau$.

As

[TABLE]

[TABLE]

uniformly in $\tau$ by (v3), so, by (v4) $M_{+}(a,b)p_{-}(u,T)(U_{+}^{*}(\tau)-U_{-}^{*}(\tau))\sim 0$ uniformly in $\tau$. By (v4),

[TABLE]

uniformly in $\tau$, so

[TABLE]

uniformly in $\tau$.

The last summand, $M_{+}(a,b)U_{+}^{*}(\tau)(U_{+}(\tau)-U_{-}(\tau))$, vanishes as $t\to\infty$ uniformly in $\tau$ by similar argument. So, $M^{\tau}_{+}(a,b)(\Psi^{+}(c)-\Psi^{-}(c))\sim 0$ uniformly in $\tau$. Similarly one can show that the same holds also for $M^{\tau}_{-}(a,b)(\Psi^{+}(c)-\Psi^{-}(c))$ and for $M^{\tau}_{\pm}(a,b)\Psi^{\pm}(c)(\Psi^{+}(d)-\Psi^{-}(d))$.

∎

Thus, we have a homotopy connecting $(\Psi^{+}(0),\Psi^{-}(0))$ with $(\Psi^{+}(1),\Psi^{-}(1))$. As $U_{\pm}(0)=\left(\begin{matrix}1&0\\ 0&1\end{matrix}\right)$, so $\Psi^{\pm}(0)=\Psi^{\pm}$. For $\tau=1$, $\gamma(1)=s=\sqrt{f(u)}$, $\delta(1)=c=\sqrt{1-f(u)}$, and both $s$ and $c$ asymptotically commute with $\Phi(a)$, i.e. $[s,\Phi(a)]\sim 0$, $[c,\Phi(a)]\sim 0$. Then

[TABLE]

As $s[T,\Phi(a)]\sim 0$, we can skip the off-diagonal elements. Similarly,

[TABLE]

hence the first summand also vanishes as $t\to\infty$, so

[TABLE]

therefore, $\Psi^{+}_{t}(a)(1)=w_{r(t)}\sum_{|i-j|\leq 1}\alpha_{i}(u_{t})\left(\begin{matrix}T\Phi(a)T^{*}&0\\ 0&0\end{matrix}\right)\alpha_{j}(u_{t})\epsilon_{ij}\ w_{r(t)}$.

Similarly, we have $\Psi^{-}_{t}(a)(1)=w_{r(t)}\sum_{|i-j|\leq 1}\alpha_{i}(u_{t})\left(\begin{matrix}\Phi_{t}(a)&0\\ 0&0\end{matrix}\right)\alpha_{j}(u_{t})\epsilon_{ij}\ w_{r(t)}$.

Third step of homotopy.

Set $u(\tau)=1-((1-\tau)v+\tau 1)$. This is the linear homotopy connecting $u=u(0)=1-v$ with [math].

Set

[TABLE]

[TABLE]

Norm-continuity of $\alpha_{i}(u_{t}(\tau))$ in $(t,\tau)$ for each $i=0,1,\ldots$ implies norm-continuity of $\Psi^{\pm}(a)$ in $\tau$ for each $t\in[0,\infty)$, and strict continuity in $t$ uniformly in $\tau$, so (h2) holds for $\Psi^{\pm}(a)$. (h3) follows from $\lim_{i\to\infty}\sup_{\tau\in[0,1]}\|\alpha_{i}(u(\tau)(T\Phi(a)T^{*}-\Phi(a))\|=0$. Finally, (h4) follows from Lemma 5.2, so $(\Psi^{+}(\tau),\Psi^{-}(\tau))$ is an asymptotic $KK$-cycle from $A$ to $IB\otimes\mathbb{K}^{2}$.

When $\tau=1$, $u_{t}(1)=0$, hence $\alpha_{0}(u_{t}(1))=1$, $\alpha_{i}(u_{t}(1))=0$ for any $i\geq 1$. As $\lambda_{0}(t)=1$, we get

[TABLE]

We can write $(\Lambda^{+},\Lambda^{-})=(T\Phi T^{*}\oplus 0,\Phi\oplus 0)=\left(\left(\begin{matrix}\Phi^{+}&0\\ 0&\Phi^{\prime}\end{matrix}\right),\left(\begin{matrix}\Phi^{-}&0\\ 0&\Phi^{\prime}\end{matrix}\right)\right)$, where $\Phi^{\prime}$ is the infinite direct sum of copies of $\Phi^{+}$, $\Phi^{-}$ and [math]. By Lemma 2.1, $[(\Lambda^{+},\Lambda^{-})]=[(\Phi^{+},\Phi^{-})]$. ∎

Claim 2. $\alpha\circ\beta:E(A,B)\to E(A,B\otimes\mathbb{K}\otimes\mathbb{K}\otimes M_{2})$ coincides with the isomorphism induced by an isomorphism $B\cong B\otimes\mathbb{K}\otimes\mathbb{K}\otimes M_{2}$.

Let $\varphi:S^{2}A\to C_{b}(\mathbb{R};B)$ represent an element of $E(A,B)$, and let $\Phi^{\pm}$ be defined as in (4.1). Denote $\alpha\circ\beta(\varphi)$ by $\psi:S^{2}A\to C_{b}(\mathbb{R}_{+};B)$. Then

[TABLE]

where $\Phi=\sum_{i\geq 0}\Phi^{+}e_{ii}+\sum_{i<0}\Phi^{-}e_{ii}$, $f\in C_{0}(0,1)$, $h(x)=e^{2\pi ix}$, and $v\in D\otimes\mathbb{K}\subset C_{b}(\mathbb{R}_{+};B)$ satisfies

• •

$\lim_{t\to\infty}f(v)d=0$ for any $d\in D\otimes\mathbb{K}$;

• •

$\lim_{t\to\infty}f(v)M(a,b)=0$ for any $a,b\in A$, where $M(a,b)$ is either $\Phi(ab)-\Phi(a)\Phi(b)$ or $\Phi(a+b)-\Phi(a)-\Phi(b)$.

Let $\phi^{\pm}(g\otimes a)=\sum_{i,j\geq 0}\varphi(\alpha_{i}\alpha_{j}g\otimes a)\epsilon_{ij}$, where $(\epsilon_{ij})_{i,j=0,1,2\ldots}$ is a set of matrix units for $M(B\otimes\mathbb{K})$ and $g\otimes a\in\Sigma^{2}A=C(\mathbb{S}^{2};A)$, and let $\phi^{\pm}_{\infty}(g\otimes a)=\sum_{i=0}^{\infty}\phi^{\pm}(g\otimes a)e_{ii}$. ($e_{ij}$ are the matrix units in $M(B\otimes\mathbb{K}\otimes\mathbb{K})$ with respect to the second copy of $\mathbb{K}$.) Set $W=\sum_{i=0}^{\infty}e_{i,i+1}$.

Let $E$ be the $C^{*}$-subalgebra of $C_{b,s}(\mathbb{R}_{+};M(B\otimes\mathbb{K}^{2}))$ generated by $\phi^{\pm}_{\infty}(\Sigma^{2}A)$ and by $W$, and let $D\subset E$ be the ideal generated by $\phi^{+}_{\infty}(g\otimes a)-\phi^{-}_{\infty}(g\otimes a)$, $g\otimes a\in\Sigma^{2}A$, and by $\phi^{+}(\Sigma^{2}A)e_{00}$. Let $(w_{r})_{r\in[0,\infty)}$ be a quasicentral aproximate unit in $D$. Without loss of generality, we may assume that $w_{r}$ is diagonal, $w_{r}=\sum_{i=0}^{\infty}w_{r;i}e_{ii}$. As before, we may find a reparametrization $r(t)$ such that, for any $f\in C_{0}(0,1)$ one has

• (w1)

$\lim_{t\to\infty}f(w_{r(t),t})d(t)=0$ for any $d\in D$;

• (w2)

$\lim_{t\to\infty}[f(w_{r(t),t}),e(t)]=0$ for any $e\in E$,

where $w_{r(t),t}=w_{r(t)}(t)$.

Set $u^{\prime}_{t}=1-\sum_{i\in\mathbb{Z}}w_{r(t),t;|i|}e_{ii}$, $u^{\prime}=\sum_{i\in\mathbb{Z}}u^{\prime}_{i}e_{ii}$, and

[TABLE]

It is easy to check that $\psi^{\prime}$ defines an element of $[[S^{2}A,B]]$, and the linear homotopy $\tau u+(1-\tau)u^{\prime}$ shows that $\psi^{\prime}$ is homotopic to $\psi$.

Let us write $\psi^{\prime}$ as a matrix with respect to the decomposition $1=\sum_{i=1}^{\infty}e_{ii}+e_{00}+\sum_{i=-1}^{-\infty}e_{ii}$,

[TABLE]

[TABLE]

[TABLE]

Set also

[TABLE]

[TABLE]

[TABLE]

Lemma 5.4**.**

$f(u^{\prime})T^{n}\Phi(a)-f(u^{\prime})_{0}T^{n}_{0}\Phi(a)_{0}\in C_{0}(\mathbb{R}_{+};B\otimes\mathbb{K}^{2})$.

Proof.

This follows from the definition of $u^{\prime}$ (see (w2)).

∎

Set $\psi^{\prime\prime}(f\otimes h_{n}\otimes a)=f(u^{\prime})_{0}T^{n}_{0}\Phi(a)_{0}$. Then $\psi^{\prime\prime}$ is homotopic to $\psi$, and can be written as a direct sum, $\psi^{\prime\prime}=\psi_{1}\oplus\psi_{2}\oplus 0$, where

[TABLE]

[TABLE]

$\tilde{u}=\sum_{i=1}^{\infty}u^{\prime}_{i}e_{ii}$. Therefore, $[\psi]=[\psi_{1}]+[\psi_{2}]=[\psi_{1}]-[\psi_{0}]$, where

[TABLE]

Let us recall the construction of the Bott isomorphism in $E$-theory. Let $u_{0}\in C_{b}(\mathbb{R}_{+};M(B\otimes\mathbb{K})$ be defined by $u_{0}=\sum_{i=1}^{\infty}a_{i}(t)e_{ii}$, where $t\in\mathbb{R}_{+}$ and $a_{i}(t)=\min\{\frac{t}{i},1\}$. For $g,g^{\prime}\in C_{0}(\mathbb{D}^{2})$, and for $\varphi:S^{2}A\to C_{b}(\mathbb{R}_{+};B)$, $[\varphi]\in[[S^{2}A,B]]$, set

[TABLE]

Consider the commuting diagram

[TABLE]

where $i^{*}$ is induced by the canonical inclusion $S^{2}A\subset\Sigma^{2}A$, $\mathcal{B}(\chi)(g\otimes a)=\chi(g\otimes p_{+}\otimes a)\oplus\chi(\overline{g}\otimes p_{-}\otimes a)$, where $g\in S^{2}\mathbb{C}$, $\overline{g}(re^{\phi})=g(re^{-\phi})$ in polar coordinates, $p_{\pm}\in\Sigma^{2}\mathbb{C}$, $a\in A$.

Note that $\psi_{k}=\mathcal{B}(\chi_{k})$, $k=0,1$, where

[TABLE]

[TABLE]

where $g^{\prime}\in C(\mathbb{D}^{2})$. Thus, $[\psi]=[\mathcal{B}(\tilde{\chi})]$, where

[TABLE]

The linear homotopy connecting $\tilde{u}$ with $u_{0}$ shows that $[\psi]=[\mbox{Bott}(\hat{\varphi})]$, where

[TABLE]

Finally, $[\hat{\varphi}]=[\varphi]$ by a homotopy similar to the homotopy via the functions $\alpha^{\tau}$ as in ($\alpha$1’)-($\alpha$4’).

So, we have $[\psi]=[\mathcal{B}(\tilde{\chi})]$ and $[i^{*}(\tilde{\chi})]=[\mbox{Bott}(\varphi)]$. Since Bott is an isomorphism, we conclude from (5.8) that $[\psi]=[\varphi]$.

∎

6. Surjections and asymptotic $KK$-cycles

Let $A$ be a separable $C^{*}$-algebra. Consider all surjective $*$-homomorphisms $p:C\to A$, where $C$ is a separable $C^{*}$-algebra, with a partial order given by $(C,p)\leq(C^{\prime},p^{\prime})$ if there exists a surjective $*$-homomorphism $\lambda:C^{\prime}\to C$ such that $p^{\prime}=p\circ\lambda$. Denote the set of all such $C$ by $\mathcal{E}_{A}$. Abusing the notation, we shall write $C$ in place of $(C,p)$. The set $\mathcal{E}_{A}$ is directed. Indeed, if $p_{1}:C_{1}\to A$, $p_{2}:C_{2}\to A$ are surjections then the pull-back $C=\{(c_{1},c_{2}):c_{1}\in C_{1},c_{2}\in C_{2},p_{1}(c_{1})=p_{2}(c_{2})\}$ obviously surjects onto $C_{1}$, $C_{2}$ and $A$, hence satisfies $C\geq C_{1}$, $C\geq C_{2}$.

This directed set has a minimal and a maximal elements. The minimal element is $C=A$, and the maximal element was constructed in [5], Section 2, under the name of universal extension.

Let $C\in\mathcal{E}_{A}$, and let $J=\operatorname{Ker}p$. Let $EN(C,J;B)$ be the semigroup of homotopy equivalence classes of pairs $(\psi^{+},\psi^{-})$ of asymptotic homomorphisms from $C$ to $B$ that agree on $J$, with the semigroup structure defined using stability of $B$. This semigroup need not be a group, as $(\psi^{-},\psi^{+})$ may not be the inverse for $(\psi^{+},\psi^{-})$. Set $EN(A,B)=\injlim_{C\in\mathcal{E}_{A}}EN(C,J;B)$.

Lemma 6.1**.**

$EN(A,B)$* is a group.*

Proof.

Due to the standard rotation argument, it suffices to prove that $(\psi,\psi)$ represents the zero element in $EN(A,B)$, where $C\in\mathcal{E}_{A}$, and $\psi$ is an asymptotic homomorphism from $C$ to $B$. Let $Cone(C)$ be the cone over $C$, $ev_{1}:Cone(C)\to C$ the evaluation map at 1, then $p\circ\operatorname{ev}_{1}:Cone(C)\to A$ is a surjection, so $Cone(C)\in\mathcal{E}_{A}$, $Cone(C)\geq C$. As $Cone(C)$ is contractible, $\psi\circ\operatorname{ev}_{1}$ is homotopic to zero, hence the pair $(\psi,\psi)$ is zero in $EN(A,B)$.

∎

Let $C\in\mathcal{E}_{A}$, $(\psi^{+},\psi^{-})\in EN(C,J;B)$. Let $s:A\to C$ be a $\mathbb{C}$-homogeneous $*$-respecting continuous map such that $p(s(a))=a$ for any $a\in A$, which exists by [1]. Set $\Phi^{\pm}(a)=\psi^{\pm}(s(a))$, $a\in A$.

Lemma 6.2**.**

The pair $(\Phi^{+},\Phi^{-})$ is a compact asymptotic $KK$-cycle.

Proof.

The property (a1) is obvious, so we need to check (a3). Note that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and that $s(ab)-s(a)s(b)\in J$, so $(s(ab)-s(a)s(b))s(c_{1})\cdots s(c_{n})\in J$, and as $\psi^{+}$ and $\psi^{-}$ agree on $J$, so we are done.

∎

If $s,s^{\prime}:A\to C$ are two maps satisfying the above assumptions then the linear homotopy connecting $s$ and $s^{\prime}$ gives a homotopy between the corresponding compact asymptotic $KK$-cycles, so the above construction gives a well-defined map

[TABLE]

Let $\kappa:C^{\prime}\to C$ be a surjection, $p^{\prime}=p\circ\kappa:C^{\prime}\to A$, $J^{\prime}=\operatorname{Ker}p^{\prime}$. Then we have $\gamma_{C^{\prime}}=\kappa^{*}\circ\gamma_{C}$, where the map $\kappa^{*}:EN(C,J;B)\to EN(C^{\prime},J^{\prime};B)$ is induced by $\kappa$, and we may pass to the direct limit.

Let us check homotopy invariance of $\gamma$. Let $[(\psi^{+}_{i},\psi^{-}_{i})]$, $i=0,1$, where $\psi^{\pm}_{i}$ are asymptotic homomorphisms from $C$ to $B$, represent the same element in $EN(A,B)$. This means that there exists some $C^{\prime}\geq C$ in $\mathcal{E}_{A}$ and a homotopy $(\varphi^{+},\varphi^{-})$, where $\varphi^{\pm}$ are asymptotic homomorphisms from $C^{\prime}$ to $IB$ such that $\operatorname{ev}_{i}\circ\varphi^{\pm}=\psi^{\pm}_{i}\circ\kappa$, $i=0,1$, and $\varphi^{+}$ coinsides with $\varphi^{-}$ on $\operatorname{Ker}(C^{\prime}\to A)$. Let $s^{\prime}:C\to C^{\prime}$ be a continuous section for the surjection $\kappa$. Then $\gamma_{C^{\prime}}(\varphi^{+},\varphi^{-})=(\varphi^{+}\circ s^{\prime}\circ s,\varphi^{-}\circ s^{\prime}\circ s)$. As

[TABLE]

we see that the map $\gamma:EN(A,B)\to EM(A,B)$ is well defined as the direct limit of the maps $\gamma_{C}$.

Now let us construct a map $\delta:EM(A,B)\to EN(A,B)$. Let $[(\Phi^{+},\Phi^{-})]\in EM(A,B)$. Recall that $E\subset C_{b}(\mathbb{R}_{+};B)$ is the $C^{*}$-algebra generated by $\Phi^{\pm}(A)$, and that the compact asymptotic $KK$-cycle $(\Phi^{+},\Phi^{-})$ determines an ideal $D\subset E$ generated by $\Phi^{+}(a)-\Phi^{-}(a)$, $a\in A$. It also determines another ideal $K\subset E$ generated by $\Phi^{\pm}(a+b)-\Phi^{\pm}(a)-\Phi^{\pm}(b)$ and $\Phi^{\pm}(ab)-\Phi^{\pm}(a)\Phi^{\pm}(b)$, $a,b\in A$. By definition of asymptotic $KK$-cycles, $KD\subset C_{0}(\mathbb{R}_{+};B)$, hence $K\cap D\subset C_{0}(\mathbb{R}_{+};B)$.

Let $E_{+}$ (resp. $E_{-}$) be the $C^{*}$-algebra generated by $\Phi^{+}(A)$ (resp. by $\Phi^{-}(A)$). Denote the quotient map $C_{b}(\mathbb{R}_{+};B)\to C_{b}(\mathbb{R}_{+};B)/C_{0}(\mathbb{R}_{+};B)$ by $Q$, and for a $C^{*}$-subalgebra $C\subset C_{b}(\mathbb{R}_{+};B)$ set $\underline{C}=Q(C)$. Then $\underline{D}$ and $\underline{K}$ are ideals in $\underline{E}$ with $\underline{D}\cap\underline{K}=0$, and $\underline{E}$ is generated by $\underline{E}_{+}$ and $\underline{E}_{-}$.

Lemma 6.3**.**

For a compact asymptotic $KK$-cycle $(\Phi^{+},\Phi^{-})$ there exists $[\psi]\in EN(A,B)$ such that $\gamma([\psi])=[(\Phi^{+},\Phi^{-})]$.

Proof.

Let $q:\underline{E}\to\underline{E}/\underline{K}$, $r:\underline{E}\to\underline{E}/\underline{D}$. Note that the compositions

[TABLE]

are genuine $*$-homomorphisms. Set

[TABLE]

This is a $C^{*}$-algebra that surjects onto $A$, $p(a,e_{+},e_{-})=a$. It has also two surjections, $p_{+}(a,e_{+},e_{-})=e_{+}$, $p_{-}(a,e_{+},e_{-})=e_{-}$, onto $\underline{E}_{+}$ and $\underline{E}_{-}$ respectively. Let $\sigma:\underline{E}\to E$ be a continuous Bartle–Graves selection map. As $p_{\pm}$ are $*$-homomorphisms, $\psi^{\pm}=\sigma\circ p_{\pm}:C_{\Phi}\to E$ are asymptotic homomorphisms.

Note that

[TABLE]

as $\underline{D}\cap\underline{K}=0$. Then $\psi^{+}|_{J_{\Phi}}=\psi^{-}|_{J_{\Phi}}$. Define the map

[TABLE]

Then $s$ is a continuous right inverse for $p$, and $\psi^{\pm}(s(a))\sim\Phi^{\pm}(a)$.

If $C\in\mathcal{E}_{A}$, $C\geq C_{\Phi}$, then there is a $*$-homomorphism $\kappa:C\to C_{\Phi}$, and we can define asymptotic homomorphisms $\psi_{C}^{\pm}:C\to B$ by $\psi_{C}^{\pm}=\psi^{\pm}\circ\kappa$. As $\psi_{C}^{+}|_{J}=\psi_{C}^{-}|_{J}$, where $J=\operatorname{Ker}p$, $p:C\to A$, so $(\psi_{C}^{+},\psi_{C}^{-})$ represents an element of $EN(C,J;B)$, and $\gamma([(\psi^{+},\psi^{-})])=[(\Phi^{+},\Phi^{-})]$. ∎

∎

Set $\delta(\Phi^{+},\Phi^{-})=(\psi^{+},\psi^{-})$, and let us check homotopy invariance of $\delta$. Let $(\Psi^{+},\Psi^{-})$ be a compact asymptotic $KK$-cycle from $A$ to $IB$, $\Phi^{\pm}_{i}=\operatorname{ev}_{i}\circ\Psi^{\pm}$, $i=0,1$. Apply the construction of Lemma 6.3 to this compact asymptotic $KK$-cycle. Let $E_{I}\subset C_{b}(\mathbb{R}_{+};IB)$ be the $C^{*}$-algebra generated by $\Psi^{\pm}(A)$,and let $K_{I}\subset E_{I}$ be the ideal generated by $\Phi^{\pm}(a+b)-\Phi^{\pm}(a)-\Phi^{\pm}(b)$ and $\Phi^{\pm}(ab)-\Phi^{\pm}(a)\Phi^{\pm}(b)$, $a,b\in A$. Similarly, Let $K_{i}\subset E_{i}\subset C_{b}(\mathbb{R}_{+};B)$ be the corresponding ideal and $C^{*}$-subalgebra for $\operatorname{ev}_{i}\circ\Phi^{\pm}$, and let underlining denotes the image of these $C^{*}$-algebras in $C_{b}(\mathbb{R}_{+};IB)/C_{0}(\mathbb{R}_{+};IB)$ and in $C_{b}(\mathbb{R}_{+};B)/C_{0}(\mathbb{R}_{+};B)$ respectively.

Let $q_{I}:\underline{E}_{I}\to\underline{E}_{I}/\underline{K}_{I}$, $q_{i}:\underline{E}_{i}\to\underline{E}_{i}/\underline{K}_{i}$, $r_{I}:\underline{E}_{I}\to\underline{E}_{I}/\underline{J}_{I}$, $r_{i}:\underline{E}_{i}\to\underline{E}_{i}/\underline{J}_{i}$ be the quotient maps, and let, as in Lemma 6.3, $\underline{\Psi}^{\pm}:A\to\underline{E}_{I}/\underline{K}_{I}$ be the $*$-homomorphisms induced by $\Psi^{\pm}$. Let $C_{\Psi}$ and $C_{\Phi_{i}}$, $i=0,1$, be the $C^{*}$-algebras in $\mathcal{E}_{A}$ constructed as in Lemma 6.3, with surjections $p_{I}:C_{\Psi}\to A$ and $p_{i}:C_{\Phi_{i}}\to A$. Note that $\operatorname{ev}_{i}:\underline{E}_{I}\to\underline{E}_{i}$ is surjective, and $\operatorname{ev}_{i}(\underline{K}_{I})=\underline{K}_{i}$, hence $\operatorname{ev}_{i}$ gives rise to a surjective $*$-homomorphism $\widetilde{\operatorname{ev}}_{i}:\underline{E}_{I}/\underline{K}_{I}\to\underline{E}_{i}/\underline{K}_{i}$.

This gives a commuting diagram

[TABLE]

where $\kappa_{i}:C_{\Psi}\to C_{\Phi_{i}}$ is given by $\kappa_{i}(a,e_{+},e_{-})=(a,\operatorname{ev}_{i}(e_{+}),\operatorname{ev}_{i}(e_{-}))$. Recall that $(a,e_{+},e_{-})\in C_{\Psi}$ if $a\in A$, $e_{\pm}\in\underline{E}_{I}$, $\underline{\Psi}^{\pm}(a)=q_{I}(e_{\pm})$, $r_{I}(e_{+})=r_{I}(e_{-})$.

Let $\sigma_{I}:\underline{E}_{I}\to E_{I}$, $\sigma_{i}:\underline{E}_{i}\to E_{i}$. Then $\delta(\Psi^{+},\Psi^{-})=(\psi^{+},\psi^{-})$, where $\psi^{\pm}=p_{\pm}^{\Psi}\circ\sigma_{I}$, $\delta(\Phi^{+}_{i},\Phi^{-}_{i})=(\varphi^{+}_{i},\varphi^{-}_{i})$, where $\varphi^{\pm}_{i}=p_{\pm}^{\Phi_{i}}\circ\sigma_{i}$. But the asymptotic homomorphisms $\operatorname{ev}_{i}\circ\psi^{\pm}$ are asymptotically equivalent to $\varphi^{\pm}_{i}\circ\kappa$, $i=0,1$, so $[(\varphi^{+}_{0},\varphi^{-}_{0})]=[(\varphi^{+}_{1},\varphi^{-}_{1})]$. Thus, the map $\delta:EM(A,B)\to EN(A,B)$ is well-defined.

Theorem 6.4**.**

The map $\gamma:EN(A,B)\to EM(A,B)$ is an isomorphism.

Proof.

Lemma 6.3 proves surjectivity of $\gamma$, so it suffices to show that $\delta\circ\gamma$ is the identity map.

Let $C\in\mathcal{E}_{A}$, $\varphi^{\pm}$ be asymptotic homomorphisms from $C$ to $B$ equal on $J=\operatorname{Ker}(C\to A)$. Then $\gamma(\varphi^{+},\varphi^{-})=(\Phi^{+},\Phi^{-})$, where the latter is a compact asymptotic $KK$-cycle defined by $\Phi^{\pm}=\varphi^{\pm}\circ s$, where $s:A\to C$ is a continuous section. Using notation from Lemma 6.3, we have two commuting diagrams

[TABLE]

where $[\varphi^{\pm}]$ denotes the composition of $\varphi^{\pm}$ with the quotient map $Q:E\to\underline{E}$. There exists a $*$-homomorphism $\kappa:C\to C_{\Phi}$ given by $\kappa(c)=(p(c),[\varphi^{+}](c),[\varphi^{-}](c))$ (as $\varphi^{+}$ and $\varphi^{-}$ are equal on $J$, $r([\varphi^{+}](c))=r([\varphi^{-}](c))$), so $\kappa(c)\in C_{\Phi}$. As $p$ is surjective and $[\varphi^{\pm}]$ are surjective too, by the definition of $\underline{E}_{\pm}$, $\kappa$ is surjective. Thus, $[\varphi^{\pm}]=\kappa\circ p_{\pm}$, hence, passing to their lifts, $\varphi^{\pm}$ is asymptotically equivalent to $\psi^{\pm}=\sigma\circ\kappa\circ p_{\pm}$, where $\sigma:\underline{E}\to E$ is a continuous lift for $Q$. But $\delta\circ\gamma(\varphi^{+},\varphi^{-})=(\psi^{+},\psi^{-})$.

∎

7. Full surjections

Definition 7.1**.**

Let $C\in\mathcal{E}_{A}$. We call it full if the canonical map $EN(C,J;B)\to EN(A,B)$ is an isomorphism for any $B$.

Note that $\mathcal{E}_{A}$ has the minimal element $A$. For this minimal element, $EN(A,0;B)$ is the semigroup of homotopy classes of pairs of asymptotic homomorphisms from $A$ to $B$, hence $EN(A,0;B)\cong[[A,B]]\oplus[[A,B]]$, and thus $A$ very rarely is full.

Lemma 7.2**.**

The cone extension $CA$ and the universal extension are full.

Proof.

For the universal extension the statement follows from its universality, so let us prove that $CA$ is full. Essentially, this is a special case of Theorem 5.3 of [12], but we give more details here.

We shall construct a map $\delta_{CA}:EM(A,B)\to EN(CA,SA;B)$ inverse to $\gamma_{CA}$. Given $(\Phi^{+},\Phi^{-})$ in $EM(A,B)$, let $v\in C_{b}(\mathbb{R}_{+};B)$ be as before, i.e. let $v$ satisfy

• (v1’)

$0\leq v\leq 1$;

• (v2’)

$[f(v),\Phi^{\pm}(a)]\sim 0$ for any $f\in C[0,1]$ and any $a\in A$;

• (v3’)

$(f(v)-1)(\Phi^{+}(a)-\Phi^{-}(a))\sim 0$ for any $f\in C_{0}[0,1)$ and any $a\in A$.

Define $\psi^{\pm}:CA\to C_{b}(\mathbb{R}_{+};B)$ by $\psi^{\pm}(f\otimes a)=f(v)\Phi^{\pm}(a)$, where $f\in C_{0}[0,1)$, $a\in A$.

It is easy to see that $\psi^{\pm}$ are asymptotic homomorphisms from $CA$ to $B$, and (v3’) provides that $\psi^{+}$ and $\psi^{-}$ agree on $SA$, i.e. when $f\in C_{0}(0,1)$. Thus we get a well-defined map $\delta_{CA}:EM(A,B)\to EN(CA,SA;B)$, $\delta_{CA}(\Phi^{+},\Phi^{-})=(\psi^{+},\psi^{-})$.

Then $\delta_{CA}\circ\gamma_{CA}(\Phi^{+},\Phi^{-})=(v\Phi^{+},v\Phi^{-})$, and, as in the proof of Lemma 2.2, $[v\Phi^{+},v\Phi^{-}]=[\Phi^{+},\Phi^{-}]$, hence $\delta_{CA}\circ\gamma_{CA}=\operatorname{id}$.

Let us show that $\gamma_{CA}\circ\delta_{CA}=\operatorname{id}$. Let $(\psi^{+},\psi^{-})\in EN(Cone(A),SA;B)$, i.e. $\psi^{\pm}$ are two asymptotic homomorphisms from $Cone(A)$ to $B$ that agree on $SA$. Let $s:A\to CA$ be the map defined by $s(a)(\tau)=\tau a$, $a\in A$, $\tau\in[0,1]$. Then $\gamma_{CA}\circ\delta_{CA}(\psi)=(\phi^{+},\phi^{-})$, where $\phi^{\pm}(f\otimes a)=f(v)\psi^{\pm}(s(a))$, $a\in A$, $f\in C_{0}[0,1)$.

Let us show that the pairs $(\psi^{+},\psi^{-})$ and $(\phi^{+},\phi^{-})$ of asymptotic homomorphisms are homotopic, i.e. that there exist two homotopies, connecting $\psi^{+}$ with $\phi^{+}$, and $\psi^{-}$ with $\phi^{-}$, such that they agree on $SA$.

Define asymptotic homomorphisms $\xi^{\pm}$ (resp. $\eta^{\pm}$) from $C_{0}([0,1)\times[0,1])\otimes A$ (resp. from $C_{0}([0,1]\times[0,1))\otimes A$) to $B$ by

[TABLE]

[TABLE]

Note that on the common domain $C_{0}([0,1)^{2}\otimes A)$, these asymptotic homomorphisms are homotopic via the linear homotopy

[TABLE]

and if $g\in C_{0}(0,1)$ then the two maps $\zeta_{\tau}^{+}$ and $\zeta_{\tau}^{-}$ agree.

Let $(x,y)$ be coordinates in $[0,1]^{2}$, and let $p_{1},p_{2},p:[0,1]^{2}\to[0,1]$ be defined by $p_{1}(x,y)=x$, $p_{2}(x,y)=y$, p(x,y)=\left\{\begin{array}[]{cc}x+y&\mbox{\ if\ }x+y\leq 1;\\ 1&\mbox{\ if\ }x+y\geq 1.\end{array}\right. There are obvious linear homotopies $p_{1}(\tau)$ and $p_{2}(\tau)$, $\tau\in[0,1$, connecting $p_{1}$ and $p_{2}$ with $p$. By $q^{*}$ we denote the $*$-homomorphism $C[0,1)\otimes A\to C[0,1]^{2}\otimes A$ induced by a map $q:[0,1]^{2}\to[0,1]$.

Note that $p_{1}^{*}\circ\xi^{\pm}(f\otimes a)=f(v)\psi^{\pm}(s(a))=\varphi^{\pm}(f\otimes a)$, and $\operatorname{Im}p_{1}^{*}=C_{0}([0,1)\times[0,1])$. The maps $p_{1}^{*}\circ\xi^{\pm}$ and $p_{2}^{*}\circ\xi^{\pm}$ are homotopic via the homotopy $p_{1}^{*}(\tau)\circ\xi^{\pm}$, $\tau\in[0,1]$, and $p_{1}^{*}(\tau)\circ\xi^{+}$ and $p_{1}^{*}(\tau)\circ\xi^{-}$ agree on $f\otimes a$ when $f\in C_{0}(0,1)$, for any $\tau\in[0,1]$. Hence $[(\varphi^{+},\varphi^{-})]=[(p^{*}\circ\xi^{+},p^{*}\circ\xi^{-})]$.

Similarly, $p_{2}^{*}\circ\eta^{\pm}(f\otimes a)=\psi^{\pm}(f\otimes a)$, and $[(\psi^{+},\psi^{-})]=[(p^{*}\circ\eta^{+},p^{*}\circ\eta^{-})]$.

As $p^{*}(f\otimes a)\in C_{0}([0,1)^{2})$ for any $f\in C_{0}[0,1)$, we may use the homotopy $\zeta^{\pm}$ to connect $p^{*}\circ\xi^{\pm}$ with $p^{*}\circ\eta^{\pm}$.

∎

Lemma 7.3**.**

Let $C\in\mathcal{E}_{A}$. If there exists a splitting $*$-homomorphism $\lambda:A\to C$ and if $A\in\mathcal{E}_{A}$ is not full then $C$ is not full.

Proof.

Any continuous map $s:A\to C$ can be connected to $\lambda$ by a linear homotopy, therefore, any compact asymptotic $KK$-cycle is homotopic to a compact asymptotic $KK$-cycle, which is a pair of $*$-homomorphisms, hence $EN(C,J;B)\cong EN(A,0;B)$.

∎