Tagged Questions

1
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0answers
21 views

Conditions under which $A$ is a W*-algebra for a positive map between C*-algebras $\phi : A \rightarrow B$

Let $A$ and $B$ be a C*-algebra. Let $\phi : A \rightarrow B$ be a positive map. Suppose that $B$ is a W*-algebra. Under what conditions on $\phi$ can we ensure that $A$ is also a W*-algebra?
2
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1answer
57 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
3
votes
2answers
96 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 ...
3
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1answer
50 views

What is $\sigma_{\mathfrak{M}_{p}}(p\cdot a\cdot p)$ when $p=\chi_{S}(a)$ for closed $S\subseteq \sigma(a)$?

Let $\mathfrak{M}$ be a von-Neumann algebra and $a\in\mathfrak{M}$ a self-adjoint element. Let $S\subseteq\sigma(a)$ be a non-empty, closed subset. Let $p=\chi_{S}(a)$, which is a non-zero, ...
2
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1answer
50 views

Counterexample for a polar decomposition in von Neumann and $C^\ast$ algebras

For a von Neumann algebra, we have that partial isometry and positive operator of an operator in its polar decomposition belongs to the algebra, but in a $C^\ast$ algebra this may not be true. Can ...
1
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1answer
52 views

Trace-preserving $\Rightarrow$ Norm-preserving?

Let $\theta\in\text{Aut}(\frak{M},\tau)$, where $\frak{M}$ a von-Neumann algebra and $\tau$ a faith, finite, normal trace. Does $\theta$ preserve the norm structure? (Here assumed, that $\frak{M}$ ...
3
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0answers
49 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 ...
1
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1answer
62 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 ...
8
votes
1answer
140 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
6
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1answer
344 views

Some examples in C* algebras and Banach * algebras

I would like an example of the following things. A Banach * algebra that is not a C* algebra for which there exists a positive linear functional (it takes $x^*x$ to numbers $ \geq 0$) that is not ...
4
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0answers
152 views

Abstract Von Neumann Algebras

I have just read this question Is a von Neumann algebra just a C*-algebra which is generated by its projections? and am wondering about Robert Israel's answer when he says that a subalgebra of $C(X)$ ...
3
votes
0answers
114 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 ...
4
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4answers
187 views

Applications of Operator Algebras to modern physics

I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics ...
0
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1answer
79 views

Tensor product of Hilbert Algebras

A Hilbert algebra is an inner product space that is also a *-algebra where the various operations and structures interact according to some axioms. One of those axioms is that the linear operation ...
0
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1answer
103 views

Two questions from Dixmier's book on Von Neumann algebras

It seems something is going wrong with the preview I linked in some of my previous questions, so I will just type out the question. I am having trouble with Dixmier's proof of Corollary 5 on p. 46. ...
0
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0answers
83 views

weak closures of ideals [duplicate]

Possible Duplicate: Two questions from Dixmier's book on Von Neumann algebras On p. 46-47 in Dixmier's book on Von Neumann Algebras, which I just realized can be accessed through this ...
1
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0answers
72 views

When the ultrastrong closure of a *-algebra contains the double commutant

As lemma 6 on p.44 of Dixmier's book on Von Neumann algebras, he states that if $A$ is a *-algebra (i.e. possibly without identity, not necessarily closed in any topology) of operators in $B(H)$ such ...
3
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1answer
139 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 ...
3
votes
1answer
118 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 ...
1
vote
1answer
43 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
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1answer
137 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 ...
6
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1answer
159 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 ...
6
votes
1answer
124 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 ...
4
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1answer
159 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 ...
3
votes
1answer
126 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 ...
2
votes
1answer
87 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
1answer
292 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 ...
11
votes
1answer
321 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...