All Questions
23 questions
3
votes
1
answer
148
views
positive functional on Banach *-algebra (with appro. identity) is continuous?
Theorem (N. Th.Varopoulos): Let $\mathcal{B}$ be a Banach *-algebra
with a bounded approximate identity. Then every positive functional $T$ on $\mathcal{B}$ is continuous.
I think this theorem is ...
- 325
1
vote
0
answers
49
views
Banach algebras satisfying $pq=qp=q \Rightarrow \|q\|\leq\|p\|$ for all idempotents $p$ and $q$
This question could be way below the level of MO, so apologies in advance. I posted the same question in MS about 10 days ago without a definitive answer so far.
Let $A$ be a Banach algebra with the ...
- 2,605
3
votes
0
answers
282
views
Left ideals of $\ell^{\infty}(A)$ containing all weakly null sequences in a Banach algebra $A$
Let $A$ be a Banach algebra. $\ell^{\infty}(A)$, the space of all bounded sequences in $A$, is a Banach algebra with pointwise operations.
Let $w_0(A)$ be the subspace of all weakly null sequences in $...
- 2,605
0
votes
1
answer
163
views
Regarding socle of a C* algebra
I wanted to know if the socle of a complex C*-algebra is essential?
Can anyone suggest a text where the socle is studied in detail. I tried reading it from the book by Bernard Aupetit, A Primer in ...
- 149
3
votes
0
answers
111
views
Infinite ordered products (reference request)
While writing arXiv:1510.05757v2, I found myself proving some basic facts about products of Banach algebra elements over an infinite totally ordered set. (Statements and proofs are in Appendix C.) The ...
- 2,284
1
vote
0
answers
74
views
Good source for Jordan Fréchet algebras
Is there any good source for Jordan Fréchet (or more generally, Jordan locally convex) algebras?
I'm looking for something on the level similar to the level of the book "Banach and Locally Convex ...
6
votes
1
answer
252
views
Arens regularity of Banach algebras
I was trying to learn the concept of Arens regularity of Banach algebras from T.W Palmers book -"Banach algebras and the general theory of $*$-algebras". There he have discussed the Arens regularity ...
- 243
5
votes
1
answer
766
views
When are homomorphisms between Banach algebras contractions?
When are homomorphisms between Banach algebras contractions?
I recall from my student days that there are results which show that a positive answer to the above question holds under very general ...
- 2,472
0
votes
1
answer
289
views
The definiton of a multiplier on a Banach algebra
Let $A$ be a Banach algebra. Some textbooks define a (left ) multiplier as a map $T:A\rightarrow A$ satisfying $T(ab)=T(a)b$ for all $a,b\in A$ and assume that $A$ needs to be a without order Banach ...
- 167
4
votes
0
answers
264
views
Is the Gelfand transform strictly continuous?
Let $M$ be the Banach algebra of measures on the circle with $L_1$ naturally sitting as a closed ideal of $M$. Then $M$ carries the strict topology implemented by the family of seminorms $\|\mu\|_f = \...
- 113
5
votes
0
answers
330
views
The second dual of $C(X)$ with the compact-open topology
Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
- 233
6
votes
4
answers
1k
views
Resource recommendation: Spectral theory and $C^*$ algebras
I have formally studied functional analysis, both as university courses, and by myself, but this is one area of mathematics I find so huge and complicated, I have a hard time properly getting into it.
...
- 2,792
2
votes
1
answer
481
views
Ideals of $L^1(G)$
I want to study the closed ideal structure of $L^1(G)$. Is there a good paper or book which characterizes closed ideals and maximal ideals of $L^1(G)$?
- 565
3
votes
2
answers
435
views
A possible norm on a subspace of $C^\infty([0,1])$?
I have posted the following question (with minimal differences) on MSE some days ago, without receiving a satisfactory answer, so let me try here again.
Take the vector space of infinitely ...
- 3,388
1
vote
0
answers
233
views
Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
DEFINITION 1 ...
- 7,500
0
votes
3
answers
365
views
Specific Reference? Noncommutative topology and C^* algebras [closed]
I stumbled across this "dictionary for noncommutative topology" http://planetmath.org/noncommutativetopology
and I would be very interested in learning more on the subject, particularly I'd like to ...
- 11
1
vote
0
answers
269
views
$L^{p}(\mathbb R)\subset L^{1}(\mathbb R) \ast L^{p}(\mathbb R), (1< p< \infty)$?
Let $\mathbb T$ be a circle group.
In 1939, Salem, has shown that, every member of $L^{1}(\mathbb T)$ can written as a product(convolution) some other two members of $L^{1}(\mathbb T),$ that is, $L^...
- 1,051
4
votes
1
answer
1k
views
$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$
I have not managed to find a reference for the following fact:
$H^s(\mathbb T)$ is a Banach algebra for $s>1/2$.
In particular, I need reference for the following inequality:
$$
\|uv\|_{H^s} \,\...
- 2,933
6
votes
2
answers
460
views
Terminology: Banach spaces equipped with continuous associative product?
This is admittedly a low-interest question mathematically, and is arguably a question I could resolve if I had time over the next few days to go and look through a large number of the Banach algebra/...
8
votes
3
answers
2k
views
Measures on general topological groups
I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...
- 1,152
7
votes
3
answers
814
views
Preduals of B(E)
For a Hilbert space $H$ it is well known that the algebra $B(H)$ has a unique predual; the Banach space of trace class operators.
If $E$ is a Banach space then is it known whether
$B(E)$ is always a ...
- 1,411
0
votes
0
answers
301
views
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 ...
- 1,222
4
votes
3
answers
1k
views
Set of invertible operators in B(H) is connected. Is it true? Is there a reference?
Suppose $H$ is a Hilbert space, $B(H)$ is the algebra of bounded linear operators on it, $K(H)$ is ideal of compact operators in $B(H)$, $Inv(B(H)/K(H))$ is the topological group of invertible ...
- 1,284