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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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When is the set of positive elements in a C* algebra a totally ordered set

As the title suggests I was wondering if there are any properties that ensure a C*-algebra has all positive elements comparable to each other. ( Recall $a\leq b$ if $b-a$ is a positive element in the ...
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a question on the definition of direct sum of $C^*$ algebras

According to the definition in the Olsen's book,if $A=B\bigoplus C$,the intersection of $B$ and $C$ should be zero.But in other reference books ,when talking about direct sum of matrix algebras,there ...
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GNS representation of a nuclear $C^*$-algebra

Suppose $A$ is a nuclear $C^*$-algebra with a tracial state $\psi$, $(\pi_{\psi},H_{\psi})$ is the GNS reprsentation with respect to $\psi$. My question: Does there exist $A$ which satisfy the above ...
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1answer
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find all ideals of direct sum of simple $C^*$ algebras

Suppose $A=\bigoplus A_n$ where each $A_n$ is a simple $C^*$ algebra(The direct sum is $c_0$ direct sum).I guess all the ideals of $A$ are precisely the direct sum of ideals of $A_n$ .It is easy that ...
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1answer
17 views

GNS representation

Suppose $A$ has a tracial state $\psi$, I want to prove $A/ker(\pi_{\psi})$ has a faithful tracial state,where $\pi_{\psi}$ is the $GNS$ respresentation with respect to $\psi$. My thought: define $\...
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existence of maximal ideals in $C^*$ algebras [on hold]

If $A$ is a $C^*$ algebra,$A$ has a non-trivial ideal,can we conclude that there exists a maximal ideal in $A$?If it exists,is it closed?
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Automorphisms of a particular abelian group

I am trying to solve Exercise 7.9 of An Introduction to K-theory of C*-algebras, It remains for me to show that the Automorphisms of $\mathbb{Q}\oplus\mathbb{Q}$ which send $ \{(x,y) \in \mathbb{Q}\...
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1answer
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Spectrum invariance under the passage to a sub Banach Algebra

Let $\mathcal{B}$ be a unital Banach Algebra, fix $A \in \mathcal{B}$. $\sigma_\mathcal{B}(A) = \{ \lambda \vert \lambda I - A \, not \,invertible \, in \, \mathcal{B}\}$ the specturm of $A$ in $\...
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1answer
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closed ideal in a $C^*$- algebra

Suppose $A$ is a non-simple $C^*$-algebra, let $x_0$ be a nonzero element in $A$, and let $S=\{x_0y-yx_0:y\in A\}$. If $I$ is the closed ideal generated by the set $S$. I think there is a possibility ...
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1answer
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How to show matrix algebras are Nuclear [closed]

How would I show that $M_n(\mathbb{C})$ is nuclear? I know that $M_n(B)$ already is a C*-algebra with its obvious norm why would that show that then the min and max norm are the same?
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The inner structure of finite-dimensional $C^*$-algebras

This is just to complete the questions about the inner structure of finite-dimensional $C^*$-algebras. Please correct the statements and help me with the appropriate links if it is not very difficult. ...
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Reference request unital normal *-homomorphisms $B(H) \to B(K)$

It is known that if $\varphi \colon B(H) \to B(K)$ is a unital non-zero normal $*$-homomorphism (for Hilbert spaces $H$ and $K$), then there exists a Hilbert space $K'$ and a unitary $U' \colon K \to ...
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1answer
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ideal of an non-nuclear $C^*$ algebra

If $A$ is a nuclear $C^*$-algebra,$I$ is a closed ideal of $A$,then $A/I$ is nuclear. My question :Does there exist an non-nuclear $C^*$-algebra whose quotient is nuclear?
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1answer
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connection between unital $C^*$ algebra and finite von neumann algebra

Let $A$ be a unital $C^*$-algebra with a tracial state $\tau$, $L^2(A,\tau)$ is the Hilbert space induced by the GNS constructtion.Suppose $\lambda$ is the left action of $A$ on $L^2(A,\tau)$ ,does ...
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1answer
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post-liminal/GCR algebras

from Theorem 5.6.2 [Murphy-C* algebras and operator theory]: let (H , φ) be an non zero irreducible representation of C* algebra A, and let I be a closed ideal for A. Denote (H',φ') the restriction ...
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Is the unitary group connected in infinite dimensions?

Let $H$ be an infinite dimensional Hilbertspace and $U(H)$ the group of unitaries endowed with the norm topology. Is $U(H)$ connected? The following is a generalisation of the proof in the finite-...
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1answer
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Two isomorphic finite-dimensional $C^*$-algebras with infinite eigenspaces

Let $H_1$, $H_2$ be two separable Hilbert spaces. Let $\mathscr{A}_1$, $\mathscr{A}_2$ be two isomorphic finite-dimensional $C^*$-algebras of operators acting on $H_1$, $H_2$ respectively. Suppose ...
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2answers
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Tracial states and the GNS construction

If a $C^*$-algebra $A$ has a tracial state $\tau$, can we construct a nonzero representation $\pi: A\rightarrow B(H_{\tau})$ such that $\pi(ab)=\pi(ba),\forall a,b \in A$ through the GNS construction?
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action of $C^*$ algebras

In group theory,given a group $G$,we can define a group action of $G$ on $G$.Can we define a $C^*$ algebra action similarly?To be more precise,suppose we have a $C^*$ algebra $A$,can we define a map ...
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2answers
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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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1answer
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Two isomorphic $C^*$-algebras. What is the isomorphism between corresponding Hilbert spaces?

Let $H$ be a separable Hilbert space. Suppose that $\mathscr{A}$ and $\mathscr{B}$ are some unital $C^*$-algebras of operators acting on $H$, not necessary coinciding with $C^*$-algebra of all the ...
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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
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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
32 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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Distributivity of direct sum over maximal tensor product

Let $A,B$ and $C$ be $C^{*}$-algebras.Does the following identity always holds: $(A \oplus B) \otimes^{max} C \cong (A \otimes^{max}C) \oplus (B \otimes^{max} C)$ My intuition is that this should ...
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1answer
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Modern introduction to $C^*$-algebras

Well, I am asking for the references on the subject for those who can't stand the Murphy's book. I have a background in functional analysis (including Banach algebras, functional calculus, Gelfand ...
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1answer
31 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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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
21 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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1answer
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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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1answer
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Question about Nate and Taka

When introducing central covers in this book the following is stated: For every Non-degenerate representationt $\pi:A\rightarrow B(H)$ there exists a unique normal extension $\gamma:A^{**}\rightarrow ...
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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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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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Doubt from B Blackadar's Operator Algebras

I am reading Blackadar's book on Operator algebras. In $\Pi 9.6.5$ Blackader says that Maximal Tensor products commute with arbitrary limits. In the same book the proof of this fact is not given....
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Norm on a certain quotient C$^*$-algebra

Let $\{A_i\}_{i\in I}$ be an arbitrary family of C$^*$-algebras, we may define their product as $$\prod_{i\in I}A_i=\{(a_i);~\sup_{i\in I}\|a_i\|<\infty\}.$$ We can also define $$\sum_{i\in I}A_i=\...
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1answer
18 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
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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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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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1answer
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How to show that this short exact sequence does not split

Consider the Short Exact sequence $0\rightarrow C_0((0,1))\rightarrow C([0,1])\rightarrow \mathbb{C}\bigoplus\mathbb{C}\rightarrow 0$ where the map from $C([0,1])\rightarrow \mathbb{C}\bigoplus\...
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1answer
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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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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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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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1answer
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7 doubts about the von Neumann algebra [closed]

A von Neumann algebra, or $W^*$-algebra, is a $*$-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type ...
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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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Finding the Spectrum of an element of $\ell^\infty$

I have done Q.1.2.1 already and it is quite clear.But I am not sure about the next one. How does the closure come into the picture. I have a feeling that it should be $f(S)$ only. Am I missing ...
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1answer
81 views

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
54 views

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
39 views

Almost having invariant vectors vs having almost invariant vectors?

Let $\Gamma$ be a discrete and countable Group and let $\pi:\Gamma\to \mathcal{B(H)}$ be a unitary representation. We say that $\pi$ almost has invariant vectors if for every compact (=finite) subset ...