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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$\bigoplus_\lambda A_\lambda$ is strongly closed in $B\left(\bigoplus_\lambda H_\lambda\right)$.

Let $(H_{\lambda})_{\lambda\in \Lambda}$ is a family of Hilbert spaces and $A_{\lambda}$ is a von Neumann algebra on on $H_\lambda$ for each index $\lambda$. Then prove that the direct sum $\bigoplus_\...
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Confused on two parts of Proof in C* algebras by Murphy

I'm working through the proof of the following theorem from C* algebra by Murphy. For context, $T_{\varphi}: H^2(T) \longrightarrow H^2(T)$ is given by $T_{\varphi}(f) = p(\varphi f)$ for $p: L^2(T) \...
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representation of a linear bounded operator

Let $T\in B(H)$ and $H=H_1\oplus H_2$, then $T$ can be expressed by the following matrix: \begin{pmatrix}A & B\\C & D\end{pmatrix} , where $A\in B(H_1), B\in B(H_2,H_1), C\in B(H_1,H_2),D\in B(...
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Determine whether a self-adjoint operator is negative definite or not

Let $Q$ be an idempotent in $L(H)$. Suppose there exists two sequences of unitaries $\{U_n\}$ and $\{V_n\}$ such that $X$ is the norm limit of $U_nQU_n^*+V_nQV_n^*$, and suppose that $Q+Q^*\cong 2P\...
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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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1 answer
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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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an inequality for completely positive maps

Let $A$ and $B$ be $C$*-algebras and $\phi:A\to B$ a completely positive contractive map. I want to show that, for any $a,b\in A$; $$\Vert \phi(ab)-\phi(a)\phi(b)\Vert\leq\Vert\phi(aa^*)-\phi(a)\phi(a^...
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Strong limit from the unitary orbit of $A$

Let $A,X\in L(H)$ with $\Bbb D\subset W_e(A)$ ,where $W_e(A)$ is the essential numerical range of $A$,and $\|X\|\leq 1$. Then there exists a sequence of unitaries $(U_n)_n$ in $L(H)$ such that wot $...
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On double dual of C* algebra

Can anyone provide examples of double dual of $C^{*}$-algebras except $K(\mathcal{H})$, commutative cases? Thanks in advance!
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2 answers
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Questions on Theorem 5.5.7. in Brown-Ozawa

I am currently trying to digest the proof of Theorem $5.5.7.$ in the book "$C^*$-algebras and Finite-Dimensional Approxmiations" by N. Brown and N. Ozawa. Background: Let $(X,d)$ be a metric ...
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Two von Neumann algebras not isomorphic as C*-algebras, can they be isomorphic as von Neumann algebras? [duplicate]

This might be a silly question. Consider two von Neumann algebras $M, N$, given that they are not isomorphic as C*-algebras to each other, is there a chance that they are isomorphic as von Neumann ...
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Jordan Decomposition of Self Adjoint Functionals

Im reading over the following theorem in C* algebras by Murphy, and I'm confused on two particular parts: 1: How is Hahn Banach being applied here exactly? What linear functional are we extending ...
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Spectrum of a Schwartz space of cotangent bundle

Let $E \to X$ be a smooth vector bundle over a $C^\infty$-manifold. There is an isomorphism between the algebra $(\mathcal{S}_c(E),*)$ and $(\mathcal{S}_c(E^*),.)$ by the Fourier transformation. My ...
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Estimation of commutator norms in Davidson's book “C star algebra by example”

In ChapterII Davidson's book "$C^{*}$-algebra by example" he used a estimation of norms to prove Thm II.5.3, that is $$ ||A^{\frac{1}{2}}T - TA^{\frac{1}{2}}|| \le 2 ||T||^{\frac{1}{2}}||AT-...
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KMS states on Toeplitz algebra

Consider the Toeplitz algebra which is the universal $C^*$ algebra generated by a single isometry $S$ denoted by $C^*(S)$. Define a dynamics on $C^*(S)$ by $\mathbb{R}\to Aut(C^*(S))$ by $t\mapsto \...
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Extension of a continuous and equivariant map to compactification

Given a locally compact (topological) group $G$ we denote by $C_b^{lu}(G)\subseteq C_b(G)$ the bounded continuous functions on $G$ such that $f\in C_b^{lu}(G)$ whenever the map $G\to C_b(G)$ given by $...
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1 answer
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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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1 answer
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Why does this theorem imply this next result?

I'm reading through $C^{*}$ algebras by Murphy, and the following theorem is presented. Let $I \subset A$ be a closed ideal of a $C^{*}$ algebra $A$. Then there exists a unique $*$ homomorphism $\...
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Extension of slice map to WOT closure

Given Hilbert spaces $H_1,H_2$ and a functional in the predual $\psi\in B(H_1)_*$ we may consider the slice map $S:B(H_1)\otimes B(H_2)\to B(H_2)$ defined on the spatial tensor product given by ...
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0 answers
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Trying to understand maximal tensor product of Ternary rings of operators

Let $V$ and $W$ be ternary rings of operators (TROs). In section 5 of Kaur and Ruan - Local Properties of Ternary Rings of Operators and Their Linking $C^*$-Algebras, the maximal tensor product $\...
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1 answer
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Doubt on max tensor product of $C^{\ast}$-algebras

Im trying to understand proof of corollary $11.34$ from here. The corollary goes as follows: Let $A_1$ and $A_2$ be $C^{\ast}$-algebras. Given any $C^{\ast}$-norm $\vert \vert . \vert \vert$on $A_1 \...
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1 vote
1 answer
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Compression map is an isomorphism from $pB(H)p$ to $B(K)$ via $u \to u_K$

I was reading a note on Von Neumann Algebra, and I am not able to understand this phrase as: Let $K$ be a closed vector subspace of a Hilbert space $H$ and let $p$ be the projection of $H$ onto $K$. ...
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3 votes
1 answer
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Why do we need this extra step in this proof?

For some background information, given sets $I_1,\dots,I_n \subset A$ of a $C^{*}$ algebra $A$, we define $\prod_{k=1}^n I_k$ to be the closed linear span of all products of the form $\prod_{k=1}^n ...
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3 votes
1 answer
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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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1 vote
1 answer
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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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2 votes
0 answers
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Convergence of vector states

Let $\mathfrak{A} \subset B(H)$ be a C*-algebra. For $x \in H$, $\|x\|=1$, define a vector state $\omega_x$ on $\mathfrak{A}$ by $\mathfrak{A} \ni A \mapsto \langle x, Ax \rangle$. Assume $(x_n)_{n\in ...
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3 votes
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Is every AM space a $C^*$-algebra?

A Banach lattice $E$ is said to be an AM-space if $$\|\sup\{x,y\}\|=\sup\{\|x\|,\|y\|\}$$ for all positive $x,y\in E$. My question is as follows: Is every AM-space (which is a $*$-algebra) a $C^*$-...
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1 vote
1 answer
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Finding idempotent elements of a $C^{*}$-algebra

Let $T \in B(\mathcal{H})$ (bounded operators on a complex Hilbert space $\mathcal{H}$) and suppose $T$ is normal. Suppose furthermore $\sigma(T)=\{-1\}\cup[2,3]$. Then I want to find a complete list ...
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1 vote
1 answer
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Quotient of function spaces is function space on set difference

I have seen it stated that for an open subset $Y\subseteq X$ such that $X$ is a compact Hausdorff space we get an identification of the $C^*$-algebras : $C(X\setminus Y)\cong C(X)/C_0(Y)$. I suppose ...
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1 vote
1 answer
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What's the norm in the given space?

Currently I'm reading the paper by Pluta and Russo titled Ternary operator categories. According to section $1.2$ of the paper, an associative triple system is a vector space $V$ with a trilinear map ...
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0 votes
1 answer
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a condition for a $C^*$ algebra to have a unit?

I am looking for a hint or a methodology to approach this problem, showing in Arveson's A short course on spectral theory: Let $X$ be a compact Hausdorff space and let $F$ be a proper closed subset of ...
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3 votes
1 answer
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Non Self Adjoint ideal in C(D)?

I'm currently working on a problem that's asking me to give an example of a non self adjoint ideal of $C(D) = \{f: D \longrightarrow \mathbb{C} \: | \: \text{$f$ is continuous}\}$ with $||\cdot||_{\...
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1 answer
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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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1 vote
1 answer
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Clarification on exercise in C* Algebras By Murphy

I'm working through some exercises from Ch.2 of C* algebras By Murphy, and I'm stuck on the very first part of a problem. The problem is the following: Let $A$ be a unital $C^{*}$ algebra. If $r(a) &...
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3 votes
1 answer
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Schroeder-Bernstein theorem for representations of C*-algebras

I am trying to work on an exercise which claims that If two representations ρ and σ, on Hilbert spaces H and G respectively, are each unitarily equivalent to a subrepresentation of the other, then ...
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1 vote
1 answer
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$C^\ast$ -algebra : Can we assume image of identity is identity?

In Conway's A Course in Operator Theory, proposition 1.7 (e) is If $\rho:A \to B$ is $\ast$-homomorphism, then $\|\rho(a)\| \leq \|a\|$ for all $a$ in $A$. and the beginning of the proof is First ...
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2 votes
1 answer
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A question about a part in a proof regarding approximate identities in $C^\ast$-algebra

Let $A$ be a $C^\ast$-algebra with approximate identity $\{e_\lambda\}_{\lambda\in\Lambda}$. The author states the following comment in the trenches of a proof: Since $t^2\leq t$ on $[0,1]$, it ...
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1 vote
0 answers
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Let $A$ be a $C^\ast$-algebra and $f\colon A\to\mathbb{C}$ a positive linear functional. Suppose $a\leq b^{[1]}$, is it true that $f(a)\leq f(b)$?

Let $A$ be a $C^\ast$-algebra and $f\colon A\to\mathbb{C}$ a positive$^[2]$ linear functional. Suppose $a\leq b^{[1]}$, is it true that $f(a)\leq f(b)$? This is used in a proof, but for approximate ...
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2 votes
0 answers
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Understanding the use of Hilbert modules in the definition of $K$-theory and $KK$-theory

Let $A$ be a unital $C^*$-algebra. Then elements of the $K$-group $K_0(A)$ are usually defined as equivalence classes of projections in matrix algebras over $A$. Such a projection, say $p\in M_n(A)$, ...
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0 votes
0 answers
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the norm of a state on the full matrix

Let $M$ be a 2 by 2 full matrix. $\phi$ is a state on $M$ with density $(\beta, 1-\beta)$, then we can find only one projection $e\in M$ with $\|\phi_{eM(1-e)}\|=|1-2\beta|$. How to choose the ...
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  • 1,117
1 vote
1 answer
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Support of a subalgebra of a $C^*$ algebra

Let $A$ be a finite dimensional $C^*$ algebra and let $I$ be a two sided ideal in $A$. What is meant by the notion of support of $A/I$. I have heard of support of a self adjoint operators.Can any one ...
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1 vote
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Why is this an equality? Proof Assistance In C* Algebras By Murphy

Here is the result I'm covering: Let $X$ be a compact Hausdorff space and $H$ a Hilbert space, and suppose that $\varphi: C(X) \longrightarrow B(H)$ is a unital $*$ homomorphism. Then there exists a ...
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1 vote
1 answer
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Reference request: isomorphism in $K$-theory induced by inclusion of a full corner of a $C^*$-algebra

Let $A$ be a $C^*$-algebra $p$ a projection such that $ApA$ is dense in $A$. Let $B=pAp$. Then it is known that the inclusion $$i\colon B\hookrightarrow A$$ induces an isomorphism on operator $K$-...
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1 vote
0 answers
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Identifying projections underlying projective modules over $C^*$-algebras

Let $A$ be a $C^*$-algebra and $B=pAp$ for some projection $p\in A$. Let $N=A^n q$ be a finitely generated projective (left) $A$-module, where $q$ is a projection in $M_n(A)$. Then there is a $B$-...
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