# Lifting of product of a Banach algebra

Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.

A lifting of $T$ is a bounded linear map $L:A\to A \otimes_{\gamma} A$ such that $TL=Id_{A}$.

Do you know a charaterization or examples of algebras which admits a such lifting?

Any reference on this problem or explicit examples is welcome.

You certainly want to impose further restrictions on $L$ and $A$. For instance, if $A$ has a unit $1$, then you can take $L(a) = 1 \otimes a$, so the unital case isn't very interesting. A necessary condition of course is that $T$ is onto (if $A$ has a bounded approximate unit then this follows automatically from Cohen-Hewitt). In particular, if $A$ is factorable and is isomorphic as a Banach space to $\ell^{1}(S)$ then you have such a section as well. You might want to google for 'virtual diagonal' and 'amenability' and browse in the books of Runde and Helemskii for more interesting results. – Theo Buehler Jan 30 '11 at 12:34
(I would call it a section of T.) Any Banach algebra with identity would be an example, for example $L^\infty$ or spaces of bounded continuous functions with respect to sup norm. Or, any Banach algebra obtained by adjoining a unit to another Banach algebra. – Todd Trimble Jan 30 '11 at 12:39
Thanks for the comments. @Theo Buehler: Where is written the result: "if $A$ is factorable and is isomorphic as a Banach space to $\ell_1(S)$ then there exist a section?? – BigBill Jan 30 '11 at 12:59
I don't know where it's written but it's rather trivial: factorable is equivalent to surjectivity of the multiplication (and hence of $T$). A bounded linear map on $\ell^{1}(S)$ is determined by the images of the standard basis (Dirac's $\delta_{s}$). Abusing notation, we may choose a preimage under $T$ of each $\delta_{s}$ inside a bounded set of $A \otimes_{\gamma} A$ (you can take $\|T\|+\varepsilon$ as a bound), this determines a linear map $L: A \to A \otimes_{\gamma} A$ such that $TL(\delta_{s}) = \delta_{s}$. – Theo Buehler Jan 30 '11 at 13:26
Big Bill, without some further motivation this question seems a bit too wide-ranging to me. Mere curiosity will not get us far I fear. (I also think that you should call this a splitting problem rather than a lifting problem.) There are two aspects to the problem, it seems: first one has to find non-unital algebras for which $A\hat{\otimes}_\gamma A \to A$ is surjective, and then ask if there is a linear section as Theo has said. Note that it is not necessary for the first question that A be factorable: try $\ell^1$ with pointwise multiplication. (Try Googling "biprojective".) – Yemon Choi Jan 30 '11 at 19:03