1
vote
1answer
14 views

Showing the $C^*$ identity

I'm working through a proof in Dixmier's book on $C^*$-algebras and I'm stuck on part of a proof. I'm given a Banach algebra $\mathcal{A}$ which has norm $\lVert\cdot\rVert$ and a semi-norm ...
1
vote
0answers
22 views

Biprojective $C^*$-algebra

Let $A$ be a Banach algebra. Define $\Delta:A\hat{\otimes}A\to A$ with $\Delta(\sum_{n=1}^\infty a_n\otimes b_n)=\sum_{n=1}^\infty a_nb_n$. Now $A$ is called biprojective if there exists a bounded ...
0
votes
1answer
20 views

Classification of Banach Algebras?

Is there a classification theorem for Banach algebras, or even for Banach *algebras, similar to the GNS representation theorem for $C^*$-algebras? If yes, please provide a reference where I can read ...
2
votes
0answers
29 views

Find an example of product of operators not Jointly Continuous in strong topology.

I'm trying to find an example of the fact that the product of operators is not jointly continuous in the strong topology. I know the example of the unilateral shift (that is on wikipedia), but I ...
2
votes
1answer
39 views

How do we show that prime C* algebras have trivial center

A prime C* algebra is a C* algebra with the property that the product of any two of its non zero ideals is non zero. The claim is that it has trivial center, i.e., the only central elements are ...
3
votes
1answer
112 views

A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form ...
4
votes
2answers
110 views

Commutative unital Banach algebra with nilpotent elements

What would be a concrete example of a commutative unital Banach algebra that contains nilpotent elements?
2
votes
1answer
101 views

Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge (or monotonous converge) to $1$ in $A^+$, does $\{x_n\}$ must be the ...
1
vote
0answers
51 views

$\ast$-homomorphism

Let $\phi: C(X,M_{4}(\mathbb{C})) \rightarrow C(Y,M_{8}(\mathbb{C})) $ be a $\ast$-homomorphism where $X$ and $Y$ are compact Hausdorff spaces. Let $M_{2}(\mathbb{C})$ be the C*-subalgebra of ...
2
votes
1answer
51 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
3
votes
0answers
116 views

C* algebra of bounded Borel functions

Let $T\in B(H)$ is normal, and $B(\sigma(T))$ denote the $C^*$ algebra of all bounded Borel functions on $\sigma(T)$. Then is it true that $B(\sigma(T))$ is a closed $C^*$ algebra under the sup. norm ...
3
votes
1answer
84 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
1
vote
0answers
153 views

Proving properties of exponential map on a Banach algebra

$$\exp(a) := \sum\frac {a^k}{k!}$$ Can you help me prove that: $\exp$ is well defined (i.e. converges for all $a$ in $A$) $\exp$ is continuous $\exp(A)$ is a subset of $A_0$ (where $A_0$ is the ...
3
votes
1answer
191 views

Polar decomposition of invertible elements in a unital C$ ^{*} $-algebra.

If $ A $ is a unital C$ ^{*} $-algebra and $ a $ is invertible, then $ a = u|a| $ for a unique unitary element $ u $ of $ A $. If $ \| a \| = \| a^{-1} \| = 1 $, what can you say about $ |a| $? I ...
4
votes
0answers
108 views

Biduals generated by projections

This question is motivated by a similar question recently posed at MO: http://mathoverflow.net/questions/122091/masas-in-second-duals-of-banach-algebras In this setting, let $B$ be a Banach algebra ...
3
votes
1answer
71 views

Double centralizers in the Murphy book

I've been into this for days and days and I still can't see why, given the definition of $L^\ast$ as $L^\ast =(L(a^\ast))^\ast$ we get that $(LM)^\ast =L^\ast M^\ast$. Where is my mistake: ...
6
votes
1answer
490 views

Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
5
votes
2answers
185 views

Why are compact operators 'small'?

I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it. I know that compact operators map bounded sets to totally bounded ones, that ...
1
vote
1answer
93 views

References on Algebraic Operators

Let $\mathcal{H}$ be a Hilbert space and $d$ is an inner derivation on $\mathcal{L}(\mathcal{H})$. An operator $T\in\mathcal{L}(\mathcal{H})$ is algebraic if $p(T)=0$ for some polynomial $p$. In ...
4
votes
2answers
299 views

Non-$C^{*}$ Banach algebras?

It suddenly occurred to me almost every Banach algebra I know is actually a $C^{*}$ algebra. Several kinds of function algebras are definitely $C^{*}$ algebras. So is the matrix algebra. Although one ...
2
votes
1answer
138 views

Subadditive, submultiplicative and scalar-multiplication-invariant functions

Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...
12
votes
2answers
625 views

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...