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)

1
vote
1answer
175 views

What are operators, commutators and anti commutators algebra?

What is the proof for the fact that the product of two operators is generally not commutative? $$\hat A\hat {\vphantom{A}B}\not=\hat{\vphantom{A}B} \hat A.$$ What is the difference between $\hat ...
1
vote
2answers
131 views

Does the inequality $0\leq a\leq b$ in a C*-algebra imply $\|a\|\leq\|b\|$?

In relation to this question of mine: C* algebra inequalities I am wondering if it is true that if $0\leq a \leq b$ in a C* algebra, does one have $||a||\leq||b||$? If you need the C* algebra to be ...
1
vote
1answer
221 views

Kaplansky's density theorem proof

I will assume familiarity with the statement, which can also be found here, and I will use the notation there too. http://en.wikipedia.org/wiki/Kaplansky_density_theorem I have a problem with the ...
2
votes
1answer
75 views

An expression for the sup of projections

In $B(H)$ where $H$ is a Hilbert space, we have that if p and q are (orthogonal) projections that Inf{$p$, $q$} is in $B(H)$. This also holds if we replace $B(H)$ by the phrase "the strong closure of ...
13
votes
1answer
328 views

How does $\sigma(T)$ change with respect to $T$?

Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane. I wonder whether there is some result concerning how ...
1
vote
1answer
79 views

Showing that the WO closure of a *-algebra is a Von Neumann Algebra

I think it's best to defer to the source that I'm reading for a statement of exactly what I need to prove. Please refer to statement EP6 found on p.20 of this source. The trouble is I don't follow ...
2
votes
1answer
143 views

Subadditive, submultiplicative and scalar-multiplication-invariant functions

Let $\mathcal{A}$ be an algebra. $d: \mathcal{A}\to \mathbb{N}$ is a function satisfying 1) $d(S+T)\le d(S)+d(T)$, 2)$d(ST)\le d(S)+d(T)$ and 3) $d(aS)=d(S)$ for all $a\in \mathbb{C}, ...
1
vote
1answer
155 views

Positive Linear Functionals on Von Neumann Algebras

Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal ...
2
votes
1answer
71 views

Classification of Type 1 factors

In the proof of this theorem, which says all of the type 1 factors (factors with minimal projections) are isomorphic to $B(\ell^2(I))$ for some $I$, I want to know a few things: The supposed ...
1
vote
1answer
112 views

Properties of an algebra of operators that are invariant under scalar multiplication

I am thinking about the invariant subspace problem or some related problems like the almost invariant half-space problem. In this type of problems one has the following If a statement holds for an ...
6
votes
1answer
260 views

Weak-* continuity of the adjoint map on a $W^*$-algebra

Let $\mathcal{M}$ be a $W^*$-algebra, i.e. a $C^*$-algebra with a Banach space predual $\mathcal{M}_*$. I'm trying to show that the adjoint map $x \mapsto x^*$ on $\mathcal{M}$ is weak-* (aka ...
8
votes
1answer
308 views

Ideals in $C(X)$

Let $X$ be a Hausdorf Compact topological space. Please help me to show, for the purpose of understanding an example in some of my lecture notes, that the closed ideals in $C(X)$ are of the following ...
2
votes
0answers
119 views

Is inversion sequentially continuous in SOT?

Let $A_n \overset{SOT}{\to} A$ where $A$ is invertible. Does $A_n^{-1} \overset{SOT}{\to} A^{-1}$? Does $A_n^{-1} \overset{WOT}{\to} A^{-1}$? EDIT: Forgot to mention $\{A,A_n\}\in\mathscr{B(H)}$ ...
3
votes
1answer
215 views

One particular application of the Cauchy Schwarz Inequality

A document I am reading on Von-Neumann algebras (VNA) asserts that it follows from Cauchy-Schwarz that if $M$ is a VNA, and $w$ is a positive linear functional on M that is merely norm continuous, ...
2
votes
1answer
146 views

Generation of Von Neumann Algebras

Suppose $M$ is a Von Neumann Algebra. (VNA) For me, these are subsets of some $B(H)$ that are $*$-algebras, containing the $1$ of $B(H)$, that are Weak Operator (WO) closed, or equivalently Strong ...
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 ...
1
vote
1answer
2k views

Why call this a spectral projection?

Regarding this question, Why do spectral projections give norm approximations? I have figured out what is meant by spectral projection, and have thus found the answer as well. A spectral projection ...
0
votes
1answer
386 views

Why do spectral projections give norm approximations?

First off, I'd like to ask: If $H$ is a Hilbert space, and we have $A$ a bounded operator from $H$ to itself, $A$ being self adjoint (or normal), then if $A$ is compact there is a eigenspace ...
4
votes
0answers
75 views

Products in $C^*$-algebra $K$-theory

Let $A_1$ and $A_2$ be unital $C^*$-algebras. If $p_1 \in M_{n_1}(A_1)$ and $p_2 \in M_{n_2}(A_2)$ are projections then $p_1 \otimes p_2 \in M_{n_1 n_2}(A_1 \otimes A_2)$ is also a projection, ...
3
votes
1answer
186 views

Strongly-Continuous linear functionals on $\mathcal{B}(H)$

Suppose $H$ is a complex Hilbert space and $$w: \mathcal{B}(H) \longrightarrow \mathbb{C}$$ is a bounded linear functional on $\mathcal{B}(H)$ such that $w$ is continuous even if $\mathcal{B}(H)$ is ...
0
votes
2answers
173 views

In a C*-algebra, does $a \leq b$ imply $a^2 \leq b^2$?

While attempting to fill in the gaps in a proof of the Gelfand-Naimark-Segal representation theorem that I was given in a course in operator algebras, I found myself wondering whether, if ...
3
votes
2answers
299 views

Examples of not completely bounded maps

Let $\phi:\mathcal{A}\longrightarrow\mathcal{B}$ be a bounded map between $C^*$ algebras. $\phi$ is said to be completely bounded if the natural extension map \begin{eqnarray} ...
1
vote
1answer
1k views

Taylor expansion for matrices

Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ? More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ ...
4
votes
0answers
103 views

Direct limits of completely positive maps on $C^*$-algebras vs. operator systems

I believe I've heard, as part of the "lore," that the category (operator systems, completely positive maps) has direct limits, whereas the category ($C^*$-algebras, completely positive maps) does not. ...
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) ...
5
votes
1answer
181 views

Separating vectors for $C^*$-algebras

Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the map $A\rightarrow H, a\mapsto a(\xi_0)$ is injective ...
1
vote
1answer
279 views

Orthogonal Projectors and Operators

If we have a projection T on a finite-dimensional inner product V, and we know that ||Tx|| = ||x||, can we conclude that T is an orthogonal projection? The equality with the norms is enough to ...
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 ...
1
vote
0answers
91 views

Affine Homeomorphism between a compact set K and the state space on A(K)

Let $V$ be a locally convex space, and let $K$ be compact set in $V$. Define $A(K)\subset C(K)$ as $A(K)=\{\phi:K\rightarrow \mathbb{C}\; |\; \phi\; \text{is continuous and affine}\}$. Then we know ...
1
vote
0answers
93 views

Extreme points and Matrix Extreme Points

With reference to this paper. Let $V$ be a locally convex space, and $K=(K_n)$ be a compact matrix convex set in $V$. Then as proved in Cor 3.6 in the above paper, we see that if $v\in K_n$ is a ...
4
votes
2answers
122 views

How to prove compactness of matrix convex sets?

I am reading a paper - "the Krein Milman theorem in Operator Convexity"; and the third section there deals with compact matrix convex sets. The first example there states that the matrix interval ...
3
votes
2answers
760 views

Is a von Neumann algebra just a C*-algebra which is generated by its projections?

von Neumann algebras have the nice property that they are generated by their projections (the elements satisfying $e = e^{\ast} = e^2$) in the sense that they are the norm closure of the subspace ...
8
votes
2answers
573 views

$C^*$-algebra which is also a Hilbert space?

Does there exist a nontrivial (i.e. other than $\mathbb{C}$) example of a $C^*$-algebra which is also a Hilbert space (in the same norm, of course)? For $\mathbb{C}^n$ with $n > 1$ the answer is ...
9
votes
2answers
206 views

Why can we classify the W*algebra?

Many operator algebra books discuss the classifiation of W*algebra(von Neumann algebra),but not the C*algebra,why? I think a direct reason is that we have the projection comparison theorem in the ...
1
vote
1answer
45 views

Where can I find an English version of this paper by Gel'fand and Raikov?

It is titled 'Irreducible unitary representations of locally bicompact groups' and the original version is in Russian. Google scholar shows it has been translated into English and once pubulished in ...
6
votes
1answer
244 views

Visualize operator algebras?

It seems to me that to study mathematics is to convert the abstract language into diagrams, graphs and images. It does depend on the subject how much this technique can ease the struggle yet most of ...
6
votes
1answer
237 views

Homomorphic conditional expectations?

To clarify, I mean "conditional expectation" in the sense of $C^*$-algebras (a completely positive projection of norm 1, equivalently, a completely positive linear map onto a $C^*$-subalgebra which is ...
6
votes
1answer
304 views

Matrices with entries in $C^*$-algebra

Let $\mathcal{A}$ be a $C^*$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution on ...
0
votes
1answer
216 views

Functional calculus of a projection in a C*-algebra

Let $A$ be a C*-algebra and $p$ be a projection in $A$. Let $f$ be a continuous function defined on the spectrum of $p$. In particular let $f(\lambda)=\lambda^{-\frac{1}{2}}$. What will be the ...
3
votes
1answer
334 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 ...
8
votes
1answer
523 views

What is the use of Spectral Theorem?

Obviously the version for compact and self-adjoint linear operators on Hilbert Spaces is very useful since it decomposes the operators into orthogonal projections. However, the following more general ...
5
votes
1answer
164 views

Nonamenable subgroups of the unitary group of the hyperfinite II_1 factor

The hyperfinite $II_1$ factor arises as the group von Neumann algebra of any infinite amenable group such that every conjugacy class but that of the identity has infinite cardinality. The unitary ...
3
votes
1answer
113 views

Complemented ideals in von Neumann algebras

Let $I$ be an ultraweakly closed ideal in a von Neumann algebra $M$. For example, this can be the kernel of an ultraweakly continuous homomorphism. Is it true that there is another ideal $J\subset M$ ...
3
votes
2answers
86 views

Dimension of an operator module

Let $H$ be an infinite dimensional Hilbert space. Consider unital $C^*$-sub-algebra of $\mathcal{A}\subset\mathcal{B}(H)$ such that there exist a family of isometries $\{S_n:n\in N\}$ with pairwise ...
14
votes
1answer
439 views

Motivation for abstract harmonic analysis

I am reading Folland's A Course in Abstract Harmonic Analysis and find this book extremely exciting. However it seems Folland does not give many examples to illustrate the motivation behind much of ...
2
votes
0answers
79 views

Induced representations of topological groups

Sorry if this is a naive question-- I'm trying to learn this stuff. If $G$ is a group with subgroup $H$, then we have the restriction functor $\operatorname{Res}$ from $G-\operatorname{mod}$ to ...
1
vote
0answers
42 views

Does the locality or non-locality of operators imply matrix structure?

I understand that an operator, $\hat{O}$, is said to be non-local if $$b(x)=\hat{O}a(x)=\int dx'O(x,x')a(x')$$ that is, to find $b(x)$ at aparticular value of $x$, we need to know ...
1
vote
0answers
33 views

Find the probability that a measurement results in a certain interval

I was given this problem: Let the state $f(x)=e^{-|x|}$ and an operator $P=-i\frac{d}{dx}$. (a) What is the probability that a measure of P results in the interval $[0,1]$ (b) What is the Fourier ...
5
votes
2answers
399 views

Regarding Ladder Operators and Quantum Harmonic Oscillators

When dealing with the Quantum Harmonic Oscillator Operator $H=-\frac{d^{2}}{dx^{2}}+x^{2}$, there is the approach of using the Ladder Operator: Suppose that are two operators $L^{+}$ and $L^{-}$ and ...
3
votes
1answer
372 views

A problem on $C^\ast$-algebras and $W^\ast$-algebras

Let $I$ is a compact topological space, $m$ is a positive regular Borel measure. Then $L^\infty(m)$ is a standard example of commutative $W^\ast$-algebra (von Neumann algebra), but it is also a ...