1
vote
1answer
64 views

Unitization via “End points compactification”

We know that every compactification of locally compact Hausdorff spaces correspond to a unitization of $C^{*}$ algebras. For example the one point compactification corresponds to the minimal ...
0
votes
1answer
152 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 ...
1
vote
0answers
63 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 ...
5
votes
2answers
435 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 ...
0
votes
0answers
77 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 ...
5
votes
1answer
210 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. ...
1
vote
1answer
87 views

Banach Isomomorphic Cts Fucntion Algebras for two Non-Homeomorphic Top Spaces?

Can anyone provide me with an example of two non-homeomorphic locally-compact Hausdorff spaces $X$ and $Y$, such that $C(X)$ and $C(Y)$ are isomorphic as Banach algebras. Clearly, the ...
3
votes
2answers
220 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 ...
9
votes
0answers
330 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 ...
10
votes
2answers
531 views

C*-algebras with bizzarre structure of projections

This is probably well-known to the experts but I could not find any answer neither in my head nor in the literature: Is there a (unital) C*-algebra such that its projections do not form a lattice ...
6
votes
1answer
372 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 ...
1
vote
1answer
338 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 ...
11
votes
3answers
884 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 ...