# Questions tagged [c-star-algebras]

A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying $(ab)^*=b^*a^*$ and the C*-identity $\Vert a^*a\Vert=\Vert a\Vert^2$. Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

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### Decompose a positive contraction in a continuous masa in $L(H)$

Let $A$ be a continuous masa in $L(H)$ and $T$ be a positive contraction in $A$. Then we can assume that $0＜\|Th\|＜1$ for all unit vectors $h\in H$. Otherwise decompose $T$ as $P+T_0$ for some ...
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### $0 \leqslant a \leqslant b \Rightarrow \|a\| \leqslant \|b\|$ in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a,b\in A$. Therefore $0\leqslant a \leqslant b \Rightarrow \|a\|\leqslant \|b\|$. I'm trying to prove this claim, but apparently it's necessary to use some spectral ...
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### Essential spectrum of a projection [closed]

For any $T\in B(H)$, the essential spectrum $\sigma_e(T)$ of $T$ is a subset of the spectrum $\sigma(T)$ of $T$; namely, the $\lambda$ such that $\lambda-T$ is not Fredholm. If $P$ is a projection, we ...
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### Closed Ideal $J$ of $C(X)$ there exists $f\in J$ such that $0\leq f(x)\leq 1$ for all $x\in X$ and $f(x)=1$ for all $x\notin U$

I am having a hard time understanding all the steps in the proof of the following proposition: Proposition: Suppose that $X$ is a compact Hausdorff space and consider the algebra $C(X)$. Let $J$ be a ...
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### Commutant of corner of $*-$algebra on a hilbert space is corner of commutant

Let $A$ be a $*-$algebra on a Hilbert space $H$ and $p$ be a projection in $A'$, where $A'$ is the commutant of $A$, that is, $$A':=\{u \in B(H): ua=au~\text{ for all }~a \in A\}.$$ If also $p \in A''$...
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### Confused on a set inclusion in C* algebras by Murphy

Im stuck on the following part of this theorem: The closed vector subspaces of $L^2(T)$ invariant for the bilateral shift $v=M_{z}$ (for $z: T \longrightarrow \mathbb{C}$ the inclusion map) are ...
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### $Z_2$ action yields decomposition into a direct sum

If I have a $Z_2$ (group with two elements) action on a $C^*$-algebra $A$, i.e. $A$ is graded by the definition of Ralf Meyer for example, then how may I decompose $A$ into a direct sum $A_0\oplus A_1$...
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### Given an ideal $I\subset A$ and an element $a\in A$, is it true that $\sup\{\|ab\|:b\in I,\ \|b\|\leq 1\}=\|a\|$?

Let $I$ be a 2-sided closed ideal in a C*-algebra $A$. Given $a\in A$, is it true that $$\sup_{\substack{b\in I\\ \|b\|\leq 1}}\|ab\|=\|a\|?$$ Or do we need assumptions on $I$? Note that the left-hand ...
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### Multiplier algebra of $C_0(X)$

I'm looking at the following example from C* algebras by Murphy, and I'm totally lost on the notation they're using. "Let $X$ be a locally compact Hausdorff space. Since $C_0(X) \subset C_b(X)$ ...
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### Does any von Neumann algebra have $\sigma$-finite projections?

Let $M$ be a von Neumann algebra. Let $\Sigma$ be the set of $\sigma$-finite projections of $M$. In Takesaki's book "Theory of operator algebras II", chapter 7, p51, in the proof of theorem ...
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### Norm product inequality for unitisation of a $C^*$-algebra

Let $A$ be a $C^*$-algebra without a unit. Define $\widetilde{A}=\{(\alpha,a):\alpha \in \mathbb{C}, \;a\in A\}$ equipped with componentwise addition and scalar multiplication. Vector multiplication ...
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### What does $p+q \leq 1$ means in the context of $C^*$-algebra?

Let $A$ be a $C^*$-algebra, and let $p,q \in A$. The book I'm following(An Introduction to K-theory for $C^*$-algebra) has this notation $p+q \leq 1$. What does this mean? I looked through the book, ...
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