Definable $\mathrm{K}$-homology of separable C*-algebras

A Polish space is a second countable topological space whose topology is induced by a complete metric. A subset of a Polish space $X$ is $G_{\delta}$ if and only if it is a Polish space when endowed with the subspace topology. If $X$ is a Polish space, then the Borel $\sigma$-algebra of $X$ is the $\sigma$-algebra generated by the collection of open sets. By definition, a subset of $X$ is Borel if it belongs to the Borel $\sigma$-algebra. If $X,Y$ are Polish spaces, then the product $X\times Y$ is a Polish space when endowed with the product topology. More generally, if $\left(X_{n}\right)_{n\in\omega}$ is a sequence of Polish spaces, then the product $\prod_{n\in\omega}X_{n}$ is a Polish space when endowed with the product topology. The class of Polish spaces includes all locally compact second countable Hausdorff spaces. We denote by $\omega$ the set of natural numbers including $0$. We regard $\omega$ and any other countable set as a Polish space endowed with the discrete topology. The Baire space $\omega^{\omega}$ is the Polish space obtained as the infinite product of copies of $\omega$.

A standard Borel space is a set $X$ endowed with a $\sigma$-algebra (the Borel $\sigma$-algebra) that comprises the Borel sets with respect to some Polish topology on $X$. A function between standard Borel spaces is Borel if it is measurable with respect to the Borel $\sigma$-algebras. A subset of a standard Borel space $X$ is analytic if it is the image of a Borel function $f:Z\rightarrow X$ for some standard Borel space $Z$. This is equivalent to the assertion that there exists a Borel subset $B\subseteq X\times\omega^{\omega}$ such that $B=\mathrm{proj}_{X}\left(A\right)$ is the projection of $A$ on the first coordinate. A subset of $X$ is co-analytic if its complement is analytic. One has that a subset of $X$ is Borel if and only if it is both analytic and co-analytic.

Given standard Borel spaces $X,Y$, we let $X\times Y$ be their product endowed with the product Borel structure, which is also a standard Borel space. If $\left(X_{n}\right)_{n\in\omega}$ is a sequence of standard Borel spaces, then their disjoint union $X$ is a standard Borel space, where a subset $A$ of $X$ is Borel if and only if $A\cap X_{n}$ is Borel for every $n\in\omega$. The product $\prod_{n\in\omega}X_{n}$ is also a standard Borel space when endowed with the product Borel structure. In the following proposition, we collect some well-known properties of the category of standard Borel spaces and Borel functions.

A Polish group is a topological group whose topology is Polish. If $G$ is a Polish group, and $H$ is a closed subgroup of $G$, then $H$ is a Polish group when endowed with the subspace topology. If furthermore $H$ is normal, then $G/H$ is a Polish group when endowed with the quotient topology. If $G_{0},G_{1}$ are Polish groups, and $\varphi:G_{0}\rightarrow G_{1}$ is a Borel function, then $\varphi$ is continuous. In particular, if $G$ is a Polish space, then it has a unique Polish group topology that induces its Borel structure. A subgroup $H$ of a Polish group $G$ is Polishable if it is Borel and there is a (necessarily unique) Polish group topology on $H$ that induces the Borel structure on $H$ inherited from $G$. This is equivalent to the assertion that $H$ is equal to the range of a continuous group homomorphisms $\hat{G}\rightarrow G$ for some Polish group $\hat{G}$. If $G$ is a Polish group, then a Polish $G$-space is a Polish space $X$ endowed with a continuous action of $G$. A Borel $G$-space is a standard Borel space $X$ endowed with a Borel action of $G$. Given a Borel $G$-space $X$, there exists a Polish topology $\tau$ on $X$ such that $\left(X,\tau\right)$ is a Polish $G$-space; see [becker_descriptive_1996, Theorem 5.2.1].

A standard Borel group is, simply, a group object in the category of standard Borel spaces [mac_lane_categories_1998, Section III.6]. Explicitly, a standard Borel group is a standard Borel space $G$ that is also a group, and such that the group operation on $G$ and the function $G\rightarrow G$, $x\mapsto x^{-1}$ are Borel; see [kechris_classical_1995, Definition 12.23]. Clearly, every Polish group is, in particular, a standard Borel group.

The notion of Polish topometric space was introduced and studied in [ben_yaacov_grey_2015, ben_yaacov_polish_2013, ben_yaacov_topometric_2008, ben_yaacov_continuous_2010]. A topometric space is a Hausdorff space $X$ endowed with a topology $\tau$ and a $\left[0,\infty\right]$-valued metric $d$ such that:

1. (1)

   the metric-topology is finer than $\tau$;

2. (2)

   the metric is lower-semicontinuous with respect to $\tau$, i.e. for every $r\geq 0$ the set

   $$\left\{\left(a,b\right)\in X\times X:d\left(a,b\right)\leq r\right\}$$

   is $\tau$-closed in $X\times X$.

A Polish topometric space is a topometric space such that the topology $\tau$ is Polish and the metric is complete. A Polish topometric group is a Polish topometric space $\left(G,\tau,d\right)$ that is also a group, and such that $G$ endowed with the topology $\tau$ is a Polish group, and the metric $d$ on $G$ is bi-invariant

By definition, we let a Polish category be a category $\mathcal{C}$ enriched over the category of Polish spaces (regarded as a monoidal category with respect to binary products). Thus, for each pair of objects $a,b$ of $\mathcal{C}$, $\mathcal{C}\left(a,b\right)$ is a Polish space, such that for objects $a,b,c$, the composition operation $\mathcal{C}\left(b,c\right)\times\mathcal{C}\left(a,b\right)\rightarrow\mathcal{C}\left(a,c\right)$ is continuous.

Suppose that $\mathcal{C}$ is a Polish category. For objects $a,b$ of $\mathcal{C}$, define $\mathrm{Iso}_{\mathcal{C}}\left(a,b\right)\subseteq\mathcal{C}\left(a,b\right)$ be the set of $\mathcal{C}$-isomorphisms $a\rightarrow b$. While $\mathrm{Iso}_{\mathcal{C}}\left(a,b\right)$ is not necessarily a $G_{\delta}$ subset of $\mathcal{C}\left(a,b\right)$, and hence not necessarily a Polish space when endowed with the subspace topology, $\mathrm{Iso}_{\mathcal{C}}\left(a,b\right)$ is endowed with a canonical Polish topology, defined as follows. For a net $\left(\alpha_{i}\right)$ in $\mathrm{Iso}_{\mathcal{C}}\left(a,b\right)$ and $\alpha\in\mathrm{Iso}_{\mathcal{C}}\left(a,b\right)$, set $\alpha_{i}\rightarrow\alpha$ if and only if $\alpha_{i}\rightarrow\alpha$ in $\mathcal{C}\left(a,b\right)$ and $\alpha_{i}^{-1}\rightarrow\alpha^{-1}$ in $\mathcal{C}\left(b,a\right)$. One can then easily show the following.

It is clear from the definition that, for every object $a$ of $\mathcal{C}$, $\mathrm{\mathrm{Aut}}_{\mathcal{C}}\left(a\right):=\mathrm{Iso}_{\mathcal{C}}\left(a,a\right)$ is a Polish group. Furthermore, the canonical (right and left) actions of $\mathrm{\mathrm{Aut}}_{\mathcal{C}}\left(a\right)$ and $\mathrm{\mathrm{Aut}}_{\mathcal{C}}\left(b\right)$ on $\mathcal{C}\left(a,b\right)$ are continuous.

The notion of topological equivalence of categories is the natural analogue of the notion of equivalence of categories in the context of Polish categories; see [mac_lane_categories_1998, Section IV.4]. The same proof as [mac_lane_categories_1998, Section IV.4, Theorem 1] gives the following characterization of topological equivalences.

Suppose that $C$ is a set. A $\sigma$-filter on $C$ is a nonempty family $\mathcal{F}$ of subsets of $C$ that is closed under countable intersections, and such that $\varnothing\notin\mathcal{F}$ and if $A\subseteq B\subseteq C$ and $A\in\mathcal{F}$ then $B\in\mathcal{F}$. The dual notion is the one of $\sigma$-ideal. Thus, a nonempty family $\mathcal{I}$ of subsets of $C$ is a $\sigma$-ideal if it is closed under countable unions, $C\notin\mathcal{I}$, and $A\subseteq B\subseteq C$ and $B\in\mathcal{I}$ imply $A\in\mathcal{I}$. Clearly, if $\mathcal{F}$ is a $\sigma$-filter on $C$, then $\left\{C\setminus A:A\in\mathcal{F}\right\}$ is a $\sigma$-ideal on $C$, and vice-versa. Thus, one can equivalently formulate notions in terms of $\sigma$-filters or in terms of $\sigma$-ideals.

If $\mathcal{F}$ is a $\sigma$-filter on $C$, then $\mathcal{F}$ can be thought of as a notion of “largeness” for subsets of $C$. Based on this interpretation, we use the “$\sigma$-filter quantifier” notation “$\mathcal{F}x$, $x\in A$” for a subset $A\subseteq C$ to express the fact that $A\in\mathcal{F}$. If $P(x)$ is a unary relation for elements of $C$, “$\mathcal{F}x$, $P(x)$” is the assertion that the set of $x\in C$ that satisfy $P(x)$ belongs to $\mathcal{F}$.

Suppose that $X$ is a standard Borel space. We consider an equivalence relation $E$ on $X$ as a subset of $X\times X$, endowed with the product Borel structure. Consistently, we say that $E$ is Borel or analytic, respectively, if it is a Borel or analytic subset of $X\times X$. In the following, we will exclusively consider analytic equivalence relations, most of which will in fact be Borel. For an element $x$ of $X$ we let $\left[x\right]_{E}$ be its corresponding $E$-class.

We now recall the notion of idealistic equivalence relation, initially considered in [kechris_countable_1994]; see also [gao_invariant_2009, Definition 5.4.9] and [kechris_borel_2016]. We will consider a slightly more generous definition than the one from [kechris_countable_1994, gao_invariant_2009, kechris_borel_2016]. The more restrictive notion is recovered as a particular case by insisting that the function $s$ in Definition 1.6 be the identity function of $X$. In the following definition, for a subset $A$ of a product space $X\times Y$ and $x\in X$, we let $A_{x}=\left\{y\in Y:\left(x,y\right)\in A\right\}$ be the corresponding vertical section.

Idealistic equivalence relations arise naturally as orbit equivalence relations of Polish group actions. Suppose that $G$ is a Polish group and $X$ is a Polish $G$-space. Let $E_{G}^{X}$ be the corresponding orbit equivalence relation on $X$, obtained by setting, for $x,y\in X$, $xE_{G}^{X}y$ if and only if there exists $g\in G$ such that $g\cdot x=y$. Then $E_{G}^{X}$ is an idealistic equivalence relation, as witnessed by the identity function $s$ on $X$ and the function $C\mapsto\mathcal{F}_{C}$ where $A\in\mathcal{F}_{C}$ if and only if $\mathcal{F}_{G}g$, $g\cdot x\in A$. (As in Example 1.5, $\mathcal{F}_{G}$ denotes the $\sigma$-filter of comeager subsets of $G$.) In particular, if $G$ is a Polish group, and $H$ is a Polishable subgroup of $G$, then the coset equivalence relation $E_{H}^{G}$ of $H$ in $G$ is Borel and idealistic.

Suppose that $E$ is an equivalence relation on a standard Borel space $X$. A Borel selector for $E$ is a Borel function $s:X\rightarrow X$ such that, for $x,y\in x$, $xEy$ if and only if $s(x)=s(y)$. If $E$ has a Borel selector, then $E$ is Borel and idealistic; see [gao_invariant_2009, Theorem 5.4.11]. (Precisely, an equivalence relation has a Borel selector if and only if it is Borel, idealistic, and smooth [gao_invariant_2009, Definition 5.4.1].)

Definable sets are a generalization of standard Borel sets, and can be thought of as sets explicitly presented as the quotient of a standard Borel space by a “well-behaved” equivalence relation $E$.

We now define the notion of morphism between definable sets. Let $X=\hat{X}/E$ and $Y=\hat{Y}/F$ be definable sets. A lift of a function $f:X\rightarrow Y$ is a function $\hat{f}:\hat{X}\rightarrow\hat{Y}$ such that $f\left(\left[x\right]_{E}\right)=[\hat{f}(x)]_{F}$ for every $x\in\hat{X}$.

We consider definable sets as objects of a category $\mathbf{DSet}$, whose morphisms are the definable functions. We regard a standard Borel space $X$ as a particular instance of definable set $X=\hat{X}/E$ where $X=\hat{X}$ and $E$ is the relation of equality on $X$. This renders the category of standard Borel spaces a full subcategory of the category of definable sets.

If $X=\hat{X}/E$ and $Y=\hat{Y}/F$ are definable sets, then their product $X\times Y$ in $\mathbf{DSet}$ is the definable set $X\times Y:=(\hat{X}\times\hat{Y})\left/\left(E\times F\right)\right.$, $E\times F$ being the equivalence relation on $\hat{X}\times\hat{Y}$ defined by setting $\left(x,y\right)\left(E\times F\right)\left(x^{\prime},y^{\prime}\right)$ if and only if $xEx^{\prime}$ and $yFy^{\prime}$. (It is easy to see that $E\times F$ is Borel and idealistic if both $E$ and $F$ are Borel and idealistic.)

Many of the good properties of standard Borel spaces, including all the ones listed in Proposition 1.1, generalize to definable sets.

Occasionally we will need to consider quotients $X=\hat{X}/E$ where $\hat{X}$ is a standard Borel space $E$ is an analytic equivalence relation on $\hat{X}$ that is not Borel and idealistic, or has not yet been shown to be Borel and idealistic. In this case, we say that $X=\hat{X}/E$ is a semidefinable set. Clearly, every definable set is, in particular, a semidefinable set. The notion of Borel subset and definable function are the same as in the case of definable sets. Thus, if $X=\hat{X}/E$ and $Y=\hat{Y}/F$ are semidefinable sets, $Z\subseteq X$ is a subset and $f:X\rightarrow Y$ is a function, then $f$ is definable if it has a Borel lift $\hat{f}:\hat{X}\rightarrow\hat{Y}$, and $Z$ is Borel if there is a Borel $E$-invariant subset $\hat{Z}$ of $\hat{X}$ such that $Z=\hat{Z}/E$. The category $\mathbf{SemiDSet}$ has semidefinable sets as objects and definable functions as morphisms. Notice that, in particular, an isomorphism from $X$ to $Y$ in $\mathbf{SemiDSet}$ is a bijection $f:X\rightarrow Y$ such that both $f$ and the inverse function $f^{-1}:Y\rightarrow X$ are definable.

Lemma 3.7 in [kechris_borel_2016] can be stated as the following proposition, which generalizes items (2) and (3) in Proposition 1.10.

The following result is a consequence of Lemma 1.12 and Proposition 1.13.

A definable group can be simply defined as a group in the category $\mathbf{DSet}$ in the sense of [mac_lane_categories_1998, Section III.6]. Thus, a definable group is a definable set $G=\hat{G}/E$ that is also a group, and such that the group operation $G\times G\rightarrow G$ is definable, and the function $G\rightarrow G$, $x\mapsto x^{-1}$ is also definable. As in the case of sets, we regard a standard Borel group as a particular instance of definable group $G=\hat{G}/E$ where $G=\hat{G}$ is a standard Borel group and $E$ is the relation of equality on $\hat{G}$. Thus, standard Borel groups form a full subcategory of the category of definable groups.

Naturally, a semidefinable group will be a group in $\mathbf{SemiDSets}$, i.e. a semidefinable set $G=\hat{G}/E$ that is also a group, and such that the group operation $G\times G\rightarrow G$ is definable, and the function $G\rightarrow G$ that maps every element to its inverse is definable.

Let $X$ be a Banach space. We denote by $\mathrm{\mathrm{Ball}}\left(X\right)$ its unit ball. A seminorm $p$ on $X$ is bounded if $\left\|p\right\|:=\mathrm{sup}_{x\in\mathrm{\mathrm{Ball}}\left(X\right)}p(x)<\infty$. We say that $p$ is contractive if $\left\|p\right\|\leq 1$.

Let $X$ be a seminormed space, and consider the cone $\mathcal{S}\left(X\right)$ of bounded seminorms on $X$ as a complete metric space, with respect to the metric defined by $d\left(p,q\right)=\mathrm{sup}_{x\in\mathrm{\mathrm{Ball}}\left(X\right)}\left|p(x)-q(x)\right|$. For a subset $\mathfrak{S}\subseteq\mathcal{S}\left(X\right)$, we let $\sigma\left(X,\mathfrak{S}\right)$ be the topology on $\mathrm{\mathrm{Ball}}\left(X\right)$ generated by $\mathfrak{S}$. We denote by Ball$\left(\mathfrak{S}\right)$ the set of contractive seminorms in $\mathfrak{S}$. If $p\in\mathcal{S}\left(X\right)$, $\mathfrak{S}\subseteq\mathcal{S}\left(X\right)$, and $\left(x_{n}\right)$ is a sequence in $\mathrm{\mathrm{Ball}}\left(X\right)$, then we say that:

• •

  $\left(x_{n}\right)_{n\in\omega}$ is $p$-Cauchy if for every $\varepsilon>0$ there exists $n_{0}\in\omega$ such that, for $n,m\geq n_{0}$, $p\left(x_{n}-x_{m}\right)<\varepsilon$;

• •

  $\left(x_{n}\right)_{n\in\omega}$ is $\mathfrak{S}$-Cauchy if it is $p$-Cauchy for every $p\in\mathfrak{S}$;

• •

  $\mathrm{\mathrm{Ball}}\left(X\right)$ is $\mathfrak{S}$-complete if, for every sequence $\left(x_{n}\right)_{n\in\omega}$ in Ball$\left(X\right)$, if $\left(x_{n}\right)_{n\in\omega}$ is $\mathfrak{S}$-Cauchy, then $\left(x_{n}\right)_{n\in\omega}$ is $\sigma\left(X,\mathfrak{S}\right)$-convergent to some element of $\mathrm{\mathrm{Ball}}\left(X\right)$.

The following lemma is elementary.

In view of Lemma 2.4 one can equivalently define a strict Banach space as follows.

Suppose that $\mathfrak{X}$ is a strict Banach space. We extend the strict topology of Ball$\left(\mathfrak{X}\right)$ to any bounded subset of $\mathfrak{X}$ by declaring the function

to be a homeomorphism with respect to the strict topology, where

Then we have that addition and scalar multiplication on $\mathfrak{X}$ are strictly continuous on bounded sets, and the norm is strictly lower-semicontinuous on bounded sets. In particular $n\mathrm{\mathrm{Ball}}\left(X\right)$ is a strictly closed subspace of $m\mathrm{\mathrm{Ball}}\left(X\right)$ for $n\leq m$. Notice that, if $\mathfrak{Y}$ is a norm-closed subspace of $X$ such that $\mathrm{\mathrm{Ball}}\left(\mathfrak{Y}\right)$ is strictly closed in Ball$\left(\mathfrak{X}\right)$, then $\mathfrak{Y}$ is a strict Banach space with the induced norm and the induced strict topology.

Notice that the Borel structure on $\mathfrak{X}$ is standard, as $\mathfrak{X}$ is Borel isomorphic to the disjoint union of the standard Borel spaces $\left(n+1\right)\mathrm{\mathrm{Ball}}\left(\mathfrak{X}\right)\setminus n\mathrm{\mathrm{Ball}}\left(\mathfrak{X}\right)$ for $n\geq 1$.

Clearly, strict Banach spaces form a category where the morphisms are the strict contractive linear maps. Notice that, if $T$ is a strict, bijective, and isometric linear map $T:\mathfrak{X}\rightarrow\mathfrak{Y}$ between strict Banach spaces, then the inverse $T^{-1}:\mathfrak{Y}\rightarrow\mathfrak{X}$ is not necessarily strict, whence $T$ is not necessarily an isomorphism in the category of strict Banach spaces. Nonetheless, $T:\mathfrak{X}\rightarrow\mathfrak{Y}$ is a Borel isomorphism, as both $\mathfrak{X}$ and $\mathfrak{Y}$ are standard Borel spaces.

Notice that $\mathcal{S}_{\mathrm{strict}}\left(\mathfrak{X}\right)$ is a closed subspace of the complete metric space $\mathcal{S}\left(\mathfrak{X}\right)$. A sequence $\left(x_{n}\right)$ in $\mathrm{\mathrm{Ball}}\left(\mathfrak{X}\right)$ is strictly convergent if and only if it is $\mathcal{S}_{\mathrm{strict}}\left(\mathfrak{X}\right)$-Cauchy. A bounded linear map $T:\mathfrak{X}\rightarrow Y$ is strict if and only if $p\circ T\in\mathcal{S}_{\mathrm{strict}}\left(\mathfrak{X}\right)$ for every $p\in\mathcal{S}_{\mathrm{strict}}\left(Y\right)$.

For future reference, we record the easily proved observation that a uniform limit of strictly continuous functions is strictly continuous.

A standard Baire Category argument shows that one can characterize bounded subsets in terms of bounded, strict seminorms, as follows.

A natural way to obtain strict Banach spaces is via pairings.

The following lemma is an immediate consequence of the definition of strict Banach space.

Suppose that $X$ is a norm-separable Banach space, and $\mathfrak{Y}$ is a strict Banach spaces. A linear map $T:X\rightarrow\mathfrak{Y}$ is bounded if it maps bounded sets to bounded sets or, equivalently,

This defines a norm on the space $L\left(X,\mathfrak{Y}\right)$ of bounded linear maps $X\rightarrow\mathfrak{Y}$. We also define the strict topology on Ball$\left(L\left(X,\mathfrak{Y}\right)\right)$ to be the topology of pointwise convergence in the strict topology of Ball$\left(\mathfrak{Y}\right)$. Then one can easily show the following.

We now introduce the notion of strict C*-algebra. Given a C*-algebra $A$, we let $A_{\mathrm{sa}}$ be the set of its self-adjoint elements. We also denote by $M_{n}\left(A\right)$ the C*-algebra of $n\times n$ matrices over $A$, which can be identified with the tensor product $M_{n}\left(\mathbb{C}\right)\otimes A$. We refer to [blackadar_operator_2006, davidson_algebras_1996, murphy_algebras_1990, pedersen_algebras_1979] for fundamental notions and results from the theory of C*-algebras.

Suppose that $\mathfrak{A}$ is a strict C*-algebra. Then, for every $n\geq 1$, $M_{n}\left(\mathfrak{A}\right)$ is also a strict C*-algebra. If $\mathfrak{A}$ is a strict C*-algebra, then we regard $\mathfrak{A}$ as a standard Borel space with respect to the standard Borel structure induced by the strict topology on $\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$ as in Definition 2.7. We say that a subset of $\mathfrak{A}$ is Borel if it is Borel with respect to such a Borel structure. We have that the Borel structure on $M_{n}\left(\mathfrak{A}\right)$ (as a strict C*-algebra) coincides with the product Borel structure.

We regard strict (unital) C*-algebras as objects of a category with strict (unital) *-homomorphisms as morphisms. (Recall that a bounded linear map is strict if it is strictly continuous on bounded sets.) If $\mathfrak{A}\subseteq\mathfrak{B}$, then we say that $\mathfrak{A}$ is strictly dense in $\mathfrak{B}$ if $\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$ is dense in $\mathrm{\mathrm{Ball}}\left(\mathfrak{B}\right)$ with respect to the strict topology.

It follows from the axioms of a strict C*-algebra that, if $\mathfrak{A}$ is a strict C*-algebra, and $p\left(x_{1},\ldots,x_{n}\right)$ is a *-polynomial, then $p$ defines a function $\mathfrak{A}^{n}\rightarrow\mathfrak{A}$ that is strictly continuous on bounded sets. In particular, the sets of normal, self-adjoint, and positive elements of norm at most $1$ are strictly closed in $\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$. If $f:[-1,1]^{n}\rightarrow n\mathrm{\mathrm{Ball}}\left(\mathbb{C}\right)$ is a continuous function, then $f$ induces by continuous functional calculus and Lemma 2.11 a strictly continuous functions $\left(x_{1},\ldots,x_{n}\right)\mapsto f\left(x_{1},\ldots,x_{n}\right)$ from the strictly closed set of $n$-tuples of pairwise commuting self-adjoint elements in $\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$ to $n\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$. Similarly, if $f:\mathrm{\mathrm{Ball}}\left(\mathbb{C}\right)^{n}\rightarrow k\mathrm{\mathrm{Ball}}\left(\mathbb{C}\right)$ is a continuous function, then $f$ induces by continuous functional calculus and Lemma 2.11 a strictly continuous function $\left(x_{1},\ldots,x_{n}\right)\mapsto f\left(x_{1},\ldots,x_{n}\right)$ from the strictly closed set of $n$-tuples of pairwise commuting normal elements in $\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$ to $k\mathrm{\mathrm{Ball}}\left(\mathfrak{A}\right)$.