All Questions
27 questions
14
votes
3
answers
3k
views
The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$
Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
- 1,373
10
votes
0
answers
508
views
Tensorial decomposition of $B(H)$
Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
- 101
9
votes
2
answers
298
views
Two inequalities in $C^*$ algebras
Under what conditions on a $C^*$ algebra $A$ we have the following inequality:
$$x^*a^*ax+a^*x^*xa\leq x^*x+a^*x^*ax+x^*a^*xa\;\;\; \forall x,a\in A$$
The second identity which I am looking for is ...
- 356
7
votes
1
answer
682
views
$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$
Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...
- 113
7
votes
1
answer
491
views
Projections in the tensor product of von Neumann algebras
This question seems elementary, but I have already asked an expert who does not know the answer, so I would like to post here.
Let $M$ and $N$ be von Neumann algebras, and let $M\bar{\otimes}N$ be ...
- 619
7
votes
1
answer
698
views
When $C(X)$ is an injective $C(X)$-module? Current answer is erroneous
It is an old question if every injective Banach space is isomorphic as Banach space to $C(X)$-space.
I would like to know if the weakened module version of this question is answered. More precisely: ...
- 1,697
7
votes
1
answer
429
views
Open projections and Murray-von Neumann equivalence
Let $\mathcal{A}$ be a $C^*$-algebra and $p\in\mathcal{A}^{**}$ be an open projection, that is, $p=p^*=p^2$ and $p\in\overline{(p\mathcal{A}^{**}p\cap\hat{\mathcal{A}})}^{\operatorname{w}^*}$, where $\...
- 619
7
votes
1
answer
1k
views
Inductive/Projective Limits of Topological Algebras
It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...
- 71
6
votes
2
answers
548
views
When is it $C(X)$?
Suppose that $\tilde{X}$ is a compact space. If $C(\tilde{X})$ is isometrically isomorphic to the second dual of a Banach space, does there exist a locally compact space $X$ such that $C(\tilde{X})=...
- 306
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
6
votes
1
answer
577
views
Is $\mathcal{K}(H)$ injective $\mathcal{B}(H)$-module?
Does anyone know if the right Banach $\mathcal{B}(H)$-module $\mathcal{K}(H)$ is injective? This module is not dual, so standard arguments via flatness do not work.
Injectivity is understood in the ...
- 1,697
6
votes
0
answers
107
views
Real-world example of a Banach *-algebra with a nonzero *-radical
Is there a real-world example of a Banach *-algebra with a nonzero *-radical (intersection of kernels of all *-representations)? Textbooks give examples of finite-dimensional algebras with degenerate ...
- 3,816
5
votes
1
answer
1k
views
When is a Banach Algebra $C^\star$
I know that if there are enough Hermitian elements in a Banach algebra, then the Banach algebra is stellar. In particular, I'm interested in the two spaces $B(L^1(S^1,\Sigma,\mu))$ the space of ...
- 53
4
votes
2
answers
254
views
$K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras
Let $A$ and $B$ be two $C^*$-algebras, and let $p:A \to B$ be a surjective norm-decreasing $*$-homomorphism which is injective on a dense $*$-sub-algebra of $A$. Can such a map have non-trivial kernel,...
- 969
4
votes
1
answer
775
views
Algebraically simple Banach algebras
There are plenty of semi-simple Banach algebras - this broad class includes C*-algebras and algebras of bounded operators on a given Banach space. On the other hand, it seems unlikely to me that there ...
- 113
3
votes
1
answer
388
views
Closed prime ideal in $C[0, 1]$
I know that maximal ideals of $C[0, 1]$ corresponds to singleton. Also, using Zorn's lemma one can construct a prime ideal in $C[0, 1]$ which is not maximal.
Is there any $\textbf{closed}$ prime ...
- 1,115
3
votes
1
answer
252
views
Regarding spectral radius
Let $A$ be a $C^*$ algebra. Let $a\in A$ be such that $a^*a-aa^*\geq 0$. Doe this imply that the spectral radius of $a$ is equal to $\|a\|$?
- 175
3
votes
1
answer
230
views
questions about the proof of the theorem of completely positive order zero maps
I hope my question is ok for mathoverflow. I first asked on math.stackexchange but received no answer and then delated it.
I want to understand the proof of the theorem (which you can find in the ...
user62639
2
votes
0
answers
107
views
Finding non-zero elements with $x^*x\leq\frac{1}{n}$
For a given unital Banach *-algebra $A$, let us put $A_+=\{\sum_1^n x^*_ix_i : x_i\in A, n\in \mathbb{N}\}$. We write $x\geq0$ if $x\in A_+$.
What types of (non semi-simple) unital Banach *-algebras ...
- 4,058
2
votes
0
answers
151
views
A Banach or $C^*$ algebraic analogy of a classical fact in real analysis
Let $A$ be a commutative unital Banach algebra.The maximal ideal space of $A$ is denoted by $\hat A$.
Assume that $D:A \to A$ is a derivation. Fix an element $a\in A$.
Assume that for every $\phi\in \...
- 356
1
vote
1
answer
142
views
Complemented C*-algebras
Let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
- 149
1
vote
1
answer
269
views
the space of self-adjoint trace class operators over a separable Hilbert space is separable with respect to the trace norm?
My interest is to know whether the assertion
...
- 31
0
votes
1
answer
495
views
Separability of an algebra is equivalent to separability of its spectrum
Let $A$ be a commutative C*-algebra.
I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable.
Notes ...
- 143
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
0
votes
0
answers
146
views
Non-degenerate representation of a Banach algebra
Let $\mathcal{A}$ be a non-reflexive Banach algebra. For the definition of Arens product, please refer to this link. Here we let $\square$ denote the first Arens product and $\diamond$ denote the ...
- 307
0
votes
0
answers
106
views
A noncontinous algebra map between Banach algebras
What is an example of two Banach algebras $A$ and $B$, and an algebra map $\phi:A \to B$ which is not continuous?
- 203
-1
votes
1
answer
246
views
Density of normal elements in a C*- algebra [closed]
Let $A$ be a unital C*-algebra.
I wanted to know if there is a necessary and sufficient condition for normal elements to be dense in $A$?
- 149