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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1answer
61 views

What to study after Rudin's Functional Analysis?

I am aware that there are one or two questions in this vein but I am looking for specific advice pertaining to my situation. I have completed Rudin's Functional Analysis and I'm interested in going ...
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1answer
28 views

conditional expectations on $C^*$-algebras

I am trying to get a feel for conditional expectations on arbitrary $C^*$-algebras but I am not able to find many examples. Obviously I can find conditional expectations from an $C^*$-algebra into $\...
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1answer
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Let $\phi:A\rightarrow B$. Then if $b\in B_{sa}\cap\operatorname{ Inv}(B)$, there is $a\in A_{sa}\cap \operatorname{Inv}(A)$ s.t. $\phi(a)=b$

Let $\phi:A\rightarrow B$ be a surjective $*$-homomorphism between unitary C*-agebras (not necessarily commutative). The question is that if for any $b\in B$ self-adjoint and invertible, there has to ...
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29 views

If $B\subset B(H)$ is a C*-subalgebra and $T\colon B''\to B''$ is linear, bounded and weakly continuous, then $\|T\|=\|T|_{B}\|$

Let $H$ be a Hilbert space and let $B\subset B(H)$ be a C*-subalgebra. Suppose that $T\colon M\to M$ is linear, bounded and operator-weakly continuous, then I want to prove that $\|T\|=\|T|_{B}\|$. ...
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Is it true that all representations of $C_0(X,A)$ come from representations of $A$

Let $A$ be $C^*$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\ast}$ ...
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If $A$ is a real or complex algebra and $a\in A$ is such that $ab=0$ for all $b\in A$, then $a=0$?

Let $A$ be an (not necessarily unital) algebra over $\mathbb{R}$ or $\mathbb{C}$. If $a\in A$ is an element such that $ab=0$ for all $b\in A$ (or equivalently, $aA=\{0\}$), can we then conclude that $...
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1answer
40 views

Inseparable $C^*$-algebras

In many theories about $C^*$-algebras, the $C^*$-algebras are always assumed to be separable. I have a question: Why few people discuss the inseparble $C^*$-algebras? Are they more difficult to handle?...
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2answers
48 views

Existence of non- normal element element in a non commutative C$^*$-algebra

I'm not familiar with lots of examples of non commutative C$^*$-algebras, but there are $M_n(\mathbb{C})$, and $B(H)$. These have a non trivial non-normal element. My question is : what about the ...
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1answer
35 views

Ideals of $M_2(A)$ , $A$- non commutative non unital

Let $A$ be a non unital, non commutative $C^{\ast}-$ Algebra. Let $J$ be an ideal of $M_2(A)$. Assume $$J= \begin{bmatrix}P&Q\\R &S\end{bmatrix}$$ It is easy to prove that $Q$ is an ideal of $...
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1answer
35 views

Is the center $Z$ of a C*-algebra $A$ unital if and only if $A$ is unital?

Let $A$ be a C*-algebra. Its center is defined by $$Z:=\{z\in A:az=za \ \text{for all} \ a\in A\}.$$ It is easy to verify that $Z$ is a C*-subalgebra of $A$. Also, if $A$ is unital, then it is clear ...
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1answer
22 views

Show $C^*(S)/K(l^2(\mathbb{N})) \cong C(T)$

Consider $l^2(\mathbb{N})$ and the shift operator $S: l^2(\mathbb{N}) \to l^2(\mathbb{N}): e_n \mapsto e_{n+1}$. It is easy to see that $C^*(S)$ contains the compact operators $K(l^2(\mathbb{N}))$ ...
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1answer
21 views

Algebra of compact operators is CCR algebra

In Arveson's book "Invitation to $C^*$-algebra's", it is claimed that every algebra of compact operators is a CCR algebra. Concretely, let $\mathcal{A}$ be a $C^*$-subalgebra of some $B_0(\mathcal{H})...
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1answer
28 views

Is a representation $(H,\phi)$ of a simple C*-algebra $A$ always faithful?

Suppose that $A$ is a simple C*-algebra (i.e. there is no closed ideal $I\subset A$ such that $0\neq I\neq A$) and let $(H,\phi)$ be a representation. Can we conclude that $(H,\phi)$ is faithful, i.e. ...
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1answer
24 views

Prove the following are equivalent with irreducibility of representation on $C^*$-algebras

Let $\pi: A \to B(\mathcal{H})$ be a representation where $A$ is a $C^*$-algebra and $\mathcal{H}$ a Hilbert space. I'm trying to show the following four statements are equivalent: (1) $\pi$ is ...
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10 views

Set of representations of a $C^*$-algebra

I saw the following statement from a reference book. Let $S$ be the set of representations of a $C^*$-algebra $A$. Does it mean that $S$ is the set of unitary equivalence classes of the ...
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1answer
29 views

Prove the uniqueness of the hereditary $C^*$ algebra generated by a positive element

The following question is from $C^*$-Algebras by Example written by Kenneth R. Davidson. The original question is the Problem I.11. $\mathit{Definition}:$ Say $\mathcal{W}$ is a $C^*$-subalgebra of ...
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Dimension of state space

Let A be a $\mathcal{C}$* algebra. We define state , say $\phi$ on A ( linear functional on A) such that f is positive and $\phi$( 1)= 1 . I'm trying to prove the following : If A is isomorphic to ...
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26 views

Spectrum of a Banach algebra VS spectrum of a $C^*$-algebra

The spectrum of a $C^*$-algebra $A$ is the set of unitary equivalence classes of irreducible $*$-representations of $A$. The spectrum of a unital commuative Banach algebra $B$ is the set of ...
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$beta$- KMS States

Let $(\mathcal{U},\tau)$ and $(\tilde{\mathcal{U}},\tilde{\tau})$ be two dynamical systems and let $\pi$ be a morphism between them. So I hope it should satisfy $\tilde{\tau}_t\circ \pi=\pi \circ \...
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1answer
27 views

On Direct integral decomposition of von Neumann algebras

I have a question. We know by theory that any von Neumann algebra is direct integral of factors. Then how to get the decomposition in practical situation. Basically what is the decomposition examples ...
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1answer
41 views

Number of nontrivial projection of a unital commutative C* algebra

Q.Can we construct a unital commutative C* algebra such that it admits exactly 5 non trivial projection ? I can't conclude that answer. I only know For some C* algebra , I and O is the only ...
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2answers
28 views

Finding nontivial projections corresponding to a Normal element

In a $C^*$ algebra if we consider a normal element, say $x$, such that spectrum of $x$ is $\left\lbrace-1,1 \right\rbrace$,then can we find two non-trivial projections $p$ and $q$ such that $pq=0$? ...
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42 views

Ideals and representations of $C_0(X,A)$

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...
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$*$-isomorphisms of cross product $C^*$-algebras

Let $G$ be a discrete countable group, $B$ is the CAR algebra $\otimes_{\Bbb N}M_2$. I feel confused about the statement marked green in the screenshot. How to construct the explicit bijections?
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2answers
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Extreme points of states of matrices

Consider $M_n(\mathbb{C})$ and the set $S(M_n(\mathbb{C}))$ of states on $M_n(\mathbb{C})$ (linear positive functionals on $M_n(\mathbb{C})$ that preserve the identity). I'm trying to show that the ...
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1answer
36 views

Infinite tensor product

Suppose that $X$ is an infinite set and $A$ is a unital $C^*$-algebra. The tensor product $\bigotimes_X A$ is defined to be the closed linear span of $\bigotimes_{x\in X }a_x$, where $a_x\in A$ for ...
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1answer
31 views

Elements in crossed product $C^*$-algebras

Let $A$ be a $C^*$-algebra and $G$ be a discrete group. I am quite confused about the definition of the reduced crossed product $A\rtimes_r G$ and the full crossed product $A\rtimes G$. What are ...
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52 views

Univesal Coefficient Theorem for $C^*$-algebras

The UCT theorem is shown in the sreenshot. I have a question : What is the definition of $Ext_{\Bbb Z}^1(K_{*}(A),K_{*}(B))$? Does it have a relationship with Tor functor?
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1answer
29 views

Monomorphisms in the category of $C^*$-algebras

Let $\mathcal{C}^\star$ denote the category that has $C^*$-algebras as objects and $*$-homomorphisms as morphisms. My question is the following: Are monomorphisms in $\mathcal{C}^\star$ with the ...
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0answers
25 views

Considering $l^\infty(X)$ as a C* algebra, how to show that the maximal ideal is $S(X) = \beta X$

I have read in a few sources that 'it is clear to see that' or 'it is well known that' the maximal ideal of $l^\infty (X)$ is $\beta X$ (i,e,e the space of all ultrafilters on X). But I cannot see why!...
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2answers
40 views

About an inequality between the real part and absolute value of an operator

When dealing with operators on a Hilbert Space, one can define for $A\in\mathcal{B} (\mathcal{H})$ two natural notions imitating the real part and absolute value of a complex number $$ Re(A) = \frac{...
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0answers
27 views

Bott-Periodicity for unitary matrices with entries in general $C^{*}$-algebra

When working through the proof of Bott-Periodicity (the original proof by Bott), I noticed that compactness of $U(n)$ is important as it gives us that every geodesic looks like an exponential map (...
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1answer
33 views

Show that $C_{0}(X,A)$ is a sub space of $B(H)$ for some Hilbert space $H$

Let $A$ be $C^{\ast}$- Algebra and $X$ be a locally compact Hausdorff space and $C_{0}(X,A)$ be the set of all continuous functions from $X$ to $A$ vanishing at infinity. Define $f^{\ast}(t)={f(t)}^{\...
2
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1answer
23 views

Strong operator topology on a $C^*$-algebra?

As far as I know, the Strong Operator Topology (SOT) is defined for the space of operators $B(H)$ for any Hilbert space H. The paper I am reading implicitly make references to the 'fact' that we can ...
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1answer
13 views

Tail of increasing convergent net of self-adjoint operators is bounded

Let $H$ be a Hilbert space and $(T_\alpha)$ an increasing net of self-adjoint operators that converges (in some topology) to an operator $T$. Then $(T_\alpha)$ is not necessarily norm-bounded I think (...
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1answer
29 views

Spectrum of unital, commutative C star algebra

According to the Wikipedia article on the Gelfand Represenetation (C* algebra section), the spectrum of a commutative C* algebra $A$ (the non-zero *homomorphisms $\phi : A \rightarrow \mathbb{C}$) i)...
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28 views

Simplex of states in a $C^*$- algebra

Let $A$ be a unital $C^*$- algebra. What does it mean by a simplex in the space of states on a $C^*$- algebra. I know that the space $S(A)$ of all states is a compact convex set in weak * topology ...
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1answer
44 views

Generators for $K_1(A\otimes \mathbb{K})$

I've been working computing generator for several $C^*$-algebras involved in my Master's thesis, however I've got stucked with the generators of $K_1(C(\mathbb{T})\otimes\mathbb{K})$, which is in my ...
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1answer
46 views

Inequality of Operators Preserves Hilbert Schmidt Norm

I have been trying to prove the following fact in elementary operator theory without success: For any two positive operators $0 \leq R \leq S$, we have $\Vert R \Vert_2 \leq \Vert S \Vert_2$, where $\...
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1answer
36 views

Why is this representation of $C^*\!$-algebras irreducible?

I'm focused on the text "Invitation to $C^*\!$-algebras" by Arveson. In this notation $\mathcal{C}(\mathcal{H})$ is the set of compact operators on $\mathcal{B}(\mathcal{H})$. Here is the relevant ...
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1answer
44 views

If $P$ is a projection and $T$ is positive operator such that $P\leq T$, then why $P(H)\subset\sqrt{T}(H)$?

Suppose that $H$ is a Hilbert space. If $P\colon H\to H$ is a projection (i.e. $P^{2}=P^{*}=P$) and $T\colon H\to H$ is positive operator (i.e. $T\geq0$) such that $P\leq T$, then why $P(H)\subset\...
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1answer
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Linear independence is somehow preserved by approximate unit multiplication? [duplicate]

I am reading a book on Operator Algebras and at a point the authors seem to make the following implication: Let $A$ be a $C^*$-algebra and $\{a_i\}_{i=1}^n$ a linearly independent subset. If $(u_\...
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26 views

On the definition of a positive projection onto a $\mathbb{C}^3$ subspace

Consider the following subspaces of $\mathbb{C}^3$: $$\mathfrak{A}_{1} = \mathrm{Span}\left\{t_1\begin{bmatrix}1\\1\\1\end{bmatrix},\ t_1 \in \mathbb{C}\right\},\ \mathfrak{A}_{-1} = \mathrm{Span}\...
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1answer
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Questions of B(H), the linear space of bounded linear operators in the complex Hilbert space H

I'm going through the topic C* algebra & operator algebra and facing few questions . It would be great if you people could help me to clear the doubts. Q4. If T belongs to B(H) such that the ...
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1answer
18 views

Countibility of resolvent set in C* algebra

I'm going through the topic $C^*$ algebra and facing few questions. It would be great if you people could help me to clear the doubts. Q3. Does there exists some $X$ belonging to a $C^*$ algebra such ...
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1answer
28 views

If $p^{2}=p^{*}=p$ and $pa^{*}a=paa^{*}=0$, then $a^{*}p+pa=0$.

Suppose that $p$ is a projection (i.e. $p^{2}=p^{*}=p$) in a C*-algebra $A$. Let $a\in A$ be an element such that $pa^{*}a=paa^{*}=0$. I want to prove that $$a^{*}p+pa=0.$$ I tried to express $a$ in ...
3
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0answers
68 views

Definition of property RD

In the following paper https://www.jstor.org/stable/2001458?seq=1#metadata_info_tab_contents Jolissaint introduces the property RD of a group G if the space $H_L^\infty(G)$ is contained in the reduced ...
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1answer
32 views

Application of continuous functional calculas

I'm going through the topic C* algebra and facing few questions . It would be great if you people could help me to clear the doubts. Q2. Let $x$ and $y$ be two positive elements in a C* algebra such ...
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1answer
39 views

Finding the spectrum of a C* algebra

I'm going through the topic of C$^*$-algebras and facing a few questions. Q1. Consider the C$^*$-algebra $A=\bigoplus_{j=1}^n\mathbb C$ . What is the spectrum of $A$ (the collection of multiplicative ...
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1answer
26 views

Let $p$ be a projection in a unital C*-algebra $A$. What is the kernel of the map $a\mapsto pap$?

Let $p$ be a non-zero projection in a unital C*-algebra $A$, i.e. $p$ is a self-adjoint idempotent. Can we say something about the kernel of the linear map $$\varphi\colon A\to A,\qquad a\mapsto pap?$$...

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