# Stabilization in Banach algebras

In $C^\ast$-algebras we use $K(H)$, the algebra of compact operators on a separable Hilbert space, for stabilization of a $C^\ast$-algebra, i.e. $S(A):=A\otimes K(H)$. Is there any similar stabilization functor in Banach algebras? What is the substitute of $K(H)$?

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@Z254R, are you pointing out deficiencies in a question? ;-) Well in any case I agree completely with you. @Vahid, what do you want a stabilization functor to do? Do you have in mind K-theory? –  Captain Oates Jan 14 at 23:58
After being able to define (an appropriate notion for) the stabilization of a Banach algebra, say $A$, I'd like to see if it is Morita equivalent with $A$. Of course, the equality of $K$-groups is the next. And so on. Stabilization of $C^\ast$-algebras is a elementary notion, so I thought, maybe there is a similar notion for Banach algebras too. That's why I asked this question. –  Vahid Shirbisheh Jan 15 at 0:14
Is it clear which tensor product should be used for Banach algebras? (For $C^*$-algebras, one secretly enjoys nuclearity of $K(H)$...) –  Alain Valette Jan 15 at 17:58
@Yemon: I list some instances that clarify the importance of stabilization in $C^*$-algebras: 1. Both K-theory and KK-theory are stable functors meaning $K(A)\simeq K(A\otimes K(H))$. 2. Two separable $C^*$-algebras $A$ and $B$ are Morita equivalent if and only if they are stably isomorphic, i.e. $A\otimes K(H)\simeq B\otimes K(H)$. 3. Tensoring by $K(L^2(G))$ also appears in some theorems too, for example see Takai-Takesaki duality. So, it is nice to have a similar notion in Banach algebras, for instance proving item 2 for Banach algebras would be a good start. –  Vahid Shirbisheh Jan 17 at 5:58
@Yemon: You are welcome. Following Vincent Lafforgue's works, Walter Paravicini has studied Morita equivalence of Banach algebras too. –  Vahid Shirbisheh Jan 25 at 20:58

This is not an answer, but too long for a comment: Even for $C^*$-algebras there is more than one method of stabilization. The critical feature of the compact operators is that $K(H) \otimes K(H) \cong K(H)$ for an infinite dimensional Hilbert space and that $K_*(K(H)) \cong K_*(\mathbb{C})$. But this also holds true for other $C^*$-algebras like for example the infinite Cuntz algebra $\mathcal{O}_{\infty}$ or the Jiang-Su algebra $\mathcal{Z}$. In general - again for the $C^*$-case - you might want to search for strongly self-absorbing $C^*$-algebras.

There is the following theorem:

If $A$ is a simple, separable and nuclear $C^*$-algebra, then $A \cong A \otimes \mathcal{O}_{\infty}$ if and only if $A$ is purely infinite.

And there is a famous theorem by Kirchberg, which you may read like this:

Assume $A$ and $B$ are simple, separable, nuclear and $\mathcal{O}_{\infty} \otimes K(H)$-stable $C^*$-algebras (i.e. $A \otimes \mathcal{O}_{\infty} \otimes K(H) \cong A$ and likewise for $B$), then $A$ and $B$ are isomorphic if and only if they are $KK$-equivalent.

If you are looking for a setup like this, you may want to search for Banach algebras, which have the $K$-theory of $\mathbb{C}$ and are isomorphic to the tensor product of two copies of themselves. Or one might be able to generalize the notion of strongly self-absorbing to Banach algebras.

That was a lot about $C^*$-algebras and not much about the Banach case. Sorry about that.

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Thanks Ulrich. I guess one can choose is $K(H)$ for stabilization of Banach algebras too. But I am not sure if this gives rise to similar theorems about Morita equivalence and $K$-theory of Banach algebras. So, it seems there is no obvious choose "known yet" and it is open for more investigations. –  Vahid Shirbisheh Jan 15 at 12:44
Out of curiosity, Ulrich: *what is it that one gains by tensoring a given $C^\ast$-algebra with $K(H)$? (not a rhetorical question; I am genuinely ignorant of, or have forgotten, the reason one does this). –  Captain Oates Jan 16 at 19:38
Yemon, one gains, for example idempotents. The group $K_0(A)$ can be constructed looking at the projections of $A\otimes K$ (looking at $\bigcup_n M_n(A)$ also works here, but passing to the completion may be more appealing at times). One also looses after tensoring with $K$ (which can be a good thing). E.g., Brown's theorem says that if $a,b\in A$ are positive then thr C*-algebras $\overline{aAa}$ and $\overline{bAb}$ become isomorphic after tensoring with $K$ if and only if $a$ and $b$ generate the same closed two-sided ideal. –  Leonel Robert Jan 17 at 17:05
Stabilization with K is also interesting from the point of view of Hilbert bimodules, since the Brown-Green-Rieffel theorem says that two C*-algebras with countable approximate identities are strongly Morita equivalent if and only if they are stably isomorphic. Apart from that we have an exact sequence $0 \to Inn(A \otimes \mathbb{K}) \to Aut(A \otimes \mathbb{K}) \to Pic(A \otimes \mathbb{K}) \to 0$. –  Ulrich Pennig Jan 17 at 18:22
Oops, I just realized my last comment was mentioned above already. –  Ulrich Pennig Jan 17 at 18:26