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 projections is the same as the convergent complex valued sum of $\omega$ applied to the projections). Then supposedly, $\omega$ is the pointwise convergent sum: […]

Please help me with the following problem( give some hints or references): Let $X$ be a Banach space and $B(X)$ be the algebra of bounded linear operators on $X$. Suppose that $A$ and $B$ are two operators in $B(X)$ such that for every $T \in B(X)$ we have $\sigma(AT)=\sigma(BT)$. Show that $A=B$. Here $\sigma(A)$ denotes […]

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 ? Actually I try to prove the Spectral representation Theorem of a normal operator in case the representation of $C^*(T)$ over $H$ […]

Let $\phi : M_k(\Bbb{C}) \to M_n(\Bbb{C})$ be a homomorphism of $C^*$-algebras. We know that $\phi$ decomposes as a direct sum of irreducible representations, each of them equivalent to the identity representation (because $M_k$ has no non-trivial invariant subpspaces) with some null part. So this means that $\phi(a)= u \begin{bmatrix}a&0&0&0&0…&0\\0&a&0&0&0…&0\\0&0&a&0&0…&0\\&&&…&&&\\\\\\0&0&0&0&0…&0\end{bmatrix} u^*$ where $u$ is some unitary. […]

I was reading a paper which mentioned without proof that every infinite-dimensional $C$* algebra has an infinite-dimensional commutative $C$* subalgebra. Thinking about it for 10 minutes, I didn’t see an immediately proof. It is sufficient to construct an element with infinite spectrum, but I don’t see how to construct such an element. Moreover, if one […]

While trying to solve the exercise below, I came up with a wrong conclusion, but I can’t see why it’s wrong. Also I’m accepting suggestions to get the right solution. This is the problem 17 from chapter 10 of Rudin’s Functional Analysis. Let $A$ be a Banach algebra. Suppose that the spectrum of $x\in A$ […]

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P’^2=P’=P’^*$$ Order them by: $$P\leq P’:\iff\sigma(\Delta P)\geq0\quad(\Delta P:=P’-P)$$ Then equivalently: $$P\leq P’\iff P=PP’=P’P\iff\Delta P^2=\Delta P=\Delta P^*$$ How can I check this? (Operator algebraic proof?)

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. Any hints are appreciated.

I am reading HX Lin’s book, named “An introduction to the classification of amenable C*-algebras”, I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a unital C*-algebra A, then there is an isometric *-isomorphism from $C^{\ast}(a)$ to $C_{0}(sp(a)\setminus\{0\})$, which sends $a$ to the identity function on $sp(a)$. […]

In Kadison and Ringrose’s book “FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS”, the author gives the following theorem. Theorem: If $A$ is a self-adjoint operator acting on a Hilbert space $\mathscr{H}$ and $\mathscr{A}$ is an abelian von Neumann algebra containing $A$, there is a family $\{E_\lambda\}$ of projections, indexed by $\mathbb{R}$, in $\scr{A}$ such that […]

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