The strong operator topology on a Banach space $X$ is usually defined via semi-norms: For any $x \in X$, $|\cdot|_x: B(X) \to \mathbb R, A \mapsto \|A(x)\|$ is a semi-norm, the strong topology is the weakest/coarsest topology which makes these maps continuous. Alternatively it is generated by the sub-base $\left\{B_\epsilon(A;x)=\{B\in X \mid |B-A|_x<\epsilon\}\phantom{\sum}\right\}$. If we define […]

The following is a Theorem of Conway’s operator theory: I can not understand how he proves it. I think $\phi(\text{ ball B(H)})$ is compact if $\phi(\text{ ball B(H)})$ is closed subset of compact set $X$, but why is $X$ compact?

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. […]

Let $B(H)$ be the set of bounded operators on a Hilbert space $H$. I know that $u_{\alpha}\longrightarrow u$ in S.O.T if and only if $u_{\alpha}(x)\longrightarrow u(x)$ for all$x\in H$. I know that $\cdot : B(H)\times B(H) \longrightarrow B(H)$ such that $(u,v)\longmapsto uv$ is separately continuous and jointly continuous on bounded set. […]

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 my functional analysis class I was recently met with this in the context of C* algebras: Let A be a C*-Algebra and B is a C*-subalgebra of A and I an ideal of A. We are asked to show that $ B+I $ is itself a C*-subalgebra of A (in particular it is closed). […]

Problem Given a C*-algebra $\mathcal{A}_0$ and unital extensions $1\in\mathcal{A}$ and $1’\in\mathcal{A}’$. Regard a common element: $$A_0\in\mathcal{A}_0:\quad A^{(\prime)}:=\iota^{(\prime)}(A_0)$$ Can it happen that its spectra really differ: $$\sigma(A)\cup\{0\}\neq\sigma(A’)\cup\{0\}$$ (The scenario is inspired by possible noncanonical unital extensions.) One might be tempted to conclude that they agree as: $$\langle\iota(\mathcal{A}_0)\cup\{1\}\rangle\cong\mathcal{A}_0\oplus\mathbb{C}\cong\langle\iota'(\mathcal{A}_0)\cup\{1’\}\rangle’$$ But there is a flaw as the example below […]

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