# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $*$-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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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 ...
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 ...
### 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 $$\varphi\colon A\to A,\qquad a\mapsto pap?$$...