# Tagged Questions

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

26 views

### Involution and Gelfand Transform Properties

Let $\mathcal{B}$ be a commutative unital Banach algebra, and let for each $x\in\mathcal{B}$ $\hat{x}$ be the Gelfand transform. I assume that $\mathcal{B}$ has an involution *. I want to show that: ...
30 views

### The set of all maximal ideals (Wiener Algebra)

I am trying to prove a proposition and in my proof I somehow need to find the set of all maximal ideals of a Banach Algebra. This is my working environment: Let $A(\mathbb{R}^2)$ be the (Wiener ...
9 views

24 views

14 views

27 views

### Lemma V.1.25 (page 298) from the book of Takesaki (Vol 1).

I have a problem with the following lemma (Takesaki Vol 1-page 298): Lemma 1.25. If $e$ is an abelian projection in a von Neumann algebra $\mathcal M$, then for any projection $f\in\mathcal M$ ...
45 views

### Universal $C^*$ algebras

It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for ...
35 views

### about representations of a simple $C^*$-algebra

We know that every simple $C^*$-algebra is primitive, say it has a faithful non-zero irreducible representation. The converse is not necessarily true. An counterexample is just the $B(H)$ when $H$ is ...
35 views

### Not every positive operator is positive-definite operator

According to the definitions for the operator $L: H \rightarrow H$ we have: $L$ is positive operator if the inner product $\langle Lu\mid u \rangle \geq 0$ for $\forall u \in H$ $L$ is positie-...
43 views

### What is the maximal ideal space of $H^\infty$?

What is the spectrum of $H^\infty$, the Banach algebra of all bounded holomorphic functions in the open unit disk $D=\{z\in \mathbb{C}\mid |z| <1 \}$?
44 views

15 views

### Semi-finite trace on a von Neumann algebra: Equivalent definitions

Let $(N,\tau)$ be a semi-finite von Neumann algebra. This means that $\tau$ is a normal, faithful and semi-finite trace. Normality means that $\tau(x) = \sup_i \tau(x_i)$ if $x \in N_+$ is the limit ...
25 views

### modular operator

I could calculate a modular operator $Δ$ for a state $τ_α(x_{ij})= αx_{11}+(1-α)x_{22}(0<α<1)$. But, I cannot understand an automorphism $σ_t(x)＝Δ^{it}(x)Δ^{-it}$ of $M_2(\mathbb{C})$. For ...
41 views

### Compute the positive part of $K_0(A)$ where $A$ is a simple AF algebra

I'm trying to understand the following example from my lecture notes: Define $A_n=M_{F_n}(\Bbb{C})\oplus M_{F_{n+1}}(\Bbb{C})$ where $F_n$ defined by $F_1=1, \ F_2=2, \ F_{n+2}=F_{n}+F_{n+1}$, i.e., ...
24 views

### Countable weighted shift has no invariant subspace.

Suppose I have $T(e_n)=w_ne_{n+1}$ where $w_n>0$ (and are bounded) and $\{e_n\}$ denotes the canonical basis of $l^{2}(\mathbb{N})$. I would like to prove that the only (closed) invariant ...
108 views

Let $\mathcal{A}$ be a unital $*$-algebra over $\mathbb{C}$ and let $a,b\in\mathcal{A}$ be projections, that is, $a=a^*=a^2$ and $b=b^*=b^2$. If $a+b=1$, then $ab=0$. This follows from - \begin{align*...
40 views

### Continuous family of subalgebras in a C* algebra

Let A be a separable C* algebra. For t $\in$ $\mathbb{R}$ let $A_t$ be a subalgebra of $A$ such that: $A_t \cong \mathcal{O}_n$ (Cuntz algebra for fixed n). Generators of $A_t$ depend continuously ...
34 views

### Nonunital C*-Algebra: Proper Ideals

Given a C*-algebra without unit. Does there exist a nontrivial proper ideal that does not lie in a maximally nontrivial proper ideal? (For the unital case this follows easily by Zorn's lemma.) ...
10 views

23 views

### In a C*-algebra, pure states which share the same kernel are equal

I'm reading C*-Algebras by Jacques Dixmier. And in the proof of 2.9.5, it says Let $A$ be a C*-algebra. If $f$ and $f'$ are two pure states which have the same kernel, then $f=f'$. It should ...
24 views

### Proving two stubborn inequalities for completely positive maps in C*-algebras

I recently came across this in my studies of functional analysis in C* algebras which got me stuck: For a completely positive map between C* algebras $\phi : A \to B$ we are to prove these two ...
29 views

### Non-self Adjoint Operator Algebra References

The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be ...
37 views

### Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
44 views

### $A\in B(H)$ a unital abelian $C^*$-algebra with cyclic vector then $A'$ is abelian as well

Let $A$ be a unital abelian $C^*$-subalgebra of $B(H)$ (with the same unit as that of $B(H)$), and assume there exists a vector $\xi \in H$ which is cyclic for $A$ (that is, $\{a\xi | a\in A \}$ ...
30 views

When I read some materials in topological dynamics, I met words: "canonical action of a countable discrete group $G$ on its stone-cech compactification $\beta G$" without any definition. I know that $... 1answer 51 views ### Elementary proof that$a - 1$is not invertible, for self-adjoint$a$with$\lVert a \rVert = 1$Assume$a \in A$where$A$is a unital$C^*$-algebra. If$\lVert a \rVert = 1$and$a^*=a$we know that$1 \in \sigma(a)$, the spectrum of$a$. This follows from the fact that$\lVert a \rVert = r(a) =...
Let $x$ be an operator in $B(H)$. We say a pair $(c,y)$ forms a polar decomposition for $x$ if $y$ is a positive operator, $c$ in $B(H)$ with $x=cy$ such that the restriction of $c$ on $\overline{yH}$...