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 ...

learn more… | top users | synonyms (1)

3
votes
1answer
332 views

Spatial tensor product of algebras and normed spaces

Let $A_1$ and $A_2$ be two $C^*$-algebras considered as closed $*$-subspaces of some $\mathcal{B}(H_1)$ and $\mathcal{B}(H_2)$. More preciesly there exist faithfull $*$-representations ...
3
votes
0answers
35 views

C*-algebra representation

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
3
votes
0answers
29 views

Short exact sequence involving mapping cone, cone, suspension of $C^*$-algebras

This is part of exercise 6.N in Wegge-Olsen's book '$K$-theory and $C^*$-algebras'. In the following, $A$ and $B$ are $C^*$-algebras, $\alpha:A\rightarrow B$ is a surjective $C^*$ morphism with kernel ...
3
votes
0answers
67 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 ...
3
votes
1answer
100 views

application of c*algebras

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...
3
votes
0answers
42 views

Motivations for and connections between the topologies of Vietoris, Fell and Chabauty

My main interest is in the Chabauty topology on the space of closed subgroups of a locally compact topological group, merely out of curiosity. Wikipedia states "it is an adaptation of the Fell ...
3
votes
0answers
126 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
0answers
129 views

Unitary operators - convergence problem

Let $\mathcal{U}:=\left\{ U(t) \colon t \geq 0\right\}$ be a family of unitary operators on a Hilbert space $\mathcal{H}$ where $U(0)=I$. Assume that $\left| \left<\left( \frac{U(t)-I}{t} - A ...
3
votes
0answers
66 views

maps from a convex set to itself

Suppose $S\subset \mathbb{R}^n$ be a closed convex set under Euclidean topology (but not necessarily bounded, example a closed cone). Let $\mathcal{E}(S)=\{L:\mathbb{R}^n\rightarrow \mathbb{R}^n\text{ ...
3
votes
0answers
59 views

Given $\theta(p)\neq p$ does there exist $q\leq p$, so that $\theta(q)q=0$?

Let $\mathfrak{M}$ be a vN-Alg. Let $\theta\in \text{Aut}(\mathfrak{M})$. Let $p\in\mathfrak{M}$ be a projection, so that $\theta(p)\neq p$. Is there a projection $q\in\mathfrak{M}$ with $0\neq q\leq ...
3
votes
0answers
97 views

Defining entanglement in subspaces of tensor product

Let $\mathcal{H}=\mathbb{C}^n$ be a Hilbert space. A state $\rho\in\mathcal{B(H)}$ is a positive semi-definite operator with unit trace. $\rho\in \mathcal{B(H)}$, where ...
3
votes
0answers
137 views

Minimal projections vs maximal left ideals

I've seen in some papers a statement (which is referred to a very old book of Dixmier in French which I have no access to / can't read anyway) saying that maximal left ideals of a (unital) C*-algebra ...
3
votes
0answers
130 views

Two questions about ultraweak and ultrastrong topology from Dixmier

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below: Let $M$ be an ultraweakly ...
3
votes
0answers
138 views

Basis for completely bounded maps.

The set of completely bounded (CB) maps forms can be considered as a complex span of the set of completely positive (CP) maps. Can we find a basis for this complex linear space of CB maps such that ...
3
votes
1answer
167 views

reduced crossed products

Given a discrete group $G$ and a $G$-$C^*$-algebra $A$ we can form the reduced crossed product $A\rtimes_r G$. I want to define it by the closure of $C_c(G,A)$ in $\mathcal{B}(\ell^2(G,A)$ where this ...
3
votes
0answers
65 views

Dropping homomorphisms to quotients of C$^*$-algebras

Let $A$ be a C$^*$-algebra, let $\Delta:A\rightarrow A \otimes_{\min} A$ be a $*$-homomorphism, and let $\phi$ be a state on $A$. Let $(H,\pi,\xi_0)$ be the GNS construction for $\phi$; let $B=\pi(A) ...
3
votes
0answers
241 views

When does Stinespring dilation yield a faithful representation?

Let $A$ be a $C^*$-algebra, $H$ a Hilbert space, $\phi: A \to B(H)$ a completely positive map. The Stinespring construction yields a triple $(K, V, \pi)$ where $K$ is a Hilbert space, $V: H \to K$ a ...
2
votes
3answers
362 views

Eigenvalues of operators

I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way: $T(A)=0.5(A-A^\mathrm{t})$ ($A^\mathrm{t}$: the transpose of a matrix $A$). I need to prove ...
2
votes
2answers
90 views

Commutant of bounded linear operators on a Hilbert space

Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by $$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall ...
2
votes
3answers
55 views

Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
2
votes
2answers
60 views

A simple question about operator norm

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
2
votes
3answers
139 views

universal C* algebras

Is there a standard reference which has a discussion on universal $C^*$-algebras ? (definition, properties, examples, etc) Searching on the internet has led me to tidbits of information but I would ...
2
votes
2answers
154 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
2
votes
2answers
39 views

A definition of discrete group

Definition: A discrete group $\Gamma$ is called residually finite if there exist subgroups $\Gamma\supset\Gamma_{1}\supset\Gamma_{2}\supset...$ such that each $\Gamma_{i}$ is a finite-index, normal ...
2
votes
2answers
119 views

Problem with spectral theorem and spectral measure.

There is a passage in a book that is not very clear to me: A is a C*Algebra and $a$ is selfadjoint. Then "Indeed identifying A with an algebra of operators on a Hilbert space $\mathcal{H}$, by the ...
2
votes
1answer
42 views

A question about compact Hausdorff space

Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. ...
2
votes
1answer
59 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
2
votes
1answer
58 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 ...
2
votes
2answers
197 views

Unitary operator in von Neumann algebra

Let $R\subseteq B(H)$ be a von Neumann algebra, and $U\in R$ be unitary. Prove that there is a self adjoint operator $A\in R$ such that $||A||\leq \pi$, and $U=\exp(iA)$ . Any idea how to start! Thank ...
2
votes
1answer
18 views

If $A$ is a primitive / irreducible C*-algebra, then $M(A)$ has trivial center.

Recall some definitions: a sub-C*-algebra $A$ of $B(H)$, the algebra of bounded operators on a Hilbert space $H$, is called irreducible if the only closed $A$-invariant subspaces of $H$ are $0$ and ...
2
votes
1answer
18 views

Commutative subspace lattice

I have seen an article in which there is an algebra which was named CSL-algebra (Commutative Subspace Lattice). This algebra is about projection on Banach algebra? I couldn't find any good source to ...
2
votes
1answer
58 views

Existence of invariant states in a $C^*$-algebra

Let $\mathcal{A}$ be a C*-algebra and $\{\tau_t\}_{t\in\mathbb R}$ a weakly-continuous group of *-automorphisms. I've read the claim (without proof) that for any state $\eta$ (that is $\eta$ is a ...
2
votes
1answer
79 views

A question on simple and unital $C^\star$-algebra

There is a quotation of a book "$C^\star$-algebras Finite-Dimensional Approximations" Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
2
votes
1answer
83 views

A theorem about conditional expectation in C*-algebra

Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a ...
2
votes
1answer
55 views

How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
2
votes
1answer
39 views

Exponential map in $C^{*}$-algebra and unitary invariance

Let $A$ be a unital $C^{*}$-algebra. Let $X$ be a closed vector subspace of $A$ which is unitarily invariant in the sense that $uXu^{*}\subseteq X$ for all unitaries $u$ of $A$. I want to show that ...
2
votes
1answer
79 views

When an invertible element in a $C^{*}$-algebra is unitary

I am trying to show that if $a$ is an invertible element of a unital $C^{*}$-algebra, and $||a||=||a^{-1}||=1$, then $a$ is unitary. I can do this if I think of $a$ as a Hilbert space operator using ...
2
votes
1answer
87 views

An exercise on C*-algebra

A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every ...
2
votes
1answer
126 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
2
votes
1answer
105 views

K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
2
votes
2answers
97 views

Polar decompostion for the operator algebras

I find that most of books discussing the polar decompostion at the W*-algebras, but not C*-algebras. I guess the rough reason is that the element of W*algebras has the well supported set, but I want ...
2
votes
1answer
94 views

Dense subalgebras of von Neumann algebras and increasing nets

Let $N$ be a von Neumann algebra, and $A$ be a dense $*$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that: For any $x\in N^+$, there exists a increasing net $(x_j)$ in ...
2
votes
1answer
72 views

Conjugates in integrals

Is it always true that $\int f(x)^*M^\dagger Mf(x) dx=\int (Mf(x))^*(Mf(x)) dx$? where $M$ is an operator. If so, is there a simple proof that this is so? THanks.
2
votes
1answer
23 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 ...
2
votes
2answers
32 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 ...
2
votes
1answer
19 views

An extension of representation

Let $A,~B$ be two C*-algebras, if $A$ is an ideal in $B$, then do we have that any representation of $A$ can extend to a representation of $B$?
2
votes
1answer
32 views

The canonical surjection between the full and the reduced group C^*-algebras

This might be an incredible easy question -- since any reference I've found state it as obvious -- but anyway: Given a group $G$, I can construct the full group-$C^*$-algebra $C^*(G)$ be completing ...
2
votes
1answer
28 views

Questions about multiplier algebra and corona algebra

When I read N.E. Wegge-Olsen's book K-theory and C-star-algebras_ A friendly approach I meet the following two problems about standard isomophisms: For any $C^\ast$-algebra $\mathcal{A}$, is ...
2
votes
1answer
32 views

Question about finite rank operators

Let $X$ be a normed space, $\mathcal{F}(X)$ the algebra of all operators on $X$ with finite fank, then $\mathcal{F}(X)$ is the unique minimal ideal of $\mathcal{K}(X)$ the algebra of all compact ...
2
votes
1answer
34 views

Are contractive completely positive maps trace decreasing?

Are contractive completely positive maps trace decreasing? More precisely, suppose that $f\colon M\to N$ is a normal cpc map between von Neumann algebras with normalised normal traces. (That is ...