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3 votes
1 answer
191 views

A possible spectral characterization of commutative $C^*$ algebras

Let $A$ be a $C^*$ algebra. Assume that the spectrum $Sp(a_1a_2\ldots a_{n-1}a_n)$ is unchanged as a set after a permutation of $a_i$'s. (unless possible emerge or removing 0 from the spectrum) Does ...
Ali Taghavi's user avatar
0 votes
0 answers
115 views

$C^*$ algebra generated by conjugation of an element

Assume $\mathcal{A}$ is a unital $C^*$ algebra and consider some positive-definite element $\Psi\in M_n(\mathcal{A})$. Can we say something about $C^*(\langle \Psi^{-\frac{1}{2}}E_{i,i}\Psi^{\frac{1}{...
GBA's user avatar
  • 167
9 votes
3 answers
453 views

Comparison between the operator norm and the $L^1$ norm on group algebras

Consider a discrete group $G$ and its group algebra over $\mathbb{C}$, $\mathbb{C}[G]$. There are four norms on it I wish to consider for this question: The 2-norm given by $||\sum_{g \in G} c_gg||_2^...
David Gao's user avatar
  • 2,830
0 votes
1 answer
419 views

What are the maximal ideals of $C_0 (X),$ where $X$ is a locally compact Hausdorff space?

Crossposted from MSE How do the maximal ideals of $C_0(X)$ look like where $X$ is a locally compact Hausdorff space? I know that if $X$ is a compact Hausdorff space then the maximal ideals of $C(X)$ ...
RKC's user avatar
  • 141
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 ...
user531706's user avatar
-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 $A$?
user531706's user avatar
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 ...
Ali Taghavi's user avatar
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 ...
ned grekerzberg's user avatar
1 vote
0 answers
164 views

When a finite codimensional subalgebra contains a finite codimension ideal?

What is a classification of all algebras $A$ (purely algebraic algebras, Banach or $C^*$ algebras or Lie algebras) with the following property: Every finite codimensional subalgebra $B$ of $A$ ...
Ali Taghavi's user avatar
3 votes
0 answers
156 views

Left and right topological K-theory of Banach algebras

Let us consider the topological $K$-functor on the category of Banach algebras as described in page $18$ of "Introduction to the Baum–Connes conjecture" by Alain Valette. The definition is based on ...
Ali Taghavi's user avatar
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,...
Dave Shulman's user avatar
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 \...
Ali Taghavi's user avatar
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 ...
Masayoshi Kaneda's user avatar
4 votes
1 answer
262 views

A precise definition of contractible Banach algebras

I asked this question at MSE but I did not received any answer. So I ask it here at MO I am sorry if this question is elementary: What is a precise definition of a contractible Banach ...
Ali Taghavi's user avatar
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 $\...
Masayoshi Kaneda's user avatar
3 votes
1 answer
199 views

What can be said about the algebra of continuous functions on compact countable ordinals?

Let $X$ be a compact countable Hausdorff space. By Sierpinski-Mazurkiewicz Theorem we know that $X$ is a compact countable ordinal, i.e. $$ X \simeq \omega ^{\alpha} \cdot n + 1 $$ where $\alpha$ is ...
Niki's user avatar
  • 335
6 votes
1 answer
575 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 ...
Norbert's user avatar
  • 1,697
2 votes
1 answer
357 views

Non commutative topological manifolds

Assume that $A$ is a Banach algebra with two closed two sided ideals $I$ and $J$ such that $I$ and $J$ are commutative and $A=I+J$. Does this implies that $A$ is commutative? For the $C^{*...
Ali Taghavi's user avatar
1 vote
1 answer
232 views

Injective element of a commutative Banach algebra

A revision: According to the comment of Nate Eldredge, in order to avoid the triviality, we revise the property $P$. Assume that $A$ is a commutative unital Banach algebra. Its maximal ideal ...
Ali Taghavi's user avatar
2 votes
0 answers
171 views

tensor product of the disc algebra with itself

Let $A=\mathcal{A}(\mathbb{D})$ be the disc algebra. Is there a cross norm on $A\otimes A$, which is compatible to the Banach algebra multiplication, such that the resulting Banach algebra has a $C^{*}...
Ali Taghavi's user avatar
1 vote
0 answers
198 views

Connected component of the identity in graded Banach algebras

I search for a noncommutative idempotent-less Banach algebra $A$ which is graded by a finite abelian group $G$ such that a nontrivial homogenous element lies in the same connected component as $1_{...
Ali Taghavi's user avatar
5 votes
1 answer
303 views

$Z_{2}$- graded structures for $C_{red} ^{*} (F_{2})$

Let $F_{2}$ be the free group with two generators. Then $F_{2}=\{\text{odd words}\}\sqcup\{\text{even words}\}$. This gives us a $Z_{2}$ graded structure for $C^{*}_{red} (F_{2})$, in a natural way. ...
Ali Taghavi's user avatar
3 votes
2 answers
553 views

Tensoring with a CAR-algebra

Let $A$ and $B$ be two unital infinite-dimensional simple separable nuclear $C^{\ast}$-algebras and let $C$ be a CAR-algebra. When does $A\otimes C \simeq B\otimes C$, imply $A\simeq B$? The answer ...
David's user avatar
  • 169
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 ...
TrzyTrypy's user avatar
  • 101
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 ...
Daniel's user avatar
  • 53
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 ...
Habujew's user avatar
  • 113
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 ...
Sellapan Nathan's user avatar
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 ...
yeshengkui's user avatar
  • 1,373