Questions tagged [operator-algebras]

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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

Filter by
Sorted by
Tagged with
1 vote
0 answers
23 views

If we found a norm for a subalgebra of a C*algebra, is it in fact equivalent to the original norm?

This is from the textbook "An introduction to K-theory for C*alebgra" : So I don't have a question about the problem itself but am more interested in the fact that we can define the norm ...
user avatar
  • 2,973
0 votes
0 answers
27 views

What is odd element of C*-algebra? [closed]

I cannot nowhere find a definition of an odd/even element of C*-algebra. Can someone write it here?
user avatar
3 votes
2 answers
36 views

Tensor product of multiplier algebras is contained in multiplier of tensor product.

I want to realize the following: Given two $C^*$-algebras $A$ and $B$ and with "$\otimes$" denoting the minimal (i.e., spatial) tensor product. We have that $$M(A)\otimes M(B)\subseteq M(A\...
user avatar
  • 225
3 votes
2 answers
53 views

Decompose a positive contraction in a continuous masa in $L(H)$

Let $A$ be a continuous masa in $L(H)$ and $T$ be a positive contraction in $A$. Then we can assume that $0<\|Th\|<1$ for all unit vectors $h\in H$. Otherwise decompose $T$ as $P+T_0$ for some ...
user avatar
  • 1,127
0 votes
0 answers
40 views

Books about operator algebra and number theory

Does anyone know books that covers both operator algebras and number theory. Actually, a number theory books that has operator algebraic approaches.
user avatar
1 vote
1 answer
24 views

Fredholm Index of Toeplitz operators with invertible and continuous symbol

I'm working through the following proof in C* algebras by Murphy, and I'm stuck on a step in the proof. For reference, $\epsilon_n = z^n : T \longrightarrow \mathbb{C}$, and $\Gamma = \text{span}(\...
user avatar
1 vote
0 answers
27 views

Lie-Trotter Theorem proof

Does anyone know how to prove Lie-Trotter theorem: $e^{-iA}e^{-iB}=e^{-i(A+B)}+\mathcal{O}(\delta^{2})$ whereby, $||A||, ||B||<\delta$ and $\mathcal{O}(\delta^{2})$ is shorthand for some arbitrary ...
user avatar
0 votes
1 answer
35 views

Essential spectrum of a projection [closed]

For any $T\in B(H)$, the essential spectrum $\sigma_e(T)$ of $T$ is a subset of the spectrum $\sigma(T)$ of $T$; namely, the $\lambda$ such that $\lambda-T$ is not Fredholm. If $P$ is a projection, we ...
user avatar
  • 1,127
1 vote
0 answers
39 views

dilatation analytic

I have seen that the spectrum of the operator $T=-\frac{d^2}{dx^2}+x^2$ is the $\{2n+1,n \in \Bbb{N}\}$ and by dilatation $x=r y$ the spectrum of the operator $T_r=-r^{-2}\frac{d^2}{dx^2}+r^2x^2$ is ...
user avatar
0 votes
1 answer
25 views

Maximal abelian von Neumann algebra and cyclic vectors

I know the fact that if $A\subset B(H)$ is a maximal abelian von Neumann algebra and $H$ is separable, then $A$ will have a cyclic vector. This result is proved in Conway's book "A course in ...
user avatar
  • 140
1 vote
0 answers
26 views

Am I understanding the reduced group C*-algebras correctly?

Let $G$ be a group and $\mathbb{C}G$ be its group algebra. $\mathbb{C}G$ contains finite linear combinations of group elements of $G$. Let $C^{*}_{r}(G)$ be the reduced group C-algebra. Let $a, b\in \...
user avatar
  • 899
-1 votes
0 answers
18 views

How is the restriction of the disk algebra to the boundary seen as a subalgebra?

The disk algebra is the set of continuous functions $f: D \to \mathbb C$ where $D$ is the closed unit disc in $\mathbb C$ and $f$ is analytic on the interior of $D$. It is endowed with the $\sup$-norm....
user avatar
  • 2,973
1 vote
1 answer
27 views

Confused on two parts of Proof in C* algebras by Murphy

I'm working through the proof of the following theorem from C* algebra by Murphy. For context, $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ is given by $T_{\varphi}(f) = p(\varphi f)$ for $p: L^2(T) \...
user avatar
0 votes
1 answer
44 views

Understanding the proof of Fuglede's theorem

I am trying to understand this proof of Fuglede's theorem on wikipedia : Everything was making sense until the last sentence: "Considering the first-order terms in the expansion for small λ, we ...
user avatar
  • 2,973
0 votes
0 answers
20 views

representation of a linear bounded operator

Let $T\in B(H)$ and $H=H_1\oplus H_2$, then $T$ can be expressed by the following matrix: \begin{pmatrix}A & B\\C & D\end{pmatrix} , where $A\in B(H_1), B\in B(H_2,H_1), C\in B(H_1,H_2),D\in B(...
user avatar
  • 1,127
-1 votes
0 answers
20 views

Relatinships between Non-atomic von Neumann algebras and continuous von Neumann algebras

Does there exist relationships between non-atomic von Neumann algebras and continuous von Neumann algebras? A conditional expectation onto a continuous masa vansihes on compact operators. I found ...
user avatar
  • 1,127
1 vote
0 answers
20 views

Determine whether a self-adjoint operator is negative definite or not

Let $Q$ be an idempotent in $L(H)$. Suppose there exists two sequences of unitaries $\{U_n\}$ and $\{V_n\}$ such that $X$ is the norm limit of $U_nQU_n^*+V_nQV_n^*$, and suppose that $Q+Q^*\cong 2P\...
user avatar
  • 1,127
2 votes
0 answers
38 views

Orthogonal Complement of the $\pi(H)$-invariant Vectors in a Hilbert Space

Let $H \le G$ be an inclusion of countable, discrete groups, and let $\pi$ be unitary representation of $G$ on some Hilbert space $\mathcal{K}$. Is it possible to determine the orthogonal complete $(\...
user avatar
  • 7,082
2 votes
1 answer
45 views

The enveloping $C^*$ algebra of $C^k[0,1]$.

We know that $C^k[0,1]$ is an abelian $*$-Banach algebra, but it is not a $C^*$ algebra in general unless $k=0$. I wonder what's the enveloping $C^*$ algebra of $C^k[0,1]$? I guess it might be $C[0,1]$...
user avatar
  • 140
0 votes
1 answer
21 views

Confused on a set inclusion in C* algebras by Murphy

Im stuck on the following part of this theorem: The closed vector subspaces of $L^2(T)$ invariant for the bilateral shift $v=M_{z}$ (for $z: T \longrightarrow \mathbb{C}$ the inclusion map) are ...
user avatar
1 vote
0 answers
22 views

Von Neumann algebras are generated by its projections

In a lecture about operator theory we used the claim, that the set of projections in a von Neumann algebra $\mathcal M$ is dense in $\mathcal M$, with respect to the operator norm. Sadly that claim ...
user avatar
  • 216
0 votes
1 answer
29 views

How much can the (reduced) group $C^*$ algebra maintain the subgroup structure?

I have read a theorem which says that if $H$ and $G$ are both discrete groups, $G\leq H$, then $C^*(G)$ is a subalgebra of $C^*(H)$ and $C^*_r(G)$ is a subalgebra of $C^*_r(H)$. For now, what I can ...
user avatar
  • 140
1 vote
1 answer
24 views

Strong limit from the unitary orbit of $A$

Let $A,X\in L(H)$ with $\Bbb D\subset W_e(A)$ ,where $W_e(A)$ is the essential numerical range of $A$,and $\|X\|\leq 1$. Then there exists a sequence of unitaries $(U_n)_n$ in $L(H)$ such that wot $...
user avatar
  • 1,127
1 vote
0 answers
41 views

On double dual of C* algebra

Can anyone provide examples of double dual of $C^{*}$-algebras except $K(\mathcal{H})$, commutative cases? Thanks in advance!
user avatar
  • 598
2 votes
2 answers
73 views

Questions on Theorem 5.5.7. in Brown-Ozawa

I am currently trying to digest the proof of Theorem $5.5.7.$ in the book "$C^*$-algebras and Finite-Dimensional Approxmiations" by N. Brown and N. Ozawa. Background: Let $(X,d)$ be a metric ...
user avatar
  • 225
1 vote
1 answer
26 views

Normal states on a 2 by 2 complex matrix

Denote $S$ by the set of normal states on $M_2(\Bbb C)$. Suppose $p(p\neq 0,1)$ is a projection in $M_2(\Bbb C)$ and $\rho\in S$. Define $S_p:=\{w\in S:w(p)=0\}$ and $d(\rho,S_p):=\inf_{\omega\in ...
user avatar
  • 1,127
-2 votes
0 answers
30 views

Two von Neumann algebras not isomorphic as C*-algebras, can they be isomorphic as von Neumann algebras? [duplicate]

This might be a silly question. Consider two von Neumann algebras $M, N$, given that they are not isomorphic as C*-algebras to each other, is there a chance that they are isomorphic as von Neumann ...
user avatar
  • 899
1 vote
0 answers
35 views

Spectrum of a Schwartz space of cotangent bundle

Let $E \to X$ be a smooth vector bundle over a $C^\infty$-manifold. There is an isomorphism between the algebra $(\mathcal{S}_c(E),*)$ and $(\mathcal{S}_c(E^*),.)$ by the Fourier transformation. My ...
user avatar
  • 53
1 vote
1 answer
44 views

Will the continuity of functional calculus in another aspect still be ture?

It is a well known fact that if we have a fixed normal operator $T$ on $B(H)$, $f$ and $g$ are both continuous on $\sigma(T)$, then $||(f-g)(T)||=||f-g||_{\sigma(T)}$. It tells us that functional ...
user avatar
  • 140
0 votes
1 answer
33 views

In what sense does continuity hold for unbounded functional calculus?

Let $f : \mathbb{R} \to \mathbb{R}$ be a Schwartz function. Let $I$ be the identity operator on some separable Hilbert space $\mathcal{H}$ and $A$ be an unbounded self-adjoint operator on $\mathcal{H}$...
user avatar
  • 5,360
3 votes
1 answer
47 views

How do operators of operators work?

$u,v \in V$ $M,N \in O\equiv$ {Linear operators on $V$} $A,B \in O^2\equiv$ {Linear operators on $O$} Does $O^2$ add anything new? Is it just isomorphic to $O$? $O$ and $O^2$ are both associative ...
user avatar
  • 352
1 vote
0 answers
25 views

Extension of a continuous and equivariant map to compactification

Given a locally compact (topological) group $G$ we denote by $C_b^{lu}(G)\subseteq C_b(G)$ the bounded continuous functions on $G$ such that $f\in C_b^{lu}(G)$ whenever the map $G\to C_b(G)$ given by $...
user avatar
  • 225
2 votes
1 answer
52 views

Given an ideal $I\subset A$ and an element $a\in A$, is it true that $\sup\{\|ab\|:b\in I,\ \|b\|\leq 1\}=\|a\|$?

Let $I$ be a 2-sided closed ideal in a C*-algebra $A$. Given $a\in A$, is it true that $$\sup_{\substack{b\in I\\ \|b\|\leq 1}}\|ab\|=\|a\|?$$ Or do we need assumptions on $I$? Note that the left-hand ...
user avatar
  • 2,968
1 vote
1 answer
39 views

A question about the quotient space of von Neumann algebras

It is well known that in $C^*$ algebra category, If $\mathcal{A}$ is a $C^*$ algebra and $\mathcal{I}$ is a norm closed ideal of $\mathcal{A}$, then $\mathcal{I}$ is also a $C^*$ algebra, and $\...
user avatar
  • 140
1 vote
1 answer
92 views

SOT convergence of normal operators

Let $T_n$ be a sequence of bounded normal operators on a Hilbert space which converges to a normal operator $T$ in the strong operator topology. Show that $T_n^*$ also converges to $T^*$ in SOT. I ...
user avatar
  • 31
1 vote
1 answer
27 views

Multiplier algebra of $C_0(X)$

I'm looking at the following example from C* algebras by Murphy, and I'm totally lost on the notation they're using. "Let $X$ be a locally compact Hausdorff space. Since $C_0(X) \subset C_b(X)$ ...
user avatar
0 votes
1 answer
37 views

Why does this theorem imply this next result?

I'm reading through $C^{*}$ algebras by Murphy, and the following theorem is presented. Let $I \subset A$ be a closed ideal of a $C^{*}$ algebra $A$. Then there exists a unique $*$ homomorphism $\...
user avatar
1 vote
1 answer
32 views

Extension of slice map to WOT closure

Given Hilbert spaces $H_1,H_2$ and a functional in the predual $\psi\in B(H_1)_*$ we may consider the slice map $S:B(H_1)\otimes B(H_2)\to B(H_2)$ defined on the spatial tensor product given by ...
user avatar
  • 225
0 votes
0 answers
18 views

Trying to understand maximal tensor product of Ternary rings of operators

Let $V$ and $W$ be ternary rings of operators (TROs). In section 5 of Kaur and Ruan - Local Properties of Ternary Rings of Operators and Their Linking $C^*$-Algebras, the maximal tensor product $\...
user avatar
  • 2,990
0 votes
1 answer
27 views

Doubt on max tensor product of $C^{\ast}$-algebras

Im trying to understand proof of corollary $11.34$ from here. The corollary goes as follows: Let $A_1$ and $A_2$ be $C^{\ast}$-algebras. Given any $C^{\ast}$-norm $\vert \vert . \vert \vert$on $A_1 \...
user avatar
  • 2,990
1 vote
1 answer
21 views

Compression map is an isomorphism from $pB(H)p$ to $B(K)$ via $u \to u_K$

I was reading a note on Von Neumann Algebra, and I am not able to understand this phrase as: Let $K$ be a closed vector subspace of a Hilbert space $H$ and let $p$ be the projection of $H$ onto $K$. ...
user avatar
  • 1,551
3 votes
1 answer
48 views

Why do we need this extra step in this proof?

For some background information, given sets $I_1,\dots,I_n \subset A$ of a $C^{*}$ algebra $A$, we define $\prod_{k=1}^n I_k$ to be the closed linear span of all products of the form $\prod_{k=1}^n ...
user avatar
3 votes
1 answer
96 views

Does any von Neumann algebra have $\sigma$-finite projections?

Let $M$ be a von Neumann algebra. Let $\Sigma$ be the set of $\sigma$-finite projections of $M$. In Takesaki's book "Theory of operator algebras II", chapter 7, p51, in the proof of theorem ...
user avatar
  • 985
1 vote
1 answer
40 views

Norm product inequality for unitisation of a $C^*$-algebra

Let $A$ be a $C^*$-algebra without a unit. Define $\widetilde{A}=\{(\alpha,a):\alpha \in \mathbb{C}, \;a\in A\}$ equipped with componentwise addition and scalar multiplication. Vector multiplication ...
user avatar
  • 85
0 votes
1 answer
40 views

Turning a scalar into a derivative operator (context: Schrödinger equation)

At the sight of the Schrödinger equation, most of you will think "No, you are not in the right forum !". However, I'm not interested in the physical sense of this (fabulous) equation today. ...
user avatar
2 votes
1 answer
65 views

Unique square root of a positive operator on a Hilbert space

Let $T$ be a positive operator on a Hilbert space $\mathcal{H}$. That is $\langle Tx,x\rangle\geq0$ for all $x$. Then there is a unique positive operator $S$ such that $S^2=T$ and this $S$ is called ...
user avatar
  • 1,542
2 votes
0 answers
101 views

Convergence of vector states

Let $\mathfrak{A} \subset B(H)$ be a C*-algebra. For $x \in H$, $\|x\|=1$, define a vector state $\omega_x$ on $\mathfrak{A}$ by $\mathfrak{A} \ni A \mapsto \langle x, Ax \rangle$. Assume $(x_n)_{n\in ...
user avatar
  • 186
1 vote
2 answers
52 views

Minimal tensor product of $B(H)$ and $C(G)$

Let $H$ be a finite dimensional vector space, and $G$ be a compact group. Let $B(H)$ be the bounded operators on $H$, let $C(G)$ be the complex valued continuous functions on $G$, and let $C(G;B(H))$ ...
user avatar
  • 11
3 votes
1 answer
65 views

When is a group von Neumann algebra a factor?

It is well-known that a von Neumann algebra (on a separable Hilbert space) can be written as a direct integrals of factors, i.e., von Neumann algebras with center $\mathbb C I$. As such, factors play ...
user avatar
  • 11.9k
1 vote
1 answer
34 views

Quotient of function spaces is function space on set difference

I have seen it stated that for an open subset $Y\subseteq X$ such that $X$ is a compact Hausdorff space we get an identification of the $C^*$-algebras : $C(X\setminus Y)\cong C(X)/C_0(Y)$. I suppose ...
user avatar
  • 225

1
2 3 4 5
61