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)

2
votes
1answer
292 views

Ultraweak topology

In Stratila and Zsido, as well as some other sources, the ultraweak topology on $B(H)$ is taken to be the smallest topology for which every element in the closure of the span in $B(H)$ of the elements ...
3
votes
0answers
143 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 ...
1
vote
1answer
98 views

References on Algebraic Operators

Let $\mathcal{H}$ be a Hilbert space and $d$ is an inner derivation on $\mathcal{L}(\mathcal{H})$. An operator $T\in\mathcal{L}(\mathcal{H})$ is algebraic if $p(T)=0$ for some polynomial $p$. In ...
6
votes
2answers
333 views

Non-$C^{*}$ Banach algebras?

It suddenly occurred to me almost every Banach algebra I know is actually a $C^{*}$ algebra. Several kinds of function algebras are definitely $C^{*}$ algebras. So is the matrix algebra. Although one ...
3
votes
1answer
193 views

Strong convergence of projections in $B(H)$

Let $\{e_{kj}\}$ be the canonical matrix units in $B(H)$, with $H$ separable. Define projections $q_k$ by $$ q_k=\sum_{n=1}^ke_{nn}. $$ Let $\{p_1,p_2,\ldots\}\subset B(H)$ be a sequence of ...
1
vote
1answer
56 views

An explicit example of an invariant halfspace of the unilateral shift?

In a recent talk, A. Popov stated the following fact The unilateral shift on $\ell^2$ has invariant halfspaces. Halfspaces are closed subspaces whose dimension and codimension are both infinite. ...
-1
votes
1answer
117 views

is there any * homomorphism $T$ from $A$ to $B$ …?

Is there any $*$ homomorphism $T$ from $A$ to $B$, wherein $B$ is a $*$ closed subalgebra of $C^*$ algebra $A$, containing the unit of $A$, such that $T(b)=b$ for all $b\in B$ and $\|T(a)\|=\|a\|$ for ...
1
vote
0answers
192 views

Convergence of net sums of complex numbers, as well as operators

I have some questions concerning convergence of sums where the summands are complex number, although the real motivation of my question comes from Von Neumann algebras where sometimes the summands are ...
3
votes
1answer
135 views

Entirely “Bare-hands” proof that completely additive states are ultraweakly continuous

Originally I had asked this question: Positive Linear Functionals on Von Neumann Algebras I got some responses that directed me to a variety of resources, some of which I could not understand because ...
4
votes
3answers
162 views

Lower bound for $\|A-B\|$ when $\operatorname{rank}(A)\neq \operatorname{rank}(B)$, both $A$ and $B$ are idempotent

Let's first focus on $k$-by-$k$ matrices. We know that rank is a continuous function for idempotent matrices, so when we have, say, $\operatorname{rank}(A)>\operatorname{rank}(B)+1$, the two ...
1
vote
1answer
60 views

Ultraweak continuity of power maps on $W^*$-algebras

Let $\mathcal{A}$ be a $W^*$-algebra. Is the map $a \mapsto a^2$, or more generally the map $a \mapsto a^k$, ultraweakly continuous? (Of course, products are not jointly ultraweakly continuous in ...
1
vote
1answer
153 views

An upper bound for $\|(\lambda-A)^{-1}\|$?

Let $A$ be a k-by-k matrix and $\sigma(A)$ its spectrum, or the collection of eigenvalues of $A$. If we know $\lambda\notin\sigma(A)$, then $\lambda$ is at a positive distance to all points in the ...
2
votes
1answer
135 views

Subprojection of a finite projection

The question is entirely explained here, in that I wonder why every source seems to regard as obvious the claim that subprojections of finite projections are finite. Here is the link. To me, playing ...
2
votes
1answer
146 views

The commutant of a tensor product

In this whole post beginning with the second paragraph, by "tensor product" I will mean the operation done on Von Neumann algebras where one applies the homomorphism from here Tensor products of maps ...
0
votes
1answer
86 views

what is mathematical difference between an hermitian operator $\hat A$ and a vector $\vec A$?

what is mathematical difference/relation between an hermitian operator $\hat A$ and a vector $\vec A$?
1
vote
1answer
176 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
228 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
331 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
73 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
114 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
264 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
313 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
120 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
216 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
170 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
413 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
77 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
187 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
313 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
104 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
67 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
183 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
283 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
248 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
94 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
94 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
125 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
778 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
580 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
208 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 ...