# Tagged Questions

The subject of operator algebras is primarily C*-algebras and von Neumann algebras, and associated topics. It also includes more general algebras of operators on Hilbert space, and may include algebras of operators on other topological vector spaces. It is related to but distinct from the subjects ...

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### Subordinate projections: Transitivity

Let $A$ be a $C^*$-algebra. If necessary, let us assume that $A$ is a von Neumann algebra. For projections $p,q \in A$ one writes $p \prec q$ if $p$ is Murray-von Neumann-equivalent to a subprojection ...
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### a point on trace class operators

Assume $H$ is separable Hilbert space and fix an orthonormal basis $\{e_n\}_1^{\infty}$. Let us denote $p_n$ by the projection onto the subspace generated by $\{e_1\cdots,e_n\}$. Let $a$ be a ...
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### Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
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### Characterization of normal operators on Hilbert space as function of a self-adjoint operator

My question : Suppose T is a normal operator on a Hilbert space H. Show that there exists a self-adjoint operator S on H such that T=f(S), where f is continuous function from spectrum of S into S. My ...
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### Normal element is sum of four commuting positive elements

I am stuck with the following problem : Every normal element in a $C^*$ algebra can be written as a linear combination of four commuting positive elements. I had tried along the following lines. ...
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### Introductory books for ‎ ‎$\frak{E}_p(I)$

Are there any good books different from abstract harmonic analysis by hewitt to study ‎$\frak{E}_p(I)$. where ‎$\frak{E}_p(I)$ is: ‎Let $I$ be an arbitrary index set‎. ‎For each $i\in I$ let $H_i$ ...
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### How to handle direct sums and unitizations of $L^p$ operator algebras?

Let $p\in[1,\infty)$. An $L^p$ operator algebra refers to a Banach algebra that is isometrically isomorphic to a closed subalgebra of $B(L^p(X,\mu))$ for some ($\sigma$-finite) measure space ...
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### Can $AB-BA=I$ hold if both $A$ and $B$ are operators on an infinitely-dimensional vector space over $\mathbb C$?

Of course, it can't hold if operators are over finite-dimensional spaces, as is evident from trace considerations. Can it be true for infinite-dimensional spaces? I think not, but I don't see how we ...
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### Can someone help me to give some hints? Left Hilbert-$C_0(T,K(H))$ module $C_0(T,H)$

I tried to prove example 3.4 from the book Morita Equivalence and Continuous-Trace C$^*$-Algebras by Iain Raeburn and Dana P. Williams, but I get uneasy with notations and ideas. Let me restate my ...
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### Weakly compact operator with different domains

Let $A$ be a Banach algebra. Suppose that $e\in A$ such that $e^2=e$ and $eAe$ is division algebra(i.e., $eAe$ is unital and every element of $eAe$ has inverse in $eAe$). Define $T_e:A\to A$ with ...
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### Projections: Orthogonality

Given a unital C*-algebra $1\in\mathcal{A}$. Consider projections: $$P^2=P=P^*\quad P'^2=P'=P'^*$$ Order them by: $$P\perp P':\iff\sigma(\Sigma P)\leq1\quad(\Sigma P:=P+P')$$ Then equivalently: ...
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### Meaning of kernel

If something is the 'kernel' of a transformation, say $K(x,x')$, does it mean I should take the integral $$\int K(x,x') f(x')dx'$$ There are many different meanings of kernel and I did not see their ...
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### Uniqueness of c.p.c. order zero extensions

I have a question about a passage in the proof (on page 316) of proposition 3.2 in this paper http://wwwmath.uni-muenster.de/42/fileadmin/Einrichtungen/mjm/vol_2/mjm_vol_2_14.pdf. My question is ...
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### Relation between tracial states on von Neumann algebras and their GNS representations

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$, and let $\tau$ be a faithful tracial state on $M$. What is the relation between the GNS representation of $(M,\tau)$ and the original ...
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### Partial Isometries: Positivity

Given a unital C*-algebra $1\in\mathcal{A}$. Then implication holds: $$J\in\mathcal{A}:\quad JJ^*J=J\implies\sigma(J)\geq0$$ How can I check this? (Operator-algebraically?)
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### Weakly compact left multipliers

This is Exercise 3(a) on p. 157 in Takesaki's Operator algebras. Let $A$ be a C*-algebra. Then each opeator $T_a\colon A\to A$ given by $T_ax = ax$ ($a\in A$) is weakly compact if and only if $A$ is ...
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### Flip automorphism for a $II_1$ factor is not inner

It is known that for a $II_1$ factor $M$, the flip automorphism defined on $M \overline{\otimes} M$ by $a \otimes b \mapsto b \otimes a$ is not inner. A proof can be found on Vol. IV of the books by ...
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### Some questions about Cuntz’s proof of the $K_{1}$-injectivity of purely infinite simple unital $C^{*}$-algebras

I have some questions about Joachim Cuntz’s proof of the $K_{1}$-injectivity of purely infinite simple unital $C^{*}$-algebras, which is found in this paper. For this post, let us adopt the ...
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### Ordering: Identity

Given a unital C*-algebra $1\in\mathcal{A}$. Denote selfadjoints: $$\mathcal{S}(\mathcal{H}):=\{A\in\mathcal{B}(\mathcal{H}):A=A^*\}$$ Introduce an order: $$A\leq A':\iff\sigma(A'-A)\geq0$$ ...
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### Projections: Spectrum

Given a unital C*-algebra $1\in\mathcal{A}$. For projection one has: $$P^2=P=P^*\iff\sigma(P)\subseteq\{0,1\}\quad(P=P^*)$$ And all cases can appear: ...
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### Projections: Ordering

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: ...
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### Finding minimal projections in subalgebra generated by a given set

Consider the set of complex matrices $\mathbb{C}^{n\times n}$ for some set. Suppose we have a set $\{A_1,\ldots, A_n\}$ of Hermitian matrices. We want to find minimal projections in the subalgebra ...
### If a C*-algebra $A=\overline{\bigcup S}$, where $S$ is a class of prime C*-subalgebras, then $A$ is prime.
This is question 5.6 of Murphy's C$^*$-Algebras and Operator Theory: Let $S$ be a set of C*-subalgebras of a C*-algebra $A$ that is upwards-directed, that is, if $B,C\in S$, then there exists ...
Let $CL(X,Y)$ be the set of all closed range bounded linear operators from Banach space $X$ to Banach space $Y$. Is $CL(X,Y)$ an open set of $B(X,Y)$?