# Tag Info

• 61k
Accepted

### is every element in a C* algebra a product of normal elements?

Since the question was asked wrt C$*$-algebras, I guess there is room for a general remark. Suppose that $xy = 1$ where $x$ is a product of normal elements, say $v_1v_2\cdots v_n$. Then $v_1$ is ...
• 4,669

### About the existence of characters on $B(X)$

Examples were known before the Argyros-Haydon space mentioned in Yves Cornulier's answer. For instance, if $J$ denotes the James space, then the image of the canonical map $J\to J^{**}$ has ...
• 25.5k
Accepted

### $K$-theory and surjective norm-decreasing $*$-homomorphisms between $C^*$-algebras

Yes to both.$\newcommand{\Cst}{{\rm C}^*}$ The standard example for the first is: take a discrete group $G$ and let $A$ be its full $\Cst$-algebra, $B$ its reduced $\Cst$-algebra. There is a ...
• 25.5k
Accepted

### Trace norm of operators obtained by restricting the matrix of a trace class operator

Here's an algorithm for testing an ad-hoc conjecture $C$ about Hilbert space operators. :-) Set up the runtime environment correctly by loading the information "Most conjectures are false" ...
• 11.7k

### Conceptually, what does unitization do?

I don't remember where I read this, but Gert Pedersen once said something to the effect that "When I was young, the first thing we did with any C*-algebra was to adjoin a unit, but nowadays the first ...
• 42.4k
Accepted

### Removing the interior of spectrums

The answer is no, in general. Here is a counterexample: Let $A$ be the algebra of bounded linear operators on $\ell^2(\mathbb{N})$, and let $a \in A$ be the left shift on $\ell^2(\mathbb{N})$. Then ...
• 11.7k
Accepted

### What algebras are quotients of $\ell_1(\mathbf{N})$?

(What follows is largely the result of digging around online, based on knowing a few more magic words than the OP.) Answer to the first question (I think). Let $V$ be a separable Banach space. The ...
• 25.5k