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Decomposition of a contractive representation into an orthogonal sum for the $n$-dimensional case. Has this been done yet?

I know that it has been done for the two-dimensional case. Marek Kosiek showed it in his work "Decomposition of operator representations of the algebra $R(K_1 \times K_2)$" and "...
S-F's user avatar
  • 63
1 vote
1 answer
410 views

Takesaki lemma: existence Gelfand-Pettis integral

Consider the following fragment from Takesaki's second volume of "Theory of operator algebras" (lemma 2.4, chapter VI "Left Hilbert algebras"). In another post, it was explained ...
Andromeda's user avatar
  • 175
6 votes
1 answer
574 views

Integration in Banach algebra

Let $\mu$ be a Borel measure on the real line $\mathbb{R}$ taking values in a separable Banach algebra $A$. Assume that $\mu$ is such that the total variation measure $|\mu|$ is finite. Let $f$ be a ...
user72829's user avatar
  • 552
1 vote
0 answers
135 views

Description of state space of $C(K,M_n)$?

Edit: closed convex hull added. I am trying to understand the state space of $C(K,M_n)=C(K)\otimes M_n$ for $K$ a compact space. My guess would be that these are the closed convex hull of states on $C(...
C-star-W-star's user avatar
1 vote
1 answer
198 views

Takesaki proposition 7.4 chapter 4 volume I

I initially asked on MSE, but did not get an answer there. Consider the following proposition from chapter IV of Takesaki's "Theory of operator algebras I" (more context/definitions in the ...
Andromeda's user avatar
  • 175
3 votes
0 answers
227 views

Is there a noncommutative version of von Neumann's ergodic theorem? [closed]

The two most celebrated ergodic theorems are Birkhoff's ergodic theorem and von Neumann's ergodic theorem. E. C. Lance in his remarkable work (Ergodic Theorems for Convex Sets and Operator Algebras) ...
Neil hawking's user avatar
9 votes
1 answer
384 views

Comparing two $\sigma$-algebras on $B(\ell^1)$

Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow $$w-\lim T_i=T \Longleftrightarrow \...
ABB's user avatar
  • 4,058
2 votes
0 answers
164 views

An operator valued Egoroff's theorem

The following statements suggests $B(H)$-valued Egoroff's theorem when $H$ is a separable Hilbert space. Probably it will be hold even if a von Neumann algebra $M$ whose predual is separable is ...
ABB's user avatar
  • 4,058
4 votes
1 answer
2k views

Operator topologies on $L^{\infty}(X,\mu )$

Let $(X,\mu )$ be a measure space. Then, $L^2(X):=L^2(X,\mu )$ is a Hilbert space in the usual way and we may view $L^{\infty}(X):=L^{\infty}(X,\mu )$ as a subalgebra of bounded operators on $L^2(X)$ ...
Jonathan Gleason's user avatar
4 votes
0 answers
185 views

A strongly open set which is not measurable in the weak operator topology

Let $H$ be a non-separable Hilbert space and $\{e_i\}_{i\in I}$ be an orthonormal basis for $H$. Let $J$ be a uncountable proper subset in $I$. Let us put $$E=\{x\in B(H): \lVert xe_j\rVert <1: \...
ABB's user avatar
  • 4,058
8 votes
1 answer
635 views

Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions $\...
Andre Kornell's user avatar
17 votes
3 answers
3k views

Which sigma-ideals in a sigma-algebra are ideals of null sets?

My question is motivated, to be somewhat vague, by an attempt to see how much a measure space is defined by the set of null sets. In other words, assume we are not given a concrete measure on a space ...
Super-Measurable Analyst's user avatar
3 votes
1 answer
565 views

When does a $W^*$-algebra have a standard Borel spectrum?

EDIT: André Henriques has commented below that the correct separability condition is not weak-* separability as I have written below, but separability of the predual. This post came out a bit long, ...
Super-Measurable Analyst's user avatar
5 votes
1 answer
2k views

definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011). There, they work on a Hilbert space $H$ and on the ...
Yul Otani's user avatar
  • 342
4 votes
2 answers
957 views

Do semi-continuous functions generate bounded Borel measurable functions as a $C^*$-algebra?

This question is related to Question 2 of my previous posting. Question. Let $\mu$ be a Radon measure on a compact Hausdorff space $\Omega$ and $L^{\infty}(\Omega,\mu)$ the set of essentially bounded ...
Masayoshi Kaneda's user avatar
2 votes
2 answers
867 views

Decomposition of an abelian von Neumann algebra

Hi, I came across the statement below and I couldn't figure out why it is true. I was hoping someone could explain it or give me a good reference. Thank you in advance. "Let $\pi$ be a non-degenerate ...
Wishiwere Smarter's user avatar