Abstract
We associate an algebra \(\Gamma ^\infty (\mathfrak {A})\) to each bornological algebra \(\mathfrak {A}\). Each symmetric ideal \(S\) of the algebra \(\ell ^\infty \) of complex bounded sequences gives rise to an ideal \(I_{S(\mathfrak {A})}\) of \(\Gamma ^\infty (\mathfrak {A})\). We show that all ideals arise in this way when \(\mathfrak {A}\) is the algebra of complex numbers. We prove that for suitable \(S\), Weibel’s \(K\)theory of \(I_{S(\mathfrak {A})}\) is homotopy invariant, and show that the failure of the map from Quillen’s to Weibel’s \(K\)theory of \(I_{S(\mathfrak {A})}\) to be an isomorphism is measured by cyclic homology.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\ell ^2=\ell ^2(\mathbb {N})\) be the Hilbert space of squaresummable sequences of complex numbers and \(\mathcal {B}=\mathcal {B}(\ell ^2)\) the algebra of bounded operators. Let \(\mathrm {Emb}\) be the inverse monoid of all partially defined injections
Each element \(f\in \mathrm {Emb}\) defines a partial isometry \(U_f\in \mathcal {B}\); for the canonical Hilbert basis we have \(U_f(e_n)=e_{f(n)}\) if \(n\in \text{ dom }f\) and \(0\) otherwise. Similarly, each bounded sequence of complex numbers \(\alpha \in \ell ^\infty \) defines an element \(\mathrm {diag}(\alpha )\in \mathcal {B}\) by \(\mathrm {diag}(\alpha )(e_n)=\alpha (n)e_n\). The subspace generated by all the \(U_f\) and \(\mathrm {diag}(\alpha )\) with \(f\in \mathrm {Emb}\) and \(\alpha \in \ell ^\infty \) is the subalgebra
In this article we show that the algebra \(\Gamma ^\infty \) has several remarkable properties. One of them is that the lattice of twosided ideals of \(\Gamma ^\infty \) is isomorphic to the lattice of twosided ideals of \(\mathcal {B}\). A theorem of Calkin [2], as restated by Garling [16], establishes a onetoone correspondence between twosided ideals of \(\mathcal {B}\) and the ideals of \(\ell ^\infty \) that are symmetric, that is, invariant under the action of \(\mathrm {Emb}\). Calkin’s correspondence maps a symmetric ideal \(S\vartriangleleft \ell ^\infty \) to the ideal \(J_S\) of those operators whose sequence of singular values belongs to \(S\). Consider the subspace
Note that \(I_{\ell ^\infty }=\Gamma ^\infty \); for all symmetric ideals \(S\), \(I_S\vartriangleleft \Gamma ^\infty \) is a twosided ideal. We prove (see Theorem 4.2)
Theorem 1.1
The map \(J\mapsto J\cap \Gamma ^\infty \) is a bijection between the sets of twosided ideals of \(\mathcal {B}(\ell ^2(\mathbb {N}))\) and \(\Gamma ^\infty \). If \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal, then \(J_S\cap \Gamma ^\infty =I_S\).
More generally, we define for any bornological algebra \(\mathfrak {A}\) (in particular for a Banach algebra \(\mathfrak {A}\)) an algebra \(\Gamma ^\infty (\mathfrak {A})\). The algebra \(\Gamma ^\infty (\mathfrak {A})\) contains an ideal \(I_{S(\mathfrak {A})}\) for any symmetric ideal \(S\vartriangleleft \ell ^\infty \), and \(S\mapsto I_{S(\mathfrak {A})}\) is a lattice homomorphism. Thus the smallest nonzero \(I_{S(\mathfrak {A})}\) occurs when \(S\) is the symmetric ideal \(c_f\vartriangleleft \ell ^\infty \) of finitely supported sequences; we get
Hence the inclusion \(\mathfrak {A}\rightarrow M_\infty \mathfrak {A}\) into the upper left corner gives a stability homomorphism
If \(\mathfrak {A}\) is unital then \(\iota _{c_f}\) induces an isomorphism in algebraic \(K\)theory, by matrix stability. At the other extreme, \(I_{\ell ^\infty (\mathfrak {A})}=\Gamma ^\infty (\mathfrak {A})\) is a ring with infinite sums in the sense of [22] (see Proposition 5.1); this permits the Eilenberg swindle and we have
For \(c_f\subsetneq S\subsetneq \ell ^\infty \), the \(K\)theory of \(I_{S(\mathfrak {A})}\) is more interesting. We study it for
Here \(c_0\) is the ideal of sequences vanishing at infinity, \(\ell ^q\) consists of the \(q\)summable sequences, and
Let \(\mathrm {BAlg}\) be the category of bornological algebras. We consider several variants of \(K\)theory. We write \(K\) for algebraic \(K\)theory, \(\textit{KH}\) for Weibel’s homotopy algebraic \(K\)theory and \(K^{\mathrm {top}}\) for topological \(K\)theory. A bornological algebra is a bornolocal \(C^*\)algebra if it is a filtering union of \(C^*\)algebras. The following result follows from Theorem 8.2.
Theorem 1.2

i)
The functor \(\mathrm {BAlg}\rightarrow \mathfrak {Ab}\), \(\mathfrak {A}\mapsto \textit{KH}_*(I_{c_0(\mathfrak {A})})\) is invariant under continuous homotopy.

ii)
If \(\mathfrak {A}\) is a bornolocal \(C^*\)algebra and \(n\ge 0\), then there is a natural split monomorphism

iii)
If \(n\le 0\), then the comparison map
$$\begin{aligned} K_n(I_{c_0(\mathfrak {A})})\rightarrow \textit{KH}_n(I_{c_0(\mathfrak {A})}) \end{aligned}$$(1.2)is an isomorphism for every \(\mathfrak {A}\in \mathrm {BAlg}\).
The results above should be compared with Karoubi’s conjecture (Suslin–Wodzicki’s theorem [21, Theorem 10.9]) that for a \(C^*\)algebra \(\mathfrak {A}\), the comparison map
is an isomorphism. Hence we may think of \(\mathfrak {A}\rightarrow I_{c_0(\mathfrak {A})}\) as a smaller version of the stabilization \(\mathfrak {A}\mapsto \mathfrak {A}\overset{\sim }{\otimes }\mathcal {K}\) whose homotopy algebraic \(K\)theory is continuously homotopy invariant and contains \(K_*^{\mathrm {top}}(\mathfrak {A})\) as a direct summand. Next let \(p\ge 1\) and consider the Schatten ideal \(\mathcal {L}^p\vartriangleleft \mathcal {B}\). Notice that \(\mathcal {L}^p\) is the ideal corresponding to \(\ell ^p\) under Calkin’s correspondence. We have
Recall from [9, Theorem 6.2.1] that if \(\mathfrak {A}\) is a locally convex algebra and \(\mathfrak {A}\hat{\otimes }\mathcal {L}^p\) is the projective tensor product then
In the present article (Theorem 8.1) we prove the following analogue of the latter result.
Theorem 1.3
Let \(S\) be one of \(\ell ^p\), \(\ell ^{p+}\) (\(0<p<\infty \)) or \(\ell ^{p}\) (\(0<p\le \infty \)).

i)
The functor \(\mathrm {BAlg}\rightarrow \mathfrak {Ab}\), \(\mathfrak {A}\mapsto \textit{KH}_*(I_{\ell ^1(\mathfrak {A})})\) is invariant under Höldercontinuous homotopies and we have \(\textit{KH}_*(I_{S(\mathfrak {A})})=\textit{KH}_*(I_{\ell ^1(\mathfrak {A})})\) for all \(S\) as above.

ii)
If \(\mathfrak {A}\) is a local Banach algebra and \(n\ge 0\), then there is a natural split monomorphism

iii)
If \(n\le 0\), then the comparison map
$$\begin{aligned} K_n(I_{S(\mathfrak {A})})\rightarrow \textit{KH}_n(I_{S(\mathfrak {A})}) \end{aligned}$$(1.3)is an isomorphism for every \(\mathfrak {A}\in \mathrm {BAlg}\).
Both these theorems rely on a homotopy invariance theorem (Theorem 7.8) which we think is of independent interest. The theorem says that if \(F:\mathbb {C}\mathrm {Alg}\rightarrow \mathfrak {Ab}\) is an \(M_2\)stable, split exact functor and \(S\in \{c_0,\ell ^p\}\), then the functor
is homotopy invariant. For \(S=c_0\) it is continuous homotopy invariant, while for \(S=\ell ^p\) it is invariant under Hölder continuous homotopies, with Hölder exponent depending on \(p\). For \(F=\textit{KH}_*\) we have \(\textit{KH}_*(I_{\ell ^p(\mathfrak {A})})=\textit{KH}_*(I_{\ell ^1(\mathfrak {A})})\), and so it is invariant under arbitrary Hölder continuous homotopies. Furthermore, we have the following general result (see Theorem 8.5) about the comparison map \(K\rightarrow \textit{KH}\). Its proof uses the homotopy invariance theorem mentioned above applied to infinitesimal \(K\)theory.
Theorem 1.4
Let \(\mathfrak {A}\) be a bornological algebra and let \(S\) be \(c_0\), \(\ell ^p\), \(\ell ^{p+}\) (\(0<p<\infty \)) or \(\ell ^{p}\) (\(0<p\le \infty \)). Then there are long exact sequences (\(n\in \mathbb {Z}\))
and
It is shown in the companion paper [6] that \(\textit{HC}_*(\Gamma ^\infty (\mathfrak {A}):I_{S(\mathfrak {A})})=0\) when either \(S=c_0\) and \(\mathfrak {A}\) is a C*algebra or \(S=\ell ^\infty \) and \(\mathfrak {A}\) is a unital Banach algebra. Therefore, the comparison map \(K_*(I_{S(\mathfrak {A})})\longrightarrow \textit{KH}_*(I_{S(\mathfrak {A})})\) is an isomorphism in these cases. In addition, the groups \(\textit{HC}_n(\Gamma ^\infty :I_S)\) are computed in [6] for all \(S\), and it is shown that for \(S\in \{\ell ^p,\ell ^{p\pm }\}\) the map \(\textit{HC}_n(\Gamma ^\infty :I_S)\longrightarrow \textit{HC}_n(\mathcal {B}:J_S)\) is an isomorphism for those values of \(n\) for which \(\textit{HC}_n(\mathcal {B}:J_S)\) was computed by Wodzicki [24]. In summary, the ideals \(I_S\vartriangleleft \Gamma ^\infty \) and \(J_S\vartriangleleft \mathcal {B}\) and their corresponding stable algebras have very similar properties in what \(K\)theory and cyclic homology are concerned, and the cyclic homology of the former seems to be easier to describe.
We expect that these results will help shed light on the various characters and regulators which take values in the relative \(K\)theory and cyclic homology of operator ideals [9, 24]. This was our original motivation to study the ideals \(I_S\).
The rest of this paper is organized as follows. In Sect. 2 we establish some notation about sequence spaces, the inverse monoid \(\mathrm {Emb}\) and the partial isometries \(U_f\). The algebra \(\Gamma ^\infty (\mathfrak {A})\) and the ideals \(I_{S(\mathfrak {A})}\) are introduced in Sect. 3. In this section we also recall the definition of Karoubi’s cone \(\Gamma (R)\) which is \(R\)linearly generated by the \(U_f\) (\(f\in \mathrm {Emb}\)). Proposition 3.5 identifies \(I_{S(\mathfrak {A})}\) with a ring formed by certain \(\mathbb {N}\times \mathbb {N}\) matrices with coefficients in \(\mathfrak {A}\). The twosided ideals of \(\Gamma ^\infty \) are studied in Sect. 4; Theorem 1.1 is contained in Theorem 4.2. We prove in Sect. 5 that if \(\mathfrak {A}\) is unital, then \(\Gamma ^\infty (\mathfrak {A})\) is a ring with infinite sums in the sense of Wagoner (Proposition 5.1). In Sect. 6 we show that \(I_{S(\mathfrak {A})}\) can be written as a crossed product of \(\Gamma =\Gamma (\mathbb {Z})\) and \(S(\mathfrak {A})\), by using the conjugation action of \(\mathrm {Emb}\) in \(S(\mathfrak {A})\) via the partial isometries \(U_f\) (Proposition 6.4). Section 7 deals with the homotopy invariance theorem mentioned above, proved in Theorem 7.8. Applications to \(K\)theory are given in Sect. 8; see Theorems 8.1, 8.2 and 8.5.
2 Preliminaries
2.1 Sequence ideals
Throughout this paper we work in the setting of bornological spaces and bornological algebras; a quick introduction to the subject is given in [12, Chapter 2]. Recall a (complete, convex) bornological vector space over the field \(\mathbb {C}\) of complex numbers is a filtering union \(\mathbb {V}=\cup _D\mathbb {V}_D\) of Banach spaces, indexed by the disks of \(\mathbb {V}\) such that the inclusions \(\mathbb {V}_D\subset \mathbb {V}_{D'}\) are bounded. A subset of \(\mathbb {V}\) is bounded if it is a bounded subset of some \(\mathbb {V}_D\). A sequence \(\mathbb {N}\rightarrow \mathbb {V}\) is bounded if its image is a bounded subset of \(\mathbb {V}\). We write \(\ell ^\infty (\mathbb {N},\mathbb {V})\) or simply \(\ell ^\infty (\mathbb {V})\) for the bornological vector space of bounded sequences where \(X\subset \ell ^\infty (\mathbb {V})\) is bounded if \(\bigcup _{x\in X}x(\mathbb {N})\) is. We consider the following subspace
We equip \(c_0(\mathbb {V})\) with the bornology induced by that of \(\ell ^\infty (\mathbb {V})\); thus \(c_0(\mathbb {V})\subset \ell ^\infty (\mathbb {V})\) is a closed bornological subspace. We also consider the subspace (\(p>0\))
If \(p\ge 1\), we equip \(\ell ^p(\mathbb {V})\) with the following bornology: we say that a subset \(S\subset \ell ^p(\mathbb {V})\) is bounded if there is a disk \(D\) and a constant \(C\) such that \(\sum _n\alpha (n)_D^p<C\) for all \(\alpha \in S\). Notice that the inclusion \(\ell ^p(\mathbb {V})\rightarrow \ell ^\infty (\mathbb {V})\) is bounded for \(p\ge 1\). Recall a bornological algebra is a bornological vector space \(\mathfrak {A}\) with an associative bounded multiplication. If \(\mathfrak {A}\) is a bornological algebra, then pointwise multiplication makes \(\ell ^\infty (\mathfrak {A})\) into a bornological algebra, \(c_0(\mathfrak {A})\vartriangleleft \ell ^\infty (\mathfrak {A})\) is a closed bornological ideal, and \(\ell ^p(\mathfrak {A})\vartriangleleft \ell ^\infty (\mathfrak {A})\) is an algebraic ideal for all \(p>0\).
Notation 2.1
When \(\mathfrak {A}\) is \(\mathbb {C}\), we shall omit it from our notation. Thus we shall write \(\ell ^\infty \), \(\ell ^p\), \(c_0\), etc, for \(\ell ^\infty (\mathbb {C})\), \(\ell ^p(\mathbb {C})\), \(c_0(\mathbb {C})\), etc.
The space \(\mathcal {B}(\ell ^2(\mathbb {V}))\) of bounded operators \(\ell ^2(\mathbb {V})\rightarrow \ell ^2(\mathbb {V})\) on a bornological vector space \(\mathbb {V}\) is a bornological algebra with the uniform bornology [12, Def. 2.4]. If \(\mathfrak {A}\) is a bornological algebra, then
is a bounded representation. It is faithful if and only if the left annihilator of \(\mathfrak {A}\) is trivial:
This happens, for instance, when \(\mathfrak {A}\) is unital.
2.2 The monoid \(\mathrm {Emb}\)
We begin by recalling some definitions from [14]. We denote by \(\mathrm {Emb}\) the set of injective functions
Note that \(\mathrm {Emb}\) is a monoid for the composition law:
In (2.3) and elsewhere, we shall omit the composition sign \(\circ \), except when strictly necessary to avoid confusion. The monoid \(\mathrm {Emb}\) is pointed, i.e. it has a zero element, namely, the empty function \(\emptyset \rightarrow \mathbb {N}\). The antipode map \(^\dagger : \mathrm {Emb}\rightarrow \mathrm {Emb}\) is defined by
If \(A\subset \mathbb {N}\), we write \(P_A\) for the inclusion \(A\hookrightarrow \mathbb {N}\). It is easily checked that
for any \(f\in \) \(\mathrm {Emb}\). Observe that \(f^\dagger \) is characterized as the unique element of \(\mathrm {Emb}\) which satisfies simultaneously
Thus the monoid \(\mathrm {Emb}\) together with its antipode is a pointed inverse monoid that is, a pointed inverse semigroup with identity element. Note that \(\mathrm {Emb}\) is the object usually denoted \(\mathcal {I}(\mathbb {N})\) in the literature on semigroups (see [15, Def. 4.2], for instance).
If \(\mathbb {V}\) is a bornological vector space, the monoid \(\mathrm {Emb}\) acts on \(\ell ^\infty (\mathbb {V})\) via:
The subspaces \(c_0(\mathbb {V})\) and \(\ell ^p(\mathbb {V})\) defined in 2.1 are symmetric, i.e. they are invariant under the action of \(\mathrm {Emb}\). Indeed, this follows from the fact that \(c_0\) and \(\ell ^p\) are symmetric, and that if \(D\) is a bounded disk and the image of \(\alpha \) is contained in \(\mathbb {V}_D\), then the following sequences of real numbers are identical
More generally, if \(S\subset \ell ^\infty \) is any symmetric subspace, then
is symmetric. We denote by \(U\) the representation of \(\mathrm {Emb}\) by partial isometries on \(\ell ^2(\mathbb {V})\):
Straightforward computations show that
Observe that \(U_f\) is a partial isometry whose initial and final space are, respectively, the closed subspaces
This follows from (2.4), (2.7), and from the fact that if \(A\subset \mathbb {N}\), then
Remark 2.2
We will often work with sequences indexed by infinite countable sets other than \(\mathbb {N}\). A bijection \(u:\mathbb {N}\rightarrow X\) gives rise to a bounded isomorphism \(\alpha \mapsto \alpha u\) between the bornological vector space \(\ell ^\infty (X,\mathbb {V})\) of bounded maps from \(X\) to the bornological space \(\mathbb {V}\) and the space \(\ell ^\infty (\mathbb {V})=\ell ^\infty (\mathbb {N},\mathbb {V})\). If \(S\subset \ell ^\infty \) is a symmetric subspace, we define \(S(X,\mathbb {V})=\{s u^{1}:s\in S(\mathbb {V})\}\). Because \(S\) is symmetric by assumption, this definition does not depend on the choice of \(u\). If \(A\in M_{\mathbb {N}\times \mathbb {N}}(\mathbb {V})\), we will write \(A\in S({\mathbb {N}\times \mathbb {N}},\mathbb {V})\) to indicate that \(\{A_{ij}:i,j\in \mathbb {N}\}\in S({\mathbb {N}\times \mathbb {N}},\mathbb {V})\).
Notation 2.3
Let \(S\subset \ell ^\infty \) be a symmetric subspace, \(X\) an infinite countable set and \(\mathbb {V}\) a bornological vector space. We use the following abbreviated notation: \(S=S(\mathbb {N},\mathbb {C})\), \(S(X)=S(X,\mathbb {C})\) and \(S(\mathbb {V})=S(\mathbb {N},\mathbb {V})\).
3 The algebras \(\Gamma ^\infty (\mathfrak {A})\) and \(\Gamma (R)\)
Throughout this section, \(\mathfrak {A}\) will be a fixed bornological algebra, which, except in Definition 3.7, will be assumed unital. It follows straightforwardly from equations (2.2), (2.5), and (2.6) that
where \(\alpha \in \ell ^\infty (\mathfrak {A})\) and \(f\in \mathrm {Emb}\). Set
Notice that, by Eqs. (2.7) and (3.1), \(\Gamma ^\infty (\mathfrak {A})\) is a subalgebra of the algebra \(\mathcal {B}(\ell ^2(\mathfrak {A}))\). For each symmetric ideal \(S\vartriangleleft \ell ^\infty \), we write \(I_{S(\mathfrak {A})}\) for the ideal of \(\Gamma ^\infty (\mathfrak {A})\) generated by \(\mathrm {diag}(S(\mathfrak {A}))\). Because \(S\) is invariant under the action of \(\mathrm {Emb}\), then by equations (3.1) we have
Note that \(\Gamma ^\infty (\mathfrak {A})=I_{\ell ^\infty (\mathfrak {A})}\). If \(X\) is any infinite countable set, we may also consider the subalgebra \(\Gamma ^\infty (X,\mathfrak {A})\subset \mathcal {B}(\ell ^2(X,\mathfrak {A}))\) spanned by \(\mathrm {diag}(\ell ^\infty (X,\mathfrak {A}))\) and \(U_{\mathrm {Emb}(X)}\). Thus \(\Gamma ^\infty (\mathfrak {A})=\Gamma ^\infty (\mathbb {N},\mathfrak {A})\). In keeping with our notational conventions 2.1 and 2.3, we write \(\Gamma ^\infty =\Gamma ^\infty (\mathbb {C})\) and \(\Gamma ^\infty (X)=\Gamma ^\infty (X,\mathbb {C})\).
Notation 3.1
Since \(\mathfrak {A}\) is assumed to be unital, every sequence \(a=\{a_n\}\) in \(\ell ^2(\mathfrak {A})\) can be written uniquely as \(a=\sum _n a_ne_n\), where \(e_n\in \ell ^2(\mathfrak {A})\) is defined by \((e_n)_i=\delta _{n,i}\). Notice that the elements of \(\Gamma ^\infty (\mathfrak {A})\) are \(\mathfrak {A}\)linear operators on the right \(\mathfrak {A}\)module \(\ell ^2(\mathfrak {A})\). As usual, we identify an \(\mathfrak {A}\)linear operator \(A\in \mathcal {B}(\ell ^2(\mathfrak {A}))\) with the infinite matrix \((A_{ij})_{i,j\in \mathbb {N}}\) with entries in \(\mathfrak {A}\) defined by
We denote by \(E_{ij}\) the matrix \((E_{ij})_{kl}=\delta _{ik}\delta _{jl}\). Given a matrix \(A=(A_{ij})_{i,j\in \mathbb {N}}\) with entries in \(\mathfrak {A}\), and \(i,j\in \mathbb {N}\), we set:
where \(r_i(A),c_j(A),N(A)\in \mathbb {N}\cup \{\infty \}\). If \(R\) is a ring, we write \(\Gamma (R)\) for Karoubi’s cone
It was shown in [8, Lemma 4.7.1] that \(\Gamma (R)\) is isomorphic to \(R\otimes \Gamma (\mathbb {Z})\), for any ring \(R\). We shall write
Observe that definition (3.4) extends to matrices indexed by any countable infinite set \(X\); if \(f:\mathbb {N}\rightarrow X\) is a bijection, \(\Gamma (X,R)\subset R^{X\times X}\) is the image of \(\Gamma (R)\) under the map \(A\mapsto U_f AU_{f^{1}}\). Thus \(\Gamma (R)=\Gamma (\mathbb {N},R)\); we shall write \(\Gamma (X)=\Gamma (X,\mathbb {Z})\).
The following lemmas will be useful in obtaining characterizations of \(\Gamma ^\infty (\mathfrak {A})\), \(I_{S(\mathfrak {A})}\) and \(\Gamma (R)\) as rings of matrices acting on \(\ell ^2(\mathfrak {A})\) and \(R^{(\mathbb {N})}\), respectively. If \(A\in R^{\mathbb {N}\times \mathbb {N}}\) is such that \(N(A)<\infty \), we write \(\Gamma (R)A\Gamma (R)\) to denote the set
Lemma 3.2
Let \(R\) be a unital ring, \(A=(A_{ij})_{i,j\in \mathbb {N}}\in R^{\mathbb {N}\times \mathbb {N}}\) a matrix such that \(N(A)<\infty \) and \(r(A)>1\). Then

(1)
\(A=A_1+A_2+\cdots +A_k,\) where \(A_i\in \Gamma (R)A\Gamma (R)\), \(r(A_i)< r(A)\) and \(c(A_i)\le c(A)\) for all \(i=1,\ldots ,k\).

(2)
If in addition \(R\) is a unital bornological algebra and \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal such that \(A\in S(\mathbb {N}\times \mathbb {N},R)\), then \(A_l\in S(\mathbb {N}\times \mathbb {N},R)\), for all \(l=1,\ldots , k\).
Proof
(1) We first establish some notation and make some reductions. Let
For \(i\in I\), let
be the columns where the nonzero entries of row \(i\) occur. Let \(A_r\) denote the matrix obtained from \(A\) upon multiplying by zero those rows that have less than \(r\) nonzero entries. Then \(A_r\in \Gamma (R)A\Gamma (R)\), and
Thus it suffices to prove (1) for \(A_r\). Hence we may assume that \(A=A_r\), that is, that all nonzero rows of \(A\) have exactly \(r\) nonzero entries. Furthermore, since there are at most \(c(A)\) nonzero entries in each column of \(A\), the set \(I\) can be written as a disjoint union \(I=I_1\sqcup I_2\sqcup \cdots \sqcup I_s\) with \(s\le c(A)\) and such that each \(I_t\) (\(1\le t\le s\)) satisfies the following property:
Proceeding as above we see that we may assume that \(s=1\). Notice that if \(A'\) is obtained from \(A\) by permuting its rows, then \(A'=U_fA\) for some bijection \(f:\mathbb {N}\rightarrow \mathbb {N}\). Therefore, \( \Gamma (R)A\Gamma (R)=\Gamma (R)A'\Gamma (R)\), \(r(A')=r(A)\), and \(c(A')=c(A)\), so we may assume that \(A=A'\). Thus we will assume that the rows of \(A\) are ordered so that if \(i,j\in I\), then \(h_i(1)<h_j(1)\) if and only if \(i<j\).
Thus, it only remains to show (1) for matrices \(A\) such that for \(I\) and \(h_i\) as above:
We shall proceed by induction on
Notice that the righthand side of the equation above is bounded by \(c(A)\), so \(M_A\in \mathbb {N}\). First assume that \(M_A=1\). Then for all \(i,j\in I\) we have that \(A_{ih_j(1)}\ne 0\) if and only if \(i=j\). Set
Then
so the statement in (1) holds for \(A\). Assume now that \(M_A>1\) and that (1) holds for matrices \(B\) satisfying 3.5 and 3.6, and such that \(M_B<M_A\). Let
For \( n\ge 1\) such that \( \bigcup _{j=1}^{n1}K_j\ne I\), let
Let
We claim that
In fact a) follows from the inequality
and the fact that \(i_n\ne i_{n1}\) because \(i_{n}\not \in K_{n1}\) and \(i_{n1}\in K_{n1}\). It is clear that b) holds when \(\mathcal {J}\) is finite. Assume now that \(\mathcal {J}\) infinite. If \(k\in I\), then either \(k\in \{i_n:n\in \mathcal {J}\}\subset \bigcup K_j\) or, by a), there exists \(n\in \mathcal {J}\) such that
This implies that \(k\in \bigcup _1^{n1}K_j\). Thus b) holds also when \(\mathcal {J}\) is infinite, and both claims are proven. Now set
Notice that \(B\) is obtained from \(A\) by multiplying by zero the \(i^{th}\) row whenever \(i\not \in \{i_n:n\in \mathcal {J}\}\). Therefore \(B\) satisfies 3.5 and 3.6, \(r(B)=r\), and \(c(B)\le c(A)\). We next show that \(M_B=1\). We begin by noting that \(B_{i_mi_n(1)}\ne 0\) implies that \(A_{i_mi_n(1)}\ne 0\). Then \(i_n(1)\ge i_m(1)\), which implies by 3.6 that \(i_n\ge i_m\), which in turn implies, by part a) of Eq. (3.7), that \(n\ge m\). Now, if \(n>m\) we would have
Then \(i_n\not \in K_m\) and \(i_n\not \in \bigcup _1^{m1}K_j\), which implies that \(A _{i_mi_n(1)}=0\), a contradiction. Thus \(n=m\) and \(M_B=1\), as claimed. Set \(C=AB\); we have \(r(C)=r\) and \(c(C)\le c(A)\). Notice that \(C\) is obtained from \(A\) upon multiplying by zero the \(i_n^{th}\) row for all \(n\in \mathcal {J}\). Besides, the \(i^{th}\) row of \(C\) is nonzero if and only if \(i\in I_C:=I{\setminus }\{i_n:n\in \mathcal {J}\}\), and in that case it is equal to the \(i^{th}\) row of \(A\). Therefore, \(C\) satisfies 3.5 and 3.6. We next prove that \(M_C<M_A\), which will conclude the proof of part (1). If \(i, j\in I_C\), then \(A_{ih_j(1)} = 0\) implies that \(C_{ih_j(1)} = 0\). On the other hand, by part b) of Eq. (3.7), we can choose \(n\in \mathcal {J}\) such that \(j\in K_n\). Then \(A_{i_nh_j(1)}\ne 0\), whereas \(C_{i_nh_j(1)}=0\). It follows that \(M_C\le M_A1\). This concludes the proof of part (1). Part (2) holds because for \(l=1,\ldots ,k\), \(\{(A_l)_{ij}\}\) is obtained upon multiplication of \(\{A_{ij}\}\) by bounded sequences and by permutations of terms. \(\square \)
Lemma 3.3
Let \(A=(A_{ij})_{i,j\in \mathbb {N}}\) be a matrix with entries in a unital ring \(R\) such that \(N(A)<\infty \). Then

(1)
\(A=A_1+A_2+\cdots + A_k,\) where \(A_i\in \Gamma (R)A\Gamma (R)\), and \(N(A_i)\le 1\), for all \(i=1,\ldots ,k\).

(2)
If in addition \(R\) is a bornological algebra and \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal such that \(A\in S(\mathbb {N}\times \mathbb {N}, R)\), then \(A_l\in S(\mathbb {N}\times \mathbb {N},R)\), for all \(l=1,\ldots , k\).
Proof
Use Lemma 3.2 and proceed by induction on \(r(A)\) to write
Next apply the same procedure to each transpose matrix \(B_i^t\) to get the decomposition in (1). The second statement follows from the second part of Lemma 3.2. \(\square \)
Proposition 3.4
Let \(A=(A_{ij})_{i,j\in \mathbb {N}}\) be a matrix with entries in a ring \(R\). Then \(N(A)\le 1\) if and only if \(A=\mathrm {diag}(\alpha )U_f\), where \(f\in \mathrm {Emb}\) and \(\alpha \in R^\mathbb {N}\) are defined as follows:
Proof
For \(f\) and \(\alpha \) as in the proposition, the \(n\hbox {th}\) column of \(A\) is
\(\square \)
Proposition 3.5
Let \(\mathfrak {A}\) be a unital bornological algebra, \(S\vartriangleleft \ell ^\infty \) a symmetric ideal, and \(I_{S(\mathfrak {A})}\vartriangleleft \Gamma ^\infty (\mathfrak {A})\) the ideal defined in Eq. (3.3). Then
Proof
Let \(D_S\) denote the set on the right hand side of equation (3.8). By Lemma 3.3 and Proposition 3.4, a matrix \(A\) belongs to \(D_S\) if and only if \(A=\sum A_k\), with \(A_k=\mathrm {diag}(\alpha _k)U_{f_k}\in D_S\). Further, we may choose \(\alpha _k\) and \(f_k\) such that \(\text{ supp }(\alpha _k)=\text{ ran }(f_k)\). Under these conditions, \(A_k\in D_S\) if and only if \(\alpha _k\in S\). This shows that \(A\in D_S\) if and only \(A\in I_S\). \(\square \)
Corollary 3.6
Let \(\mathfrak {A}\) be a unital bornological algebra. Then Karoubi’s cone \(\Gamma (\mathfrak {A})\) is a subalgebra of \(\Gamma ^\infty (\mathfrak {A})\).
Definition 3.7
If \(\mathfrak {A}\) is a not necessarily unital bornological algebra, and \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal, \(I_{S(\mathfrak {A})}\) is defined by (3.8).
Example 3.8
Let
Then
We shall write \(M_\infty =M_\infty \mathbb {Z}\).
Remark 3.9
Let \(\mathfrak {A}\) be a unital bornological algebra, \(I\vartriangleleft \Gamma ^\infty (\mathfrak {A})\) a twosided ideal and \(T\in I\). Then by Lemma 3.3 and Remark 3.4, we can write
where \(f_i\in \mathrm {Emb}\) and \(\alpha _i\in \ell ^\infty (\mathfrak {A})\). Similarly, if \(R\) is a unital ring and \(T\in I\vartriangleleft \Gamma (R)\), then we can also write \(T\) as in (3.9) but now with \(\alpha _i\) such that the set \(\{\alpha _i(n):n\in \mathbb {N}\}\subset R \) is finite.
4 The twosided ideals of \(\Gamma ^\infty \) and those of \(\mathcal {B}(\ell ^2(\mathbb {N}))\)
Calkin’s theorem [2, Theorem 1.6]), as restated by Garling [16, Theorem 1], establishes a bijective correspondence between the set of proper twosided ideals of \(\mathcal {B}=\mathcal {B}(\ell ^2)\) and the set of proper symmetric ideals of \(\ell ^\infty \). Calkin defined this correspondence in terms of the sequence of singular values of a compact operator. It can also be described as follows: an ideal \(J\vartriangleleft \mathcal {B}\) is mapped to the symmetric ideal
The inverse correspondence maps a symmetric ideal \(S\) in \(\ell ^\infty \) to the twosided ideal
We refer the reader to [20, Theorem 2.5] for further details. Recall that, by another result of Calkin [2, Theorem 1.4], the Calkin algebra \(\mathcal {B}/\mathcal {K}\) is simple. On the other hand, it is easily checked that \(c_0\vartriangleleft \ell ^\infty \) is maximal among proper symmetric ideals. Thus, by mapping \(\ell ^\infty \) to \(\mathcal {B}\) we extend the correspondence above to a bijection between the family of symmetric ideals of \(\ell ^\infty \) and that of twosided ideals of \(\mathcal {B}\). In Theorem 4.2 below we show that Calkin’s correspondence carries over to ideals in \(\Gamma ^\infty \). We will make use of the following lemma.
Lemma 4.1
Let \(\alpha \in \ell ^\infty \), \(f\in \mathrm {Emb}\) and let \(I\vartriangleleft \Gamma ^\infty \) a twosided ideal. Consider the operator
Then
Proof
We have
Therefore, \(T=\mathrm {diag}(f^\dagger _*(\alpha ))\), and the polar decomposition of \(T \) is \(T=VT\), where
for
It is now clear that \(V\in \Gamma ^\infty \). Thus \(T\in I\) if and only if \(T\in I\), since \(\Gamma ^\infty \) is a \(*\)algebra and \(T=V^*T\). \(\square \)
Theorem 4.2

i)
The map \(S\mapsto I_S\) is a bijection between the set of symmetric ideals of \(\ell ^\infty \) and the set of twosided ideals of \(\Gamma ^\infty \). Its inverse maps an ideal \(I\vartriangleleft \Gamma ^\infty \) to the symmetric ideal \(S(I)\) defined as in (4.1).

ii)
The map \(J\mapsto J\cap \Gamma ^\infty \) is a bijection between the sets of twosided ideals of \(\mathcal {B}\) and those of \(\Gamma ^\infty \). Its inverse maps an ideal \(I\vartriangleleft \Gamma ^\infty \) to the twosided ideal of \(\mathcal {B}\) it generates.

iii)
If \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal, then \(J_S\cap \Gamma ^\infty =I_S\).
Proof
Let \(I\vartriangleleft \Gamma ^\infty \); write \(S=S(I)\). It is clear that \(I_S\subseteq I\). On the other hand, if \(T=\mathrm {diag}(\alpha )U_f\in I\), for some \(\alpha \in \ell ^\infty \) and \(f\in \mathrm {Emb}\), then, by Lemma 4.1,
Hence \(T\in I_S\), again by Lemma 4.1. In view of Remark 3.9, this implies that \(I=I_S\). We have shown that \(I_{S(I)}=I\). Let now \(S\vartriangleleft \ell ^\infty \) be a symmetric ideal. Then
the last inclusion being due to Calkin’s theorem. It follows that \(S=S(I_S)\), completing the proof of part i). Next, since the ideal \(\langle I_S\rangle \vartriangleleft \mathcal {B}(\ell ^2)\) generated by \(I_S\) is also generated by \(\mathrm {diag}(S)\) we have \(\langle I_S\rangle =J_S\), by Calkin’s theorem. Now, again by Calkin’s theorem,
Thus \(J_S\cap \Gamma ^\infty =I_S\), by part i). We have proven part iii) and also shown that \(\langle I_S\rangle \cap \Gamma ^\infty =I_S\). Moreover, by parts i) and iii) we have
It follows that \(\langle J_S\cap \Gamma ^\infty \rangle =J_S\), which ends the proof. \(\square \)
It follows from Proposition 3.5, Example 3.8 and Theorem 4.2 that
for every proper ideal \(I\vartriangleleft \Gamma ^\infty \). The next proposition shows that in fact \(M_\infty (\mathbb {C})\) is the only proper ideal of \(\Gamma (\mathbb {C})\).
Proposition 4.3
Let \(k\) be a field. Then \(M_\infty (k)\) is the only proper twosided ideal of \(\Gamma (k)\).
Proof
It is well known and easy to check that \(M_\infty (R)\vartriangleleft \Gamma (R)\) for any ring \(R\). Let \(I\ne 0\) be a twosided ideal of \(\Gamma (k)\), and let \(A\ne 0,\ A\in I\). If \(i_0\) and \(j_0\) are such that \(A_{i_0j_0}\ne 0\), then
This shows that \(M_\infty (k)\subseteq I\). Assume that the inclusion is strict. Let \(A\in I{\setminus } M_\infty (k)\). By Remark (3.9), we may assume that \(A=\mathrm {diag}(\alpha )U_f\) for \(f\in \mathrm {Emb}\) and \(\alpha \in k^\mathbb {N}\), where \(\text{ Im }(\alpha )= \{\alpha (n):n\in \mathbb {N}\}\) is finite and \(\text{ supp }(\alpha )=\text{ dom }f\subset \mathbb {N}\) is infinite. Because \(k\) is a field, we can multiply \(A\) on the left by a diagonal matrix in \(\Gamma (k)\) to conclude that \(U_f\in I\). But since \(\text{ ran }(f)\) is infinite, there are bijections \(g:\mathbb {N}\rightarrow \text{ dom }(f)\) and \(h:\text{ ran }(f)\rightarrow \mathbb {N}\) such that \(hfg=1\). Hence \(I\) must contain \(1=U_hU_fU_g\). \(\square \)
5 \(\Gamma ^\infty \) as an infinite sum ring
We begin this section by recalling some definitions from [22] and [8]. A sum ring \((R,x_0,x_1,y_0,y_1)\) consists of a unital ring \(R\) and elements \(x_0,x_1,y_0,\) and \(y_1\in R\) satisfying:
If \(R\) is a sum ring, the map
is a unital ring homomorphism. An infinite sum ring consists of a sum ring \(R\) equipped with a unital ring homomorphism
The notion of infinite sum ring was introduced by Wagoner [22]. He showed that if \(R\) is unital, then the following is an infinite sum ring:
We may regard \(\Gamma ^W(R)\) as a multiplier algebra of \(M_\infty R\). One checks that a matrix \(A\in \Gamma ^W(R)\) if and only if every row and every column of \(A\) has finite support. Let
The elements \(x_i=U_{f^\dagger _i}\), \(y_i=U_{f_i}\) satisfy conditions (5.1). The homomorphism \(\Phi \) is defined by
This map is welldefined because \((k,i)\mapsto 2^{k+1}i+2^k1\) is onetoone; Wagoner showed in [22, pp. 355] that it satisfies (5.3). Observe that the \(x_i's\) and \(y_i's\) are elements of \(\Gamma (R)\). It is not hard to check, and noticed in [8, 4.8.2], that \(\Phi (\Gamma (R))\subset \Gamma (R)\), whence \(\Gamma (R)\) is an infinite sum ring too. Now we remark that if \(\mathfrak {A}\) is a bornological algebra, then
Furthermore, \(\Phi \) also sends \(\Gamma ^\infty (\mathfrak {A})\) to itself. Thus if \(\mathfrak {A}\) is unital, then \(\Gamma ^\infty (\mathfrak {A})\) is an infinite sum ring. We record this in the following proposition.
Proposition 5.1
Let \(\mathfrak {A}\) be a unital bornological algebra, and let \(f_i\) be as in (5.4) and \(\Phi \) as in (5.5) Then \((\Gamma ^\infty (\mathfrak {A}), U_{f^{\dagger }_0},U_{f^{\dagger }_1},U_{f_0},U_{f_1},\Phi )\) is an infinite sum ring.
Corollary 5.2
Let \(F:\mathbb {C}\mathrm {Alg}\rightarrow \mathfrak {Ab}\) be a functor. Assume that the restriction of \(F\) to unital \(\mathbb {C}\)algebras is splitexact and \(M_2\)stable. Then \(F(\Gamma ^\infty (\mathfrak {A}))=0\) for any unital bornological algebra \(\mathfrak {A}\). If furthermore \(F\) is split exact on all \(\mathbb {C}\)algebras, then \(F(\Gamma ^\infty (\mathfrak {A}))=0\) for any, not necessarily unital bornological algebra \(\mathfrak {A}\).
Proof
Immediate from Proposition 5.1 and [5, Proposition 2.3.1]. \(\square \)
Example 5.3
Both Weibel’s homotopy algebraic \(K\)theory [23] and periodic cyclic homology [13] are \(M_2\)stable and excisive on all \(\mathbb {Q}\)algebras. Hence if \(\mathfrak {A}\) is a bornological algebra, then
Algebraic \(K\)theory groups \(K_n\) are split exact and \(M_2\) stable for \(n\le 0\); the same is true of Karoubi–Villamayor \(K\)groups \(KV_m\) for \(m\ge 1\) [18, Théorème 4.5]. Hence,
again for all \(\mathfrak {A}\). For positive \(n\), the groups \(K_n\) are still split exact and \(M_2\)stable on unital rings. The same is true of both the Hochschild and cyclic homology groups \(\textit{HH}_n\) and \(\textit{HC}_n\) for \(n\ge 0\); moreover these groups vanish for \(n\le 1\). Hence we have
for any unital bornological algebra \(\mathfrak {A}\).
6 The algebra \(\Gamma ^\infty (\mathfrak {A})\) as a crossed product
Let \(2^\mathbb {N}\) denote the submonoid of idempotent elements of \(\mathrm {Emb}\)
Note that if \(p\in 2^\mathbb {N}\), then for \(A=\text{ ran }(p)=\text{ dom }(p)\), we have \(U_p=\mathrm {diag}(\chi _A)\), the diagonal matrix on the sequence
We will often identify \(p\), \(U_p=\mathrm {diag}(\chi _A)\), and \(\chi _A\). Notice that
The subgroup of \(\Gamma \) generated by the image of \(2^\mathbb {N}\) under \(f\mapsto U_f\) is the subring
We also consider the monoid rings \(\mathbb {Z}[2^\mathbb {N}]\) and \(\mathbb {Z}[\mathrm {Emb}]\), and the twosided ideals
Observe that \(I\) and \(J\) contain the element
for any pair of not necessarily disjoint subsets \(A,B\subset \mathbb {N}\).
Lemma 6.1

i)
\(\mathcal {P}=\mathbb {Z}[2^\mathbb {N}]/I\).

ii)
\(\Gamma =\mathbb {Z}[\mathrm {Emb}]/J\)

iii)
If \(\mathfrak {A}\) is a unital bornological algebra, then \(\ell ^\infty (\mathfrak {A})\otimes _\mathcal {P}\Gamma \cong \Gamma ^\infty (\mathfrak {A})\) as \(\mathcal {P}\)bimodules.
Proof
It is clear that there are natural surjective ring homomorphisms
and a natural surjective \(\mathcal {P}\)bimodule homomorphism
Let \(\xi =\sum _{j=1}^n \lambda _j\chi _{A_j}\in \mathbb {Z}[2^\mathbb {N}]\) represent an element \(\in \ker \pi _1\); for each subset \(F\subset \{1,\ldots ,n\}\), let \(A_F=\bigcap _{j\in F}A_j\cap \bigcap _{j\notin F}A_j^c\). From \(\pi _1(\xi )_{A_F}=0\) we get
Next note that \(\bigcup _{i=1}^nA_i=\sqcup _{F} A_F\); hence, modulo \(I\), we have
This proves i). In order to prove ii) we have to show that \(\ker (\pi _2)=0\). Let \(\xi =\sum _{j=1}^n\lambda _jf_j\in \mathbb {Z}[\mathrm {Emb}]\) be a representative of an element in \(\ker (\pi _2)\). Let \(A_i=\text{ dom }f_i\), and let \(A_F\) be as above; then \(\xi \equiv \sum _F\xi \chi _{A_F}\). Hence we may assume that the \(A_i\) are disjoint. Furthermore, upon replacing \(\xi \) by \(\xi \chi _{A_i}\), and eliminating zero elements of \(\mathrm {Emb}\), we may assume that \(A_1=\cdots =A_n\). For each \(j\in \mathbb {N}\), we have
Let \(K=\{f_i(j):i=1,\ldots ,n\}\); for each \(k\in K\), let \(D_k=\{i:f_i(j)=k\}\). Then \(D(j):=\{D_k\}_{k\in K}\) is a partition of \(\{1,\ldots ,n\}\), and \(\sum _{i\in D_k}\lambda _i=0\). There is a finite set \(\mathcal {D}\) of partitions arising in this way, since the number of all partitions of \(\{1,\ldots ,n\}\) is finite. For each \(D\in \mathcal {D}\), let \(J_D=\{j\in \mathbb {N}: D(j)=D\}\). Then \(\sqcup _{D\in \mathcal {D}}J_D=\mathbb {N}\), and \(\xi \equiv \sum _D \xi \cdot \chi _D\). Hence, upon replacing \(\xi \) by \(\xi \chi _D\) if necessary, we may assume that \(\mathcal {D}\) has only one element \(D=\{D_1,\ldots ,D_r\}\). But \(\xi \equiv \sum _i\chi _{D_i}\xi \), so we further reduce to the case when \(r=1\). This means that \(f_1=\cdots =f_n\) and, by (6.4), \(\sum _i\lambda _if_i\) is the zero element of \(\mathbb {Z}[\mathrm {Emb}]\). We have proved ii). To prove iii) we must show that \(\pi _3\) is injective. Let \(\xi =\sum _{i=1}^n\alpha _{(i)}\otimes U_{f_i}\in \ker \pi _3\). Because
we may assume that \(\text{ supp }(\alpha _i)=\text{ ran }(f_i)\) (\(i=1,\ldots ,n\)). Proceeding as above, we may assume that \(\text{ dom }f_1=\cdots =\text{ dom }f_n\). For each \(j\in \mathbb {N}\), we have
Proceeding as above again, we may reduce to the case \(f_1=\cdots =f_n\). By (6.5), we have \(\sum _{i=1}^n\alpha _{(i)}=0\). Thus
\(\square \)
Remark 6.2
Given any monoid \(M\), a representation of \(M\) is the same thing as module over the monoid ring \(\mathbb {Z}[M]\). In view of Lemma 6.1, the modules over \(\mathcal {P}\) and \(\Gamma \) correspond to those representations of the inverse monoids \(2^\mathbb {N}\) and \(\mathrm {Emb}\) which are tight in the sense of Exel (see [15, Def. 13.1 and Prop. 11.9]).
Remark 6.3
It was proved in [8, Lemma 4.7.1] that the map
is an isomorphism. It follows from this that \(\Gamma \) is flat as an abelian group. Therefore the map \(J\otimes R\rightarrow \mathbb {Z}[\mathrm {Emb}]\otimes R\) is injective. Thus, by Lemma 6.1,
Next observe that the inclusion \(\mathcal {P}\subset \Gamma \) is a split injection. Indeed the map
is a left inverse. It follows that if \(R\) is any ring then the map \(\psi :\mathcal {P}\otimes R\rightarrow \mathcal {P}(R):=\psi (\mathcal {P}\otimes R)\) is an isomorphism. Thus using Lemma 6.1 and a similar argument as that given above for the case of \(\Gamma \), one can show that
Because \(\mathrm {Emb}\) is a monoid, if \(\mathcal {A}\) is a ring on which \(\mathrm {Emb}\) acts by ring endomorphisms we can form the crossed product \(\mathcal {A}\#\mathrm {Emb}\). As an abelian group, \(\mathcal {A}\#\mathrm {Emb}=\mathcal {A}\otimes \mathbb {Z}[\mathrm {Emb}]\) with multiplication given by
Here \(\#=\otimes \) and \(f_*(b)\) denotes the action of \(f\) on \(\mathrm {Emb}\). Now assume that the \(\mathrm {Emb}\)ring \(\mathcal {A}\) is also a \(\mathcal {P}\)algebra, that is, it is a ring and a \(\mathcal {P}\)bimodule, and these operations are compatible in the sense that
Further assume that \(\mathcal {A}\) is central as a \(\mathcal {P}\)bimodule, i.e. \(pa=ap\) (\(a\in \mathcal {A}\), \(p\in \mathcal {P}\)), and that
Under all these conditions, we say that \(\mathcal {A}\) is an \(\mathrm {Emb}\)bundle (cf. [1, Def. 2.10]). For \(J\vartriangleleft \mathbb {Z}[\mathrm {Emb}]\) as in (6.3), we have
Set
Thus, \(\mathcal {A}\#_\mathcal {P}\Gamma =\mathcal {A}\otimes _\mathcal {P}\Gamma \) as left \(\mathcal {P}\)modules, and the product is that induced by (6.6); we have
Proposition 6.4
Let \(\mathfrak {A}\) be a bornological algebra. The map
is an isomorphism of \(\mathcal {P}\)algebras. If \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal, then (6.9) sends \(S(\mathfrak {A})\#_\mathcal {P}\Gamma \) isomorphically onto \(I_{S(\mathfrak {A})}\vartriangleleft \Gamma ^\infty (\mathfrak {A})\).
Proof
Assume first that \(\mathfrak {A}\) is unital. Then the map (6.9) is the same as that of Lemma 6.1(iii). Hence, it is bijective. By (3.1) and (6.8), it is an algebra homomorphism. This proves the first assertion in the unital case; the second is immediate from the fact that (6.9) is bijective and maps \(S(\mathfrak {A})\#_\mathcal {P}\Gamma \) onto \(I_{S(\mathfrak {A})}\). For not necessarily unital \(\mathfrak {A}\), write \(\tilde{\mathfrak {A}}\) for its unitalization as a bornological algebra. We have an exact sequence
Observe that the inclusion \(\mathbb {C}\subset \tilde{\mathfrak {A}}\) induces a \(\mathcal {P}\)module homomorphism \(S\rightarrow S(\tilde{\mathfrak {A}})\) which splits the sequence (6.10). Hence we get an exact sequence
Combining this sequence with the unital case of the proposition, we obtain an isomorphism
\(\square \)
7 Homotopy invariance
7.1 Crossed products by the Cohn ring
The following two elements of \(\mathrm {Emb}\) will play a central role in what follows
We have the following relations
Following standard conventions, if \(\nu \) is a word of length \(l\) on \(\{1,2\}\), we write \(s_\nu =s_{\nu _1}\cdots s_{\nu _l}\) and \(s^\dagger _\nu =(s_\nu )^\dagger \). Thus for the empty word we have \(s_\emptyset =s^\dagger _\emptyset =1\). Observe that if \(\mu \) is of length \(l\) then
Put
We write
Thus \(\mathcal {M}_2\subset \mathrm {Emb}\) is the inverse submonoid generated by the \(s_i\). Its idempotent submonoid is
One checks, using (7.2) that \(s_\mu s^\dagger _\nu =s_{\mu '}s^\dagger _{\nu '}\) if and only if \(\mu =\mu '\) and \(\nu =\nu '\). It follows that \(\mathcal {M}_2\) is the universal inverse monoid on generators \(s_1,s_2\) subject to the relations (7.1). Write
The ring \(C_2\) is the Cohn ring on two generators [3]; it is a purely algebraic version of the Toeplitz Cuntz algebra (called \(\mathcal {E}_2\) in [11]). The assignment
defines an isomorphism between \(M_\infty \) and the ideal of \(C_2\) generated by \(1\sum _{i=1}^2s_is^\dagger _i\). We shall identify each element of \(M_\infty \) with its image in \(C_2\). If \(\mathfrak {A}\) is a bornological algebra and \(S\vartriangleleft \ell ^\infty \) is a symmetric ideal, then we can consider the action of \(\mathcal {M}_2\) on \(S(\mathfrak {A})\) coming from restriction of the \(\mathrm {Emb}\) action, and form the crossed product \(S(\mathfrak {A})\#\mathcal {M}_2\). Recall from Sect. 6 that \(S(\mathfrak {A})\#\mathcal {M}_2=S(\mathfrak {A})\otimes _\mathbb {Z}\mathbb {Z}[\mathcal {M}_2]\) equipped with the product (6.6). Put
As a vector space, \(S(\mathfrak {A})\#_{\mathcal {P}_2}C_2=S(\mathfrak {A})\otimes _{\mathcal {P}_2}C_2\); the product is defined as in (6.6). We have an algebra homomorphism
Lemma 7.1
The map (7.3) is injective.
Proof
Any nonzero element \(x\in C_2\) can be written as a finite sum of nonzero terms
Let \(l\) be the maximum length of all the multiindices \(\mu \) appearing in the expression above. Remark that we may rewrite (7.4) as another finite sum
such that
Indeed this follows from (7.2) and from the identities
Suppose that the element (7.5) is in \(\ker \rho \). Observe that \(\rho (\chi _{\{i\}}\otimes E_{i,j})=E_{i,j}\). Hence, we have
But by (7.2), for \(\mu \) as in (7.7), we have
This together with (7.6) imply that each of the summands of (7.7) vanishes. Thus
for all \(i,j\) and all \(\mu \) and \(\nu \) in (7.6). Hence,
It follows that \(\beta _{\mu ,\nu }\#s_\mu s^\dagger _\nu =0\) and therefore the element (7.5) must be zero, finishing the proof. \(\square \)
Remark 7.2
Let \(S\vartriangleleft \ell ^\infty \) be a nonzero symmetric ideal and let \(c_f\) be as in Example 3.8. Then \(S\) contains \(c_f\) and thus if we identify \(S\#_{\mathcal {P}_2}C_2\) with its image in \(I_S\), we have
In particular the completion of \(c_0\#_{\mathcal {P}_2}C_2\) with respect to the operator norm in \(\mathcal {B}(\ell ^2)\) coincides with the completion of \(M_\infty \mathbb {C}\) and of \(I_{c_0}\); it is the ideal \(\mathcal {K}=J_{c_0}\) of compact operators. Similarly, for \(p\ge 1\) the completion of \(\ell ^p\#_{\mathcal {P}_2}C_2\) for the \(p\)Schatten norm \(T_p=Tr(T^p)\) coincides with that of \(I_{\ell ^p}\); it is the Schatten ideal \(\mathcal {L}^p\).
7.2 The Cohn ring and homotopy invariance
Let \(\mathbb {V}\) be a bornological vector space, \(T\) a compact Hausdorff topological space, \(X\) a metric space, and \(1\ge \lambda >0\). Put
We refer the reader to [12, §2.1.1 and §3.1.4] for the definitions of continuity and Hölder continuity in the bornological setting, as well as for those of the canonical uniform bornologies that the above algebras carry.
Let \(S\vartriangleleft \ell ^\infty \) be a symmetric ideal and \(\mathfrak {A}\) a bornological algebra. We have a natural inclusion
Lemma 7.3
(cf. [12, Lemma 3.26]) Let \(F:\mathbb {C}\mathrm {Alg}\rightarrow \mathfrak {Ab}\) be a splitexact, \(M_2\)stable functor, \(\mathfrak {B}\) a bornological algebra, \(\mathrm {ev}_t:C([0,1],\mathfrak {B})\rightarrow \mathfrak {B}\) the evaluation map, and \(0<\lambda \le 1\).

i)
$$\begin{aligned} F\left( C([0,1],\mathfrak {B})\overset{\mathrm {ev}_t}{\rightarrow }\mathfrak {B}\overset{\mathrm {inc}}{\rightarrow }c_0(\mathfrak {B})\overset{\#1}{\rightarrow }c_0(\mathfrak {B})\#_{\mathcal {P}_2}C_2\right) \end{aligned}$$
is independent of \(t\).

ii)
If \(p>1/\lambda \), then
$$\begin{aligned} F\left( H^\lambda ([0,1],\mathfrak {B})\overset{\mathrm {ev}_t}{\rightarrow }\mathfrak {B}\overset{\mathrm {inc}}{\rightarrow }\ell ^p(\mathfrak {B})\overset{\#1}{\rightarrow }\ell ^p(\mathfrak {B})\#_{\mathcal {P}_2}C_2\right) \end{aligned}$$is independent of \(t\).
Proof
Let \(S\) be either \(c_0\) or \(\ell ^p\). In the first case, put \(\mathfrak {B}[0,1]=C([0,1],\mathfrak {B})\); in the second, let \(\lambda >1/p\) and set \(\mathfrak {B}[0,1]=H^\lambda ([0,1],\mathfrak {B})\). Let
Let \(\phi _+,\phi _,\phi _0^2\) and \(\phi _^2\) be the homomorphisms \(\mathfrak {B}[0,1]\rightarrow \ell ^\infty (X,\mathfrak {B})\) defined in the proof of [12, Lemma 3.26]. One checks that \((\phi _+,\phi _)\) and \((\phi _0^2,\phi _^2)\) are quasihomomorphisms \(\mathfrak {B}[0,1]\rightarrow S(X,\mathfrak {B})\). Furthermore, it is shown in loc. cit. that there are elements \(V,\bar{V}\in \mathrm {Emb}(X)\) such that for
we have
Consider the bijection \(\psi :X\rightarrow \mathbb {N}\)
Let \(s_1,s_2\) be the generators (7.1) of \(C_2\). Let \(v,\bar{v}\in \mathrm {Emb}\) be the conjugates of \(V\) and \(\bar{V}\) under \(\psi \). One checks that, for \(\rho \) as in (7.3), we have
Now recall that \(C_2=\mathbb {Z}[\mathcal {M}_2]\) and write \(*:C_2\rightarrow C_2\) for the involution induced by \(\dagger \). It follows from (7.11) that the element
satisfies \(f^*f=1\). Hence if \(g\) is any of \(1\# s_2,1\#f\in \ell ^\infty (\tilde{\mathfrak {B}})\#C_2\), we have an algebra homomorphism
Moreover, because \(F\) is \(M_2\)stable by assumption and \(S(\mathfrak {B})\#C_2\) is an ideal in \(\ell ^\infty (\tilde{\mathfrak {B}})\#C_2\), \(F(\mathrm {conj}(g))\) is the identity [5, Proposition 2.2.6]. Let \(\phi '^2_0\), \(\phi '^2_\), \(\phi '_+\) and \(\phi '_\) be the maps \(\mathfrak {B}[0,1]\rightarrow S(\mathfrak {B})\) obtained from \(\phi ^2_0\), \(\phi ^2_\), \(\phi _+\), and \(\phi _\) after conjugating with \(U_\psi \). Then (7.8) gives the identity
We have proved that \(F((\mathrm {inc}\circ \mathrm {ev}_0)\#1)=F((\mathrm {inc}\circ \mathrm {ev}_1)\#1)\). The proposition now follows from the fact that if \(t\in [0,1]\) then \(\mathrm {ev}_t\) and \(\mathrm {ev}_0\) are linearly homotopic. \(\square \)
Remark 7.4
The key property of \(C_2\) used in the proof of Lemma 7.3 is that it contains the elements (7.10) and (7.12). In fact it is not hard to check that they generate \(C_2\) as a ring. Hence taking crossed product with \(C_2\) may be regarded as the smallest construction which makes the proof given above work.
Remark 7.5
If \(\mathfrak {A}\) is a \(C^*\)algebra, then \(c_0(\mathfrak {A})=c_0\overset{\sim }{\otimes }\mathfrak {A}\) is the spatial \(C^*\)algebra tensor product. The inclusion \(c_0\subset I_{c_0}\subset \mathcal {K}\) is equivariant for the action of \(\mathrm {Emb}\), and so we get a map \(c_0(\mathfrak {A})\#_{\mathcal {P}_2}C_2\rightarrow \mathfrak {A}\overset{\sim }{\otimes }\mathcal {K}\). Composing the latter with the inclusion \(\mathfrak {A}\rightarrow c_0(\mathfrak {A})\#_{\mathcal {P}_2}C_2\) of Lemma 7.3 we obtain the map \(\iota :\mathfrak {A}\rightarrow \mathfrak {A}\overset{\sim }{\otimes }\mathcal {K}\), \(a\mapsto a\overset{\sim }{\otimes }E_{1,1}\). Hence, the lemma implies that if \(F:\mathbb {C}\mathrm {Alg}\rightarrow \mathfrak {Ab}\) is splitexact and \(M_2\)stable, then, for every \(C^*\)algebra \(\mathfrak {B}\), the map
is independent of \(t\). One can use this to prove that \(F\) is homotopy invariant on stable \(C^*\)algebras, thus obtaining a weak version of Higson’s homotopy invariance theorem [17, Theorem 3.2.2]. Indeed it suffices to show that \(F(\iota )\) is injective if \(\mathfrak {B}=\mathfrak {A}\overset{\sim }{\otimes }\mathcal {K}\), and this follows from the fact that there is a map \(\mathcal {K}\overset{\sim }{\otimes }\mathcal {K}\rightarrow M_2\mathcal {K}\) (in fact an isomorphism) such that the following diagram commutes
Next suppose that \(\mathfrak {B}\) is any bornological algebra. Write \(\hat{\otimes }\) for the projective tensor product. For each \(p\ge 1\) we have a map \(\ell ^p\hat{\otimes }\mathfrak {B}\rightarrow \ell ^p(\mathfrak {B})\). This map is an isomorphism if \(p=1\); using this isomorphism as above, we obtain a map
In general \(\ell ^p\hat{\otimes }\mathfrak {A}\rightarrow \ell ^p(\mathfrak {A})\) is not an isomorphism. Note, however, that for every \(p\ge 1\), the quotient \(\ell ^p(\mathfrak {A})/\ell ^1(\mathfrak {A})\) is a nilpotent ring. Assume that the functor \(F\) is strongly nilinvariant in the sense that if \(f:A\rightarrow B\) is a homomorphism with nilpotent kernel, and such that \(f(A)\vartriangleleft B\) and \(B/f(A)\) is nilpotent, then \(F(f)\) is an isomorphism. Then \(F(\ell ^1(\mathfrak {A})\#_{\mathcal {P}_2}C_2)\rightarrow F(\ell ^p(\mathfrak {A})\#_{\mathcal {P}_2}C_2)\) and \(F(\mathfrak {A}\hat{\otimes }\mathcal {L}^1)\rightarrow F(\mathfrak {A}\hat{\otimes }\mathcal {L}^p)\) are isomorphisms for all \(p\ge 1\). Moreover we also have a commutative diagram
Let \(\mathrm {BAlg}\) be the category of bornological algebras and bounded homomorphisms. Using Lemma 7.3 together with diagram (7.14) above and proceeding as before, one shows that if \(F\) is splitexact, \(M_2\)stable, and strongly nilinvariant, then the functor
is invariant under Höldercontinuous homotopies. This gives a (weak) bornological version of [9, Theorem 6.1.6]. Observe that the stability properties (7.13) and (7.14) play a crucial role in the arguments above. We do not have an analogous stability result for the uncompleted algebras \(c_0(\mathfrak {A})\#_{\mathcal {P}_2}C_2\) and \(\ell ^1(\mathfrak {A})\#_{\mathcal {P}_2}C_2\). In the next subsection we shall prove a version of stability for crossed products with \(\Gamma \). This will enable us to prove a homotopy invariance theorem in the following subsection.
7.3 Stability
Lemma 7.6

i)
There is a natural isomorphism \(\Gamma (\mathbb {N}\sqcup \mathbb {N})\cong M_2\Gamma \).

ii)
Let \(\mathfrak {A}\) be a bornological algebra and \(S\vartriangleleft \ell ^\infty \) a symmetric ideal. Then \(I_{S(\mathbb {N}\sqcup \mathbb {N},\mathfrak {A})}\cong M_2I_{S(\mathfrak {A})}\).
Proof
Let \(p_1,p_2\in \mathrm {Emb}(\mathbb {N}\sqcup \mathbb {N})\) be the inclusions of each of the copies of \(\mathbb {N}\). One checks that the map
is an isomorphism. To prove part ii) one checks that the following composite of isomorphisms of abelian groups is a homomorphism of algebras
\(\square \)
Let \(\mathfrak {A}\) be a bornological algebra and let \(\iota :\ell ^\infty (\mathfrak {A})\rightarrow \ell ^\infty (\mathbb {N}\times \mathbb {N},\mathfrak {A})\) be the inclusion
Also let \(S\vartriangleleft \ell ^\infty \) be a symmetric ideal; put
Proposition 7.7
Let \(\mathfrak {A}\) be a bornological algebra and \(S\vartriangleleft \ell ^\infty \) a symmetric ideal. Then any \(M_2\)stable functor \(F:\mathbb {C}\mathrm {Alg}\rightarrow \mathfrak {Ab}\) sends the map \(\jmath \) of (7.15) to an isomorphism.
Proof
Choose a bijection \(\mathbb {N}\times \mathbb {N}\rightarrow \mathbb {N}\sqcup \mathbb {N}\) sending \(\mathbb {N}\times \{1\}\) bijectively onto the first copy of \(\mathbb {N}\). This bijection induces an isomorphism
Composing this map with the isomorphism of Lemma 7.6, we obtain an isomorphism which fits into a commutative diagram
This concludes the proof. \(\square \)
7.4 A homotopy invariance theorem
Let \(f_0,f_1:\mathfrak {A}\rightarrow \mathfrak {B}\) be homomorphisms of bornological algebras and \(0<\lambda \le 1\). A \(\lambda \)Hölder continuous homotopy from \(f_0\) to \(f_1\) is a homomorphism \(H:\mathfrak {A}\rightarrow H^\lambda ([0,1],\mathfrak {B})\) such that \(\mathrm {ev}_i H=f_i\) (\(i=0,1\)). We say that a functor \(F:\mathrm {BAlg}\rightarrow \mathfrak {Ab}\) is invariant under \(\lambda \)Hölder homotopies if it maps \(\lambda \)Hölder homotopic homomorphisms to equal maps.
Theorem 7.8
Let \(F:\mathbb {C}\mathrm {Alg}\rightarrow \mathfrak {Ab}\) be a splitexact, \(M_2\)stable functor.

i)
The functor
$$\begin{aligned} \mathrm {BAlg}\rightarrow \mathfrak {Ab}, \mathfrak {B}\mapsto F(I_{c_0(\mathfrak {B})}) \end{aligned}$$is invariant under continuous homotopies.

ii)
If \(1\ge \lambda >0\) and \(p>1/\lambda \), then the functor
$$\begin{aligned} \mathrm {BAlg}\rightarrow \mathfrak {Ab}, \mathfrak {B}\mapsto F(I_{\ell ^p(\mathfrak {B})}) \end{aligned}$$is invariant under \(\lambda \)Hölder homotopies.
Proof
Let \(\mathfrak {A}\) be a bornological algebra. We adopt the notation of the proof of Lemma 7.3. Thus \(S\) will be either \(c_0\) or \(\ell ^p\), and \(\mathfrak {A}[0,1]\) will stand for \(C([0,1],\mathfrak {A})\) in the first case, and for \(H^\lambda ([0,1],\mathfrak {A})\) in the second. By the argument of the proof of Lemma 7.3 applied to the functor
we have the following identity
Now if \(h\in \mathrm {Emb}\) then \(G(h_*)\) is the result of applying \(F\) to the map
Here the crossed product is taken with respect to the action on the external \(S\). In addition, we consider the action of \(\Gamma \) on the inner \(S\) and take the crossed product again; we write \((S(S(\mathfrak {A}))\#_\mathcal {P}\Gamma )\#_\mathcal {P}\Gamma \) for the resulting algebra. We have an inclusion
and a commutative diagram
Because \(F\) is \(M_2\)stable, \(F(\mathrm {conj}(1\#U_h))\) is the identity map, since
Hence, by (7.17),
We have to show that
is injective. Observe that we have a natural isomorphism
For \(h\in \mathrm {Emb}\) the isomorphism (7.20) transforms \(S(h_*)\) into the action of \(1\times h\in \mathrm {Emb}(\mathbb {N}\times \mathbb {N})\), and \(h_*S\) into that of \(h\times 1\). Hence we have a map
Observe that the composite
is the map of (7.15). By Proposition 7.7, this implies that the map (7.19) is injective, concluding the proof. \(\square \)
8 \(K\)theory
8.1 Homotopy algebraic \(K\)theory
Let \(0<p\le \infty \). Put
For \(0<p<\infty \) we also consider
We say that a functor \(F:\mathrm {BAlg}\rightarrow \mathfrak {Ab}\) is Hölder homotopy invariant if it is invariant under \(\lambda \)Hölder homotopies for all \(0<\lambda \le 1\). Recall from [12, §2] that a bornological algebra is called a local Banach algebra if it is a filtering union of Banach subalgebras. Similarly we say that a bornological algebra is a bornolocal \(C^*\)algebra if it is a filtering union of \(C^*\)subalgebras. If \(\mathfrak {A}=\cup _\lambda \mathfrak {A}_\lambda \) and \(\mathfrak {B}=\cup _\mu \mathfrak {B}_\mu \) are bornolocal \(C^*\)algebras, we define their spatial tensor product as the algebraic colimit of the spatial tensor products \(\mathfrak {A}_\lambda \overset{\sim }{\otimes }\mathfrak {B}_\mu \); \(\mathfrak {A}\overset{\sim }{\otimes }\mathfrak {B}=\mathop {\mathrm {colim}}_{\lambda ,\mu }\mathfrak {A}_\lambda \overset{\sim }{\otimes }\mathfrak {B}_\mu \). For the projective tensor product of bornological spaces (and of bornological algebras) see [12, §2.1.2]. In the next theorem and elsewhere we write \(KV\) for KaroubiVillamayor’s \(K\)theory.
Theorem 8.1
Let \(S\) be one of \(\ell ^p\), \(\ell ^{p+}\) (\(0<p<\infty \)) or \(\ell ^{p}\) (\(0<p\le \infty \)).

i)
The functor \(\mathrm {BAlg}\rightarrow \mathfrak {Ab}\), \(\mathfrak {A}\mapsto \textit{KH}_*(I_{\ell ^1(\mathfrak {A})})\) is Hölder homotopy invariant and we have \(\textit{KH}_*(I_{S(\mathfrak {A})})=\textit{KH}_*(I_{\ell ^1(\mathfrak {A})})\) for all \(S\) as above.

ii)
For every bornological algebra \(\mathfrak {A}\)
$$\begin{aligned} \textit{KH}_n(I_{\ell ^1(\mathfrak {A})})= \left\{ \begin{array}{ll} KV_n(I_{\ell ^1(\mathfrak {A})})&{} n\ge 1\\ K_n(I_{\ell ^1(\mathfrak {A})})&{} n\le 0. \end{array}\right. \end{aligned}$$ 
iii)
If \(\mathfrak {A}\) is a local Banach algebra and \(n\ge 0\), then there is a natural split monomorphism \(K^{\mathrm {top}}_n(\mathfrak {A})\rightarrow \textit{KH}_n(I_{\ell ^1(\mathfrak {A})})\).
Proof
Recall that \(\textit{KH}\) satisfies excision, vanishes on nilpotent rings and commutes with filtering colimits [23]. On the other hand, \(\ell ^q(\mathfrak {A})/\ell ^p(\mathfrak {A})\) is nilpotent for \(p<q<\infty \) and
It follows that \(\textit{KH}_*(I_{S(\mathfrak {A})})=\textit{KH}_*(I_{\ell ^1(\mathfrak {A})})\) for all \(S\) as in the theorem. Recall also that \(\textit{KH}\) is \(M_2\)stable. Hence \(\textit{KH}_*(I_{\ell ^1()})=\textit{KH}_*(I_{\ell ^p()})\) is Hölderhomotopy invariant, by Theorem 7.8. This proves i). By [23, Proposition 1.5] (see also [5, Proposition 5.2.3]), in order to prove ii) it suffices to show that \(I_{\ell ^1(\mathfrak {A})}\) is \(K_0\)regular. By definition, a ring \(A\) is \(K_0\)regular if for each \(n\ge 1\) the canonical map
is an isomorphism. This is equivalent to the requirement that for \(\underline{\,t\,}=(t_1,\ldots ,t_n)\), the map
induce an isomorphism in \(K_0\). Observe that
Also note that, for the projective tensor product,
Next we borrow an argument from [19, Proposition 3.4]. Consider the homomorphism
Using the identifications (8.1) and (8.2) we have a diagram
One checks that both the outer and the inner square commute. By Theorem 7.8, \(K_0(\mathrm {ev}_{s=0}\#\Gamma )=K_0(\mathrm {ev}_{s=1}\#\Gamma )\). It follows that \(K_0(\epsilon )\) is the identity; this proves ii). Next assume that \(\mathfrak {A}\) is a local Banach algebra; then \(K_0^\mathrm {top}(\mathfrak {A})=K_0(\mathfrak {A})\). On the other hand, by the universal property of the crossed product, we have a map
Composing this map with the inclusion
we obtain the map
Since the latter map induces an isomorphism in \(K_0\), it follows that (8.4) induces a split monomorphism \(K_0(\mathfrak {A})\rightarrow K_0(I_{\ell ^1(\mathfrak {A})})\). Thus we have established iii) for \(n=0\). For the case \(n\ge 1\), we consider the simplicial algebras of \(C^\infty \) functions on the topological standard simplices and of polynomial functions on the algebraic standard simplices:
and
Set
For \(n\ge 1\), we have
Hence for \(KV(\mathfrak {A})=BGL(\Delta ^\mathrm {alg}\mathfrak {A})\), there is a map
Composing the latter map with that induced by the inclusion (8.4), and using parts i) and ii), we get a homomorphism
Composing (8.6) with the homomorphism induced by (8.3) we obtain
But by [9, Theorem 6.2.1] the comparison map
is an isomorphism. One checks that the latter map composed with (8.7) is equivalent to that induced by (8.5). But (8.5) induces an isomorphism in \(K^\mathrm {top}\) of local Banach algebras. This proves that (8.6) is a split monomorphism, concluding the proof. \(\square \)
Theorem 8.2

i)
The functor \(\mathrm {BAlg}\rightarrow \mathfrak {Ab}\), \(\mathfrak {A}\mapsto \textit{KH}_*(I_{c_0(\mathfrak {A})})\) is invariant under continuous homotopies.

ii)
For every bornological algebra \(\mathfrak {A}\)
$$\begin{aligned} \textit{KH}_n(I_{c_0(\mathfrak {A})})= \left\{ \begin{array}{ll} KV_n(I_{c_0(\mathfrak {A})})&{} n\ge 1\\ K_n(I_{c_0(\mathfrak {A})})&{} n\le 0. \end{array}\right. \end{aligned}$$ 
iii)
If \(\mathfrak {A}\) is a bornolocal \(C^*\)algebra and \(n\ge 0\), then there is a natural split monomorphism \(K^{\mathrm {top}}_n(\mathfrak {A})\rightarrow \textit{KH}_n(I_{c_0(\mathfrak {A})})\).
Proof
As in Theorem 8.1, part i) follows from Theorem (7.8). To prove part ii), first observe that
Then use the argument of the proof of part ii) of Theorem 8.1. To prove part iii) first observe that if \(\mathfrak {A}\) is a bornolocal \(C^*\)algebra, then for the spatial tensor product,
Hence if \(\mathcal {K}=\mathcal {K}(\ell ^2(\mathbb {N}))\) is the \(C^*\)algebra of compact operators then the map \(\mathfrak {A}\rightarrow \mathfrak {A}\overset{\sim }{\otimes }\mathcal {K}\), \(a\rightarrow a\otimes E_{1,1}\) factors through \(I_{c_0(\mathfrak {A})}\). Taking this into account, using the fact that, by [21, Theorem 10.9] and [19, Proposition 3.4], the comparison map \(\textit{KH}_*(\mathfrak {A}\overset{\sim }{\otimes }\mathcal {K})\rightarrow K^{\mathrm {top}}_*(\mathfrak {A}\overset{\sim }{\otimes }\mathcal {K})\) is an isomorphism, and substituting continuous functions for \(C^\infty \) functions, we may now proceed as in the prooof of part iii) of Theorem 8.1. \(\square \)
Remark 8.3
The argument of the proofs of part iii) of Theorems 8.1 and 8.2 does not work for \(n<0\). Indeed, \(K_n\) and \(K_n^\mathrm {top}\) do not agree for such \(n\), not even on algebras on which the former is homotopy invariant. For example negative \(K\)theory is homotopy invariant on commutative \(C^*\)algebras [10, Theorem 1.2] yet \(K_n(\mathbb {C})=0\) for \(n<0\), while \(K_{2m}^{\mathrm {top}}(\mathbb {C})=\mathbb {Z}\) for \(m\in \mathbb {Z}\).
Remark 8.4
The argument of the proof of Theorem 8.1 shows that if \(\mathfrak {A}\) is a local Banach algebra then \(\mathfrak {A}\rightarrow \mathfrak {A}\hat{\otimes }\mathcal {L}^1\) factors through \(I_{\ell ^1(\mathfrak {A})}\) and the map
is onto for \(n\ge 0\). Similarly, the argument of the proof of 8.2 shows that for \(\mathfrak {A}\) a bornolocal \(C^*\)algebra the map \(\mathfrak {A}\rightarrow \mathfrak {A}\overset{\sim }{\otimes }\mathcal {K}\) factors through \(I_{c_0(\mathfrak {A})}\) and
is onto for \(n\ge 0\).
8.2 \(K\)theory and cyclic homology
Theorem 8.5
Let \(\mathfrak {A}\) be a bornological algebra and let \(S\) be \(c_0\), \(\ell ^p\), \(\ell ^{p+}\) (\(0<p<\infty \)), or \(\ell ^{p}\) (\(0<p\le \infty \)). Then there are long exact sequences (\(n\in \mathbb {Z}\))
and
Proof
Let \(K^{\mathrm {nil}}={{\mathrm{hofi}}}(K\rightarrow \textit{KH})\) be the homotopy fiber of the comparison map. By [5, diagram (86)], there is a natural map \(\nu :K^{\mathrm {nil}}(A)\rightarrow \textit{HC}(A)[1]\), defined for every \(\mathbb {Q}\)algebra \(A\). Write \(K^{\mathrm {ninf}}={{\mathrm{hofi}}}(\nu )\); by [7, Proposition 8.2.4] \(K^{\mathrm {ninf}}\) is excisive, \(M_2\)stable and nilinvariant, and \(K_*^{\mathrm {ninf}}\) commutes with filtering colimits. Hence to prove the theorem it suffices to show that
Note also that if \(S\ne c_0\), then
by the same argument as that used in the proof of Theorem 8.1 to prove the analogous assertion for \(\textit{KH}\). Thus we may assume from now on that \(S\in \{c_0,\ell ^1\}\). By [9, Proposition 3.1.4], to prove (8.10) it suffices to show that \(I_{S(\mathfrak {A})}\) is \(K^{\mathrm {inf}}\)regular. Here \(K^{\mathrm {inf}}\) is infinitesimal \(K\)theory; by [4] it is excisive and \(M_2\)stable. Hence, the same argument as that used in the proof of Theorems 8.1 and 8.2 to prove that \(I_{S(\mathfrak {A})}\) is \(K_0\)regular applies to show that it is also \(K^{\mathrm {inf}}\)regular. This completes the proof. \(\square \)
Remark 8.6
By Examples 5.3, we have
for unital \(\mathfrak {A}\). Hence in the unital case, the second sequence of Theorem 8.5 can be equivalently expressed in terms of the quotient \(\Gamma ^\infty (\mathfrak {A})/I_{S(\mathfrak {A})}\); we have a long exact sequence
References
Buss, A., Exel, R.: Fell bundles over inverse semigroups and twisted étale groupoids. J. Oper. Theory 67(1), 153–205 (2012)
Calkin, J.W.: Twosided ideals and congruences in the ring of bounded operators in Hilbert space. Ann. Math. 42(2), 839–873 (1941)
Cohn, P.M.: Some remarks on the invariant basis property. Topology 5, 215–228 (1966)
Cortiñas, G.: The obstruction to excision in Ktheory and in cyclic homology. Invent. Math. 164(1), 143–173 (2006)
Cortiñas, G.: Algebraic v. topological Ktheory: a friendly match. In: Topics in Algebraic and Topological KTheory. Lecture Notes in Mathematics, vol. 2011, pp. 103–165. Springer, Berlin (2008)
Cortiñas, G.: Cyclic Homology, Tight Crossed Products, and Small Stabilizations. Available at arXiv:1304.3508
Cortiñas, G., Ellis, E.: Isomorphism conjectures with proper coefficients. Available at arXiv:1108.5196v3.
Cortiñas, G.: Bivariant algebraic Ktheory. J. Reine Angew. Math. 610, 71–123 (2007)
Cortiñas, G., Thom, A.: Comparison between algebraic and topological Ktheory of locally convex algebras. Adv. Math. 218(1), 266–307 (2008)
Cortiñas, G., Thom, A.: Algebraic geometry of topological spaces I. Acta. Mathematica. 209(1), 83–131 (2012)
Cuntz, J.: Ktheory for certain C\(^{*}\)algebras. Ann. Math. (2) 113(1), 181–197 (1981). doi:10.2307/1971137. MR 604046 (84c:46058)
Cuntz, J., Meyer, R., Rosenberg, J.M.: Topological and bivariant Ktheory. Oberwolfach Seminars. Birkhäuser Verlag, Basel (2007)
Cuntz, J., Quillen, D.: Excision in bivariant periodic cyclic cohomology. Invent. Math. 127(1), 67–98 (1997)
Dykema, K., Figiel, T., Weiss, G., Wodzicki, M.: Commutator structure of operator ideals. Adv. Math. 185(1), 1–79 (2004)
Exel, R.: Inverse semigroups and combinatorial C\(^{*}\)algebras. Bull. Braz. Math. Soc. (N.S.) 39(2), 191–313 (2008)
Garling, D.J.H.: On ideals of operators in Hilbert space. Proc. Lond. Math. Soc. 3(1), 115–138 (1967)
Higson, N.: Algebraic Ktheory of stable C\(^{*}\)algebras. Adv. Math. 67(1), 140 (1988)
Karoubi, M., Villamayor, O.: K théorie algébrique et K théorie topologique. I. Math. Scand. 28(1971), 265–307 (1972)
Rosenberg, J.: Comparison between algebraic and topological Ktheory for Banach algebras and C\(^{*}\)algebras. In: Handbook of KTheory, vol. 1, 2, pp. 843–874. Springer, Berlin (2005)
Simon, B.: Trace ideals and their applications, 2nd edn. Mathematical Surveys and Monographs, vol. 120. AMS, Providence (2005)
Suslin, A.A., Wodzicki, M.: Excision in algebraic Ktheory. Ann. Math. (2) 136(1), 51–122 (1992)
Wagoner, J.B.: Delooping classifying spaces in algebraic Ktheory. Topology 11, 349–370 (1972)
Weibel, C.A.: Homotopy algebraic Ktheory. Algebraic Ktheory and algebraic number theory (Honolulu, HI, 1987). In: Contemp. Math., vol. 83, pp. 461–488. American Mathematical Society, Providence (1989)
Wodzicki, M.: Algebraic Ktheory and functional analysis. In: First European Congress of Mathematics, vol. II (Paris, 1992), pp. 485–496. Progr. Math., vol. 120. Birkhäuser, Basel (1994)
Acknowledgments
Most of the research for this paper was carried out during visits of B. Abadie to the Universidad de Buenos Aires and of G. Cortiñas to the Universidad de la República. We are thankful to these institutions for their hospitality. G. Cortiñas wishes to thank his colleague Daniel Carando for useful discussions about topological tensor products and Ruy Exel for many useful discussions and for patiently explaining his paper [15].
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Ralf Meyer.
Both authors were supported by MathAmSud network U11MATH05, partially funded by ANII, Uruguay, and by MINCyT, Argentina. Cortiñas was partially supported by CONICET and by grants UBACyT W386, PIP 11220080100900 and MTM200764704 (FEDER funds).
Rights and permissions
About this article
Cite this article
Abadie, B., Cortiñas, G. Homotopy invariance through small stabilizations. J. Homotopy Relat. Struct. 10, 459–493 (2015). https://doi.org/10.1007/s4006201300699
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s4006201300699