Can All Banach Spaces Be Structured as Unitary Banach Algebras?

[DavideGenoa]

I find, in Kolmogorov-Fomin's Элементы теории функций и функционального анализа, at the end of § 5 of chapter IV, several statement on the spectral radius and the non-emptyness of the spectrum of a linear operator ina Banach space, which are left without proof.
Nevertheless, in Tikhomirov's appendix, the same properties are prooven for non-commutative unitary Banach algebras.
I wonder whether all Banach spaces can be provided with the structure of a unitary (not necessarily commutative) Banach algebras...
$\infty$ thanks!

 

[mathman]

I haven't looked at it in detail, but I doubt it. For example, how would you make Hilbert space into an algebra?

 

[mathwonk]

I haven't read either of your sources, but the connection between banach spaces and banach algebras seems to be that the space of continuous linear maps on a banach space is a banach algebra. maybe that suffices for your purpose.

 

[mathwonk]

If that was not clear, the continuous linear operator T on the Banach space B is itself a member of the Banach algebra of operators, and thus the spectrum of T is non empty.

 

[WWGD]

I haven't read either of your sources, but the connection between banach spaces and banach algebras seems to be that the space of continuous linear maps on a banach space is a banach algebra. maybe that suffices for your purpose.

Maybe Davide is thinking of a sort of reverse situation. Given a Banach algebra B_A can we always find a Banach space B so that B_A is the algebra of continuous linear maps on B?

 

[mathwonk]

but if you read his question, :

"I find, in Kolmogorov-Fomin's Элементы теории функций и функционального анализа, at the end of § 5 of chapter IV, several statement on the spectral radius and the non-emptyness of the spectrum of a linear operator ina Banach space, which are left without proof.
Nevertheless, in Tikhomirov's appendix, the same properties are prooven for non-commutative unitary Banach algebras."

it sounds as if he just wants to know the non emptiness of the spectrum of a linear operator. Or am I missing something?

 

[WWGD]

Maybe you're right, mathwonk, but the title says Banach spaces as Banach algebras; can you clarify for us, Davide?

 

[DavideGenoa]

I was wondering whether a Banach space $B$ can be considered a Banach unitary, not necessarily commutative, algebra by defining some canonical multiplication between the vectors of $B$...

 

[dextercioby]

Well, no, basically you're extending the Banach space which is itself a topological vector space into an algebra by adding a multiplication between the vectors. Banach space + vector multiplication =/= Banach space.

A Banach algebra is thus an enhancement of a Banach space, a richer mathematical notion.

 

[economicsnerd]

I think OP's question is clear. A Banach algebra is $\langle \mathbb A, ||\cdot||,+, *\rangle$, where $\langle \mathbb A, ||\cdot||,+\rangle$ is a Banach space and $*: \mathbb A^2\to \mathbb A$ is a binary operation satisfying some properties.

The question is: given a Banach space $\langle \mathbb A, ||\cdot||,+\rangle$, must there always exist some $*$ such that $\langle \mathbb A, ||\cdot||,+,*\rangle$ is a Banach algebra?

A rephrasing of the question is as follows.
Given any Banach algebra, we can get a Banach space by just forgetting about multiplication. If I tell you a Banach space was built this way, does that give you any information about what kind of Banach algebra you're looking at?

 

[economicsnerd]

To answer your question: I have no idea.

 

[DavideGenoa]

The question is: given a Banach space $\langle \mathbb A, ||\cdot||,+\rangle$, must there always exist some $*$ such that $\langle \mathbb A, ||\cdot||,+,*\rangle$ is a Banach algebra

Exactly what I meant.
I thank any past, presend and future poster in this thread!

 

[mathwonk]

i am puzzled since the question as clarified has absolutely nothing to do with the non emptiness of the spectrum, which apparently motivated it.

 

[DavideGenoa]

@mathwonk: Tikhomirov's appendix, which is about Banach's algebras, proves those statements in the case of Banach spaces with multiplication as unitary non-commutative algebras. Kolmogorov-Fomin's text states them without a proof for Banach spaces (without multiplication). I haven't reached those proofs yet. I will check whether those proofs can be valid for Banach spaces without assuming them to be unitary algebras and I'll let you know. Thank you again!

 

[DavideGenoa]

I haven't read either of your sources, but the connection between banach spaces and banach algebras seems to be that the space of continuous linear maps on a banach space is a banach algebra. maybe that suffices for your purpose.

That is the case. Cfr. p. 519 here, corollary 2 and theorem 2, for those knowing Russian.
Thank you all!