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Questions tagged [operator-algebras]

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 of the tags (banach-algebras), (c-star-algebras), (von-neumann-algebras), and (operator-theory).

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Functional calculus of several variables

It is well know that for a normal element $a$ of C*-algebra $A$ there exists functional calculus namely there is a *-homomorphism $C(\sigma({a})) \to A$ uniquely determined by sending $z \mapsto a$. ...
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On computing multiplicity function for self adjoint operator with nonatomic spectral measure

Suppose $T$ is a self-adjoint operator in $B(\mathcal{H})$ with $\sigma(T)$ is spectrum of $T$. $\mu$ is a spectral measure. For the operators having general continuous spectrum how to calculate the ...
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1answer
39 views

The CAR algebra is simple and nuclear

It is a fact that $CAR$ algebra is simple and nuclear.How to show this conclusion?
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1answer
30 views

Density of $\mathcal{L}^2(\mathbb{R})^+\cap\mathcal{L}^{\infty}(\mathbb{R})^+$ in $\mathcal{L}^{\infty}(\mathbb{R})^+$

I am getting in a world of confusion here. One of my problems is the nomenclature for topologies on $\mathcal{L}^{\infty}(\mathbb{R})$ (or rather $B(H)$ for $H$ a Hilbert space) so straight away I ...
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1answer
24 views

commutator of $C^*$ algebra

If $A$ is a non-commutative $C^*$ algebra,can we define $[A,A]$ as the ideal generated by the set $\{xy-yx:x,y\in A\}$such that $A/[A,A]$ is commutative?
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Why Koopman operator is defined on $L^\infty$ space?

Let $(X, \mathcal{A}, \mu)$ be a measure space, $S:X\rightarrow X$ a nonsingular transformation, and $f\in L^\infty$. The operator $U:L^\infty \rightarrow L^\infty$ defined by $ Uf(x) = f(S(x)) $ is ...
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2answers
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Finding normal operator matrix from characteristic polynomial

Let $ A \in L ( \mathbb{C}^4) $ be a normal operator with characteristic polynomial $ k_{A} = (\lambda - 1)^2 * (\lambda - 2)^2$. Is then the matrix for the operator just a diagonal matrix with ...
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tracial states of group C* algebra

Do all group $C^*$ algebras $C^*(G)$ (G is locally compact)have tracial states?
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1answer
41 views

On spectral multiplicity of left shift operators

Let $U$ be an operator defined on $l^{2}(\mathbb{Z})$ by $U(e_{n})=e_{n-1}$, where $e_{n}$ is an orthonormal basis of $l^{2}(\mathbb{Z})$. $U$ is a left shift operator. Since $U$ is unitary operator ...
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1answer
20 views

Is a linear functional which is positive on a linearly generating complemented lattice of projections necessarily positive?

Let $\mathcal{H}$ be a Hilbert space, and $A \subseteq {\rm B}(\mathcal{H})$ a unital C$^*$-algebra. Suppose there is a complemented lattice of projections $L \subset \mathcal{P}(A)$ whose linear ...
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1answer
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Two $C^*$-algebras with the same multiplier algebra

Is it possible for two non-isomorphic $C^*$-algebras $A$ and $B$ to have the same multiplier algebra? If so, what is a simple example? Remark: I am thinking that one example might be: $A$ is the Roe ...
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Is a linear functional which is positive on linearly generating set of projections positive?

Let $A$ be a unital C$^*$ algebra, and suppose there is a set of projections $P \subset \mathcal{P}(A)$ whose linear span is dense in $A$. If $\varphi \in A^*$ has $\varphi(p) \ge 0$ for all $p \in P$,...
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1answer
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Density of the orbit of a pure state.

Let $A$ be a simple unital $C^*$-algebra and denote $U(A)$ the group of unitaries in $A$. For $u\in U(A)$ lets define the $^*$-automorphism $\text{Ad}u:A\to A$ given by $a\mapsto uau^*$. It is a ...
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On correspondence between direct integral of vN algebras and conditional expectation

Let $M$ be a decomposable vN algebra. Let $E: M\mapsto N$ be a conditional expectation. Is there any relation state of $M$, with $E$?
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1answer
31 views

About the definition of cyclic operator

Based on the paper the cyclic operator is defined as: A bounded linear operator $T$ on a complex Hilbert space $H$ is cyclic if there is a vector $x$ in $H$ such that $H$ is the closed linear ...
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1answer
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K- theory of stably projectionless C* algebras

could anyone give me an example of a stably projectionless C*-algebra with non-zero $K_0$ group?
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1answer
32 views

K theory of projectionless C*-algebras

Is it possible to have a projectionless C*- algebra with non trivial K-theory? If so what would be such an example? I can't come up with any. p.s. By projectionless I mean non-unital aswell.
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Grothendieck's double limit criteria for weakly compact set

I was reading about Arens regularity of normed algebras in Palmer's book. He talks about the Grothendieck's double limit criteria for relatively weak compact sets in $X^*$ . Can some one provide me a ...
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Piecewise linear structure as generators of $C(M)$

Let $M$ be a compact topological manifold and $C(M)$ the commutative unital $C^*$-algebra of complex valued continuous functions on $M$. Suppose that $M$ admits a piecewise linear structure. In the ...
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1answer
17 views

range of two projections

If $p,q $ are two projections in $B(H)$ with $dim(pH)=dim(qH)$,then $p$ is equivalent to $q$. How to construct $v\in B(H)$ such that $p=v^*v,q=vv^*$by using the o.n.b of $pH$ and $qH$?
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1answer
35 views

Regarding group action on vN algebras

$G$ be a discrete countable group acting on $M$ via automorphisms of $M$. Does there exist a faithful normal state on $M$ which preserves the action means $\varphi(\sigma_{g}(x))=\varphi(x)$, $\sigma:...
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1answer
24 views

equivalent projections

Suppose $p,q$ are two equivalent projections in $B(H)$,do $p(H)$ and $q(H)$ have the same dimension?
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Find a type I group which has a type $I_\infty$ representation

I am looking for an example of a locally compact group $G$ that is type $I$ and has a unitary representation $(\pi, H_\pi)$ of $G$ such that $\pi(G)\cong B(H)$, as C*-algebras, where $H$ is an ...
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1answer
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Inverse of compression (von Neumann algebra)

I am stuck with this seemingly easy problem but I am having trouble showing this: Let $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ be a von Neumann algebra realized inside a subalgebra of the ...
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Suppose $A,B$ are positive operators with $AB=0$,what is the norm of $A+B$?

Suppose $A,B$ are positive operators with $AB=0$,what is the norm of $A+B$?
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Understanding Conditional Expectation and relation to Crossed Product

Let $\mathcal{A}$ be a unital $\Gamma$-$C^*$-algebra. Then one can form the reduced crossed product $C^*$-algebra $\mathcal{A}\rtimes_r\Gamma$. The reduced crossed product comes equipped with a ...
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1answer
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Traciality of compressions of von Neumann algebras

Let $\phi_1$ be a linear functional on a von Neumann algebra $\mathcal{A}.$ (I need the result in particular for $\Pi_1$-factors), satisfying traciality. With "traciality" I mean the following: For $...
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2answers
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Exponential of the product between $x$ the derivative operator of $x$ acting in a $f(x)$

The question I'm stuck here trying to figure out how to compute and prove, the following operator action in a function: $\exp(\varepsilon x \partial_x) f(x) = f(x \exp(\varepsilon) )$ where $\...
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1answer
37 views

Existence of conditional expectations onto masas.

Given an inclusion $N\subset M$ of von Neumann algebras, a conditional expectation is a map $E:M\to N$ that is a projection ($E^2=E$) and it has $\|E\|=1$. This automatically implies that $E$ is ...
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1answer
27 views

Direct limit of totally ordered system of isomorphic $C^*$-algebras

Suppose we have a directed system of $C^*$-algebras $A_i$ $$\{A_i,\phi_i\}_{i\in\mathbb{N}},$$ such that each $*$-homomorphism $\phi_i:A_i\rightarrow A_{i+1}$ is an inclusion. Furthermore, suppose ...
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Strictly positive inner product for a pair of non-zero, positive operators.

Let $ A,B $ be non-zero positive operators on a infinite-dimensional separable Hilbert space $(H , \langle \cdot, \cdot \rangle)$. I am required to prove that there exists $u' \in H$ such that \begin{...
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1answer
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Condition expectation on maximal group C*-algebra

I am trying to understand some basic $C^*$-algebraic terminology in a context specific to group $C^*$-algebras, and would appreciate some help from experts to whom the meaning of these things is clear....
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Operator K-theory and Topological K-theory

I was trying to understand the relationship between these two K-theory's when you pick your C* algebra to be $C(X)$ for $X$ a compact Hausdorff space. For this you create a function between $P_\infty (...
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center of group $C^*$ algebras

Suppose $G$ is a locally compact group,$L^1(G)$ is the group algebra,$C^*(G)$ is the group $C^*$ algebra.If the center $Z(L^1(G))$ of $L^1(G)$ is 0,can we conclude that $Z(C^*(G))=0$?
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1answer
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Inclusion preserving map between ideal spaces is continuous in the Fell topology

Let $A$ be a $C^*$-algebras with ideal space $\mathcal{I} (A)$ and equip $\mathcal{I} (A)$ with the Fell topology, i.e. the topology generated by the subbase $U_{I}:=\left\{ J\in{\cal I}\left(A\right)\...
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Showing convexity of a set in $\mathbb{C}^k$

Let $\mathcal{H}$ be an infinite dimensional separable Hilbert Space and $T\in\mathscr{B(\mathcal{H})}$. Suppose $$\mu_k(T):=\left\{ \begin{pmatrix} \lambda_1 \\ \lambda_2 \\ \vdots \\ ...
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1answer
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A natural Banach algebra where adjoint doesn't preserve the norm?

Are there any natural Banach algebras with some natural operator $A$ where $\Vert A \Vert \ne \Vert A^* \Vert$?
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1answer
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Trace of a matrix exponential with tensor products, and Von Neumann entropy

$\def\T{\operatorname{Tr}}$ $\def\1{\mathbb{1}}$ Let $H=H_1\otimes H_2\otimes H_3$ be a finite dimensional Hilbert space, and let $\rho_{123}$ be a self-adjoint matrix with $\rho_{123}\geq 0$ (...
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1answer
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Normal u.c.p extension of Schur-multiplier

I'm struggeling with the proof of a theorem in [BO08]. The first part before the line is what I think I understood. The part after that I don't understand at all. Let $\Gamma$ be a discrete group and ...
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1answer
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Integral representation of log of operators

$\def\1{\mathbb{1}}$ Suppose we're in a "good enough" (finite for example) space, and we have positive (semi)-definite operators $P$ and $Q$. Let $\log{(P)}$ and $\log{(P)}$ be logarithms of $P$ and ...
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1answer
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Does the von Neumann algebra generated by a normal operator contain all commuting projections?

Let $H$ be a Hilbert space and $T\in B(H)$ a bounded normal operator. Let $\mathscr{A}$ be the von Neumann algebra generated by $T$. Is it true that $\mathscr{A}$ contains every orthogonal projection ...
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1answer
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What is the $\text{♢-Axiom}$? (now known to be the 'Diamond Principle / Axiom')

While skimming over some research papers I found this abstract Mathematics > Operator Algebras (link here) Large irredundant sets in operator algebras Clayton Suguio Hida, Piotr Koszmider (...
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1answer
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Is a Rational Rotation Algebra a Cutdown of a Matrix Algebra?

Let $\theta=m/n$ and let $A_{\theta}$ be the rational rotation C$^{*}$-algebra with rotation angle $\theta$. I.e., $A_{\theta}=C^{*}(u,v)$, where $u$ and $v$ are unitaries such that $vu=e^{2\pi i \...
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1answer
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Verapoulous Algebra $C(K) \mathbin{\hat\otimes} C(L)$ is a subalgebra of $C(K\times L)$?

Let $K$ and $L$ be compact spaces. Consider the Banach algebra $V(K,L)=C(K)\mathbin{\hat\otimes} C(L)$ , which is the completion of the $C(K)\otimes C(L)$ with respect to the projective tensor norm. ...
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1answer
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C*-algebra without finite-dimensional representations is simple?

Suppose $A$ is an infinite dimensional simple $C^*$-algebra. Then it has no non-zero finite dimensional representations. Is the converse also true? That is to say, if a $C^*$-algebra has no finite ...
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1answer
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simple nuclear $C^*$ algebra [closed]

Does there exist an infinite dimensional simple nuclear $C^*$ algebra which admits a tracial state?
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Completely positive map is $*$-homomorphism

Suppose $A$ is a $C^*$ algebra and we have a completely positive contractive map $f \colon A\rightarrow B(H)$ such that $sup_{a,b \in A}\lVert f(ab)-f(a)f(b)\rVert =0$. Can we conclude that $f$ is a $*...
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1answer
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strict topology on multiplier algebras

Suppose $A$ is a $C^*$ algebra,$M(A)$ is the multiplier algebra.If $S$ is a subset of $M(A)$ which is compact for the strict topology on $M(A)$,is $S$ also a subset of $M(M(A))$ which is compact for ...
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1answer
63 views

Finding a Unitary to Implement the action in a Minimal Dynamical System

Let $X$ be an infinite compact Hausdorff space and let $\sigma\colon X\to X$ be a minimal homeomorphism thereof. Then $\sigma$ gives rise to an automorphism $\sigma'$ of $C(X)$ defined by $\sigma'(f):=...
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There exists a linear operator with no proper invariant subspaces

Let $A$ be a bounded operator on a Hilbert space $H$ with two invariant subspaces $M$ and $N$ s.t. $N \subset M$, dim$(M \cap N^{\perp})> 1$, and have no invariant subspaces between $N$ and $M$. ...