# Tagged Questions

41 views

### Riesz Functional Calculus vs. Holomorphic Functional Calculus

"Functional calculus" is a word used to describe the practice of taking some functions or formulas defined on complex numbers, and apply them in some way to certain kinds of operators, despite that ...
23 views

### Question about the Averson's proof of the bicommutant theorem.

In the Averson's proof of the bicommutant theorem is proved that, if $A$ is a self-adjoint algebra of operators with trivial null space and $T \in A''$, for every $\epsilon>0$, $n=1,2..$ and every ...
28 views

### Question about a passage in the Bicommutant Theorem's proof.

In the Averson's book, in the proof of the Von Neumann's Bicommutant theorem there is this passage: ($A$ is a self-adjoint algebra of operators in $L(H)$) "Let $\xi_1$ be an element of the Hilbert ...
22 views

### How to prove the following map is a c.c.p map

Here is a quotation in a book "C*-algebrass and Finite-Dimensional Approximations" by Nate and Taka (P122). Let $A$ be a C*-algebra, $\Gamma$ be a discrete group and the $\alpha$ is an action of ...
37 views

### An exercise about reduced crossed product of a C*-dynamical system

Here is an Exercise in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Exercise 4.1.3. Let $A$ and $B$ be two C*-algebras and $\Gamma$ be a discrete group. If ...
25 views

### A question about cross product of C*-dynamical system

Here is a Theorem (P123) in a book "C*-algebras and Finite-Dimensional Approximations" by N.P.Brown and N. Ozawa. Theorem 4.2.4. Let $A$ be a C*-algebra and $\mathbb{Z}$ be the set of integers. ...
24 views

### $R\mbox{ is a right multiplier and }R(a)b=a\overset{?}{\implies} A\mbox{ is unital }$

Let $A$ be a $C^*$-algebra, and $R:A\to A$ its right multilplier. Is it true that $$\exists b\in A\quad \forall a\in A \quad R(a)b=a\qquad$$ implies $A$ is unital. I know this is true if A is a ...
54 views

### Integration with values in a $C^*$-algebra

My question is quite specific to locally compact groups but I'm sure it can be generalised to locally compact Hausdorff spaces with a Borel measure. Let $G$ be a locally compact group and fix a Haar ...
28 views

### A question on nuclearity

Definition 2.1.1. If $A$, $B$ are C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
55 views

### A simple lemma in tensor product

Here is a quotation of a book: ($\otimes$ denotes the minimal tensor product) Lemma 3.9.2. Let $A$ be a C*-algebra. If $E\subset A$ is an operator system and $J\triangleleft B$ is an ideal, then ...
72 views

### No trace on $B(H)$ if $H$ is infinite dimensional

Let $H$ be an infinite dimensional Hilbert space and $B(H)$ the bounded linear operators on $H$. Then thre is no ultra weakly continous non-zero positve trace $tr:B(H)\rightarrow \mathbb{C}$. I ...
48 views

### Ultra weakly continuous trace on a von Neumann Algebra

Let $M$ be a infinite dimensional von Neumann Algebra with a positive, faithful, ultra weakly continuous trace $tr:M\rightarrow \mathbb{C}$. Is it possible to show that $tr$ is strongly continuous?
29 views

### A question on a lemma about the product map

Here is a Lemma in the book “C*-algebras and Finite-Dimensional Approximations”: Lemma 3.8.4. Let $A$ be a C*-algebra, $M\subset B(H)$ be a con Neumann algebra and $\phi: A\rightarrow M$ be a ...
19 views

### Weyl Operators: Discontinuity

Let $\mathcal{A}_{CCR}(\mathcal{H})$ be a CCR algebra over a Hilbert space $\mathcal{H}$. Then the Weyl operators are unitary and therefore $\sigma(W(f))\subseteq \mathbb{S}$ so by the spectral ...
37 views

### Bogoliubov Transformation

Let $\mathcal{A}_{CAR}(\mathcal{H})$ be a CAR algebra over a Hilbert space $\mathcal{H}$. Consider a linear $S$ and an antilinear $T$ both bounded operators acting on $\mathcal{H}$ satisfying: ...
34 views

### Ultra weakly closed *-subalgebra of B(H)

I'm currently working on a text about von Neumann algebras and the author used without further clarifying that any ultra weakly closed *-subalgebra of $B(H)$ contains a largest projection. Could ...
33 views

### Polar decomposition in a von Neumann algebra

Let $M \subseteq B(H)$ be a von Neumann algebra and $T \in M$. If $T=U|T|$ is the polar decomposition of T, why is $U \in M$? I'm thinking it's because $M$ is SOT-closed, but I'm not entirely sure.
25 views

### Analytic Vectors (Nelson's Theorem)

Is there a (simple) proof for Nelson's theorem that a symmetric operator is essentialky selfadjoint if it contains a dense subset of analytic vectors?
61 views

### C*-Algebra: Positive Operator

Let $\mathcal{A}$ be a C*-algebra. If $A\in\mathcal{A}$ is a selfadjoint element then $A^*A=A^2$ has positive spectrum since: ...
41 views

Let $\mathcal{A}$ be a C*-algebra and $\omega$ a positive linear functional. Is there a simple proof for Kadison's inequality: $$\overline{\omega(A)}\omega(A)\leq\omega(A^*A)$$
40 views

### The spectral projection of a positive operator

Let $T_{n}\in B(H)$ be a positive operator on Hilbert space $H$ and $T_{n}\rightarrow 1_{H}$ in the strong operator topology. Now fix $\delta>0$ and let $P_{n}$ be the spectral projection of ...
87 views

### Positive Operators: Definition?

Let $A$ be a self adjoint element of a C*-algebra $\mathcal{A}$ resp. a self adjoint operator of the operator algebra $\mathcal{B}(\mathcal{H})$ of bounded operators over a Hilbert space ...
49 views

### *-representations of dense subalgebras

Let $H$ be a separable Hilbert space and let $K(H)$ be the C*-algebra of compact operators on $H$. Suppose that $A$ is a *-subalgebra of $K(H)$ which contains all the finite-rank operators. Given a ...
29 views

### What is the definition of regular operator?

If $T$ is a bounded linear operator on a normed space $X$. What "$T$ is regular operator" means?
25 views

### About what happens to eigenspace under functional calculus for Unbounded Operator

Let $T$ be an unbounded self adjoint positive operator on a Hilbert Space $\mathcal{H}$. Let $x \in \mathcal{H}$ be a vector such that $Tx = x$. Is it true that $T^{\frac{1}{2}} x = x$. For what $f$ ...
Here is a quotation: Corollary 3.7.12 If $\Gamma$ is a non-amenable residually finite group, then $C^{*}(\Gamma)$ is not exact. It follows from this corollary that $B(l^{2})$ is not exact ...