A C*-algebra is a complex Banach algebra together with an isometric antilinear involution satisfying (a b)* = b* a* and the C*-identity ‖a* a‖ = ‖a‖². Related tags: (banach-algebras), (von-neumann-algebras), (operator-algebras), (spectral-theory).

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16
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1answer
608 views

What is the spectrum of the commutative C*-algebra I have constructed here?

Let $B$ and $F$ be compact Hausdorff spaces. Let $E\to B$ be a fiber bundle with fibre $F$ and structure group $\mathrm{Homeo}(F)$, the group of homeomorphisms of $F$. I think this induces a fiber ...
11
votes
2answers
1k views

Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
11
votes
2answers
707 views

$C^*$-algebra which is also a Hilbert space?

Does there exist a nontrivial (i.e. other than $\mathbb{C}$) example of a $C^*$-algebra which is also a Hilbert space (in the same norm, of course)? For $\mathbb{C}^n$ with $n > 1$ the answer is ...
11
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2answers
360 views

When two projections in a C*-algebra are “almost” Murray-von Neumann equivalent, they are equivalent

Let $A$ be a C*-algebra and $p,q \in A$ be projections. Assume there is an element $a\in A$ such that $\|aa^*-p\|<\frac{1}{4}$ and $\|a^*a-q\|<\frac{1}{4}$. Then there is a partial isometry $v$ ...
10
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1answer
496 views

Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?

When $\ell^1(\mathbb Z)$ is equipped with the convolution as multiplication and $a^{*}_{n}=\bar{a}_{-n}$, I can prove it satisfies all conditions except $\|a^{*}a\|=\|a\|^2$, which I cannot prove nor ...
10
votes
3answers
238 views

If there is already enough room to add all projections, does passing to matrices change anything?

Throughout, $A$ denotes a $*$-algebra. We always assume $A$ is representable in the sense that $A$ can be embedded into $B(H)$ for some Hilbert space $H$. The particular embedding is not important, ...
10
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1answer
149 views

Computing $K$ - theory groups of certain $C^{\ast}$- algebras

I am trying to show that the $K$-theory groups of the following $C^*$-algebra $A$ vanish: Let $\mathcal{H}$ be a separable Hilbert space. Now consider the subalgebra of ...
10
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1answer
309 views

In a C*-algebra, put $a^*a \sim aa^*$. Transitivity fails?

Idle curiosity drove me to wonder about the following question. Let $A$ be a C*-algebra. Define a binary relation $\sim$ on the cone $A^{\geq 0}$ of positive elements by putting $x \sim y$ whenever ...
10
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0answers
301 views

Maximal ideal space of $C^*$-algebra of Riemann integrable functions

Let $R([0,1])$ be the unital commutative $C^*$-algebra of complex valued Riemann integrable functions on $[0,1]$ with pointwise operations and the supremum norm. In the 1980 paper The Gelfand space ...
8
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1answer
656 views

Why is every positive linear map between $C^*$-algebras bounded?

We know that every positive linear functional on a $C^*$-algebra is bounded. How can we prove every positive linear map between $C^*$-algebras is bounded?
8
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Upper-triangular matrix is invertible iff its diagonal is invertible: C*-algebra case

Exercise 1.14 of the book Rordam, Larsen and Laustsen "An introduction to K-theory for C*-algebras" asks to prove, that upper triangular matrix with elements from some C*-algebra $A$ is invertible in ...
8
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1answer
207 views

Abelian sub-C*-algebras

Given a non-abelian C*-algebra $A$. I am wondering what are the possible abelian sub-C*-algebras of $A$. Let $K$ be the spectrum of $A$. Does $A$ contain an isomorphic copy (as a Banach space) of the ...
8
votes
1answer
90 views

Physical interpretation of C* property

From a mathematical point of view, quantum mechanics can be formulated in the language of a non-commutative, unital C*-algebra $\mathcal A$ (of observables). In this context, what does the ...
7
votes
2answers
282 views

Existence of norm for C*-Algebra

I was wondering wether one can always find/construct a norm which turns an involutive algebra into a C*-algebra. For sure, if it exists it is unique, but does it always exist. If not can you provide a ...
7
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2answers
490 views

A question about pure state

For every unit vector $x$ in a Hilbert space $H$,let $F_x$ be the linear functional on $\mathcal B(H)$ (bounded linear operators) defined by $F_x(T)=(Tx,x)$. Prove that each $F_x$ is pure state and ...
7
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1answer
371 views

What is the commutative analogue of a $C^*$-subalgebra?

Using the duality between locally compact Hausdorff spaces and commutative $C^*$-algebras one can write down a vocabulary list translating topological notions regarding a locally compact Hausdorff ...
7
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4answers
298 views

What makes irreducible representations nice?

Let $\mathcal{A}$ be a C*-algebra and $(H,\pi,\Omega)$ a cyclic representation. What does it intuitively mean if the representation is irreducible? From what I've read, irreducible representations ...
7
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1answer
361 views

C*-algebras as Banach lattices?

It seems to be trivial but I am not sure about monotonicity of the norm in the non-commutative case: Is every C*-algebra a Banach lattice with respect to its natural positive cone?
7
votes
2answers
234 views

There are 3 points in the spectrum of some self-adjoint element of a non-unital C*-algebra.

Let $A$ be a non-unital C*-algebra. I would like to know a simple way to show that $A$ contains a self-adjoint element whose spectrum has at least $3$ elements. Note that the spectrum of an ...
7
votes
1answer
81 views

Extension of character in Banach algebras

Let $A$ be a Banach algebra. The continuous linear functional $\phi:A\to\Bbb{C}$ is called character if it is non-zero multiplicative function i.e., for every $a,b\in A$ we have ...
7
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0answers
134 views

K-theory for non-separable C*-algebras

Let $\kappa$ be an uncountable cardinal. What is the K-theory for the C*-algebras $\mathcal{K}(\ell_2(\kappa))$ and $\mathcal{B}(\ell_2(\kappa))$, of, respectively, compact and bounded operators on ...
6
votes
1answer
198 views

equivalent? algebraic definition of a partial isometry in a C*-algebra

An element $a\in\mathfrak{A}$ (unital C*-algebra) is a partial isometry if $a^*\cdot a $ is projection. Can one recover the equivalent caracterizations of a partial isometry in ...
6
votes
2answers
361 views

Unitisation of $C^{*}$-algebras via double centralizers

In most of the books I read about $C^{*}$-algebras, the author usually embeds the algebra, say, $A$, as an ideal of $B(A)$, the algebra of bounded linear operators on $A$, by identifying $a$ and ...
6
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1answer
307 views

Cube root in $ C^{*}$-algebra.

Let $A$ be a $C^*\text{-algbera}$ and $x\in A$. I'm trying to show thata)for $0<\alpha<\frac{1}{2}$, there exists $u\in A$ with $x=u(x^*x)^{\alpha}$ and $u^*u=(x^*x)^{1-2\alpha}$. b) there ...
6
votes
1answer
86 views

Can we say $TT^{*}=T^{2}$ implies $T=T^{*}$?

Let $A$ be a $C^{*}$-algebra, Can we say $TT^{*}=T^{2}$ implies $T^{*}=T$? for $T\in A$ I am looking for a counterexample! Thanks
6
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1answer
201 views

Uniqueness of the involution on a $C^*$-algebra

indication please Let $A$ be a C*-algebra. Suppose that there exists on $A$ another involution $x\rightarrow x^{\#}$ such that $\|xx^{\#}\|=\|x\|^2$, for all $x\in A$. Prove that $x^{\ast}=x^{\#}$, ...
6
votes
2answers
235 views

Example of a *-homomorphism that is faithful on a dense *-subalgebra, but not everwhere

Let $A,B$ be C*-algebras and let $\varphi: A \to B$ be a $*$-homomorphism. Suppose that $\ker( \varphi) \cap D = \{0\}$ where $D$ is a dense $*$-subalgebra of $A$. Does it follow that $\varphi$ is ...
6
votes
1answer
358 views

Matrices with entries in $C^*$-algebra

Let $\mathcal{A}$ be a $C^*$-algebra. Consider vector space of matrices of size $n\times n$ whose entries in $\mathcal{A}$. Denote this vector space $M_{n,n}(\mathcal{A})$. We can define involution on ...
6
votes
1answer
176 views

In how far are CCR and GCR C*-algebras interesting?

I have some basic familiarity with C*-algebras (from the mathematical physics side - Bratteli-Robinson), and while having a look at Arveson's "Invitation to C* algebras", I came across so-called CCR ...
6
votes
1answer
255 views

Homomorphic conditional expectations?

To clarify, I mean "conditional expectation" in the sense of $C^*$-algebras (a completely positive projection of norm 1, equivalently, a completely positive linear map onto a $C^*$-subalgebra which is ...
6
votes
1answer
182 views

Effect of permuting coordinates on K-theory

Let $A$ be a C*-algebra and let $f: \mathbb{R}^n \to \mathbb{R}^n$ be the linear map which permutes the coordinates via a permutation $\sigma$. There is an induced map $K_0(C_0(\mathbb{R}^n) \otimes ...
6
votes
1answer
308 views

Weak-* continuity of the adjoint map on a $W^*$-algebra

Let $\mathcal{M}$ be a $W^*$-algebra, i.e. a $C^*$-algebra with a Banach space predual $\mathcal{M}_*$. I'm trying to show that the adjoint map $x \mapsto x^*$ on $\mathcal{M}$ is weak-* (aka ...
6
votes
1answer
161 views

Invertibility in subalgebra

I have some trouble proving the following statement: Let $A$ be a self-adjoint element of a $C^*$-algebra $\mathcal{B}$ and let $\mathcal{A}$ denote the unital subalgebra of $\mathcal{B}$ that is ...
6
votes
1answer
126 views

Why need two directions to make $\sim_{\rm wa}$ an equivalence relation?

Let $\pi$ and $\sigma$ be representations of a $C^*$-algebra $\mathcal{A}$. They are weak approximately equivalent ($\pi\mathbin{\sim_{\rm wa}}\sigma$) if there are sequences of unitary operators ...
6
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0answers
151 views

How to prove this element is strictly positive? [duplicate]

Let $A$ be a $C^*\text{-algebra}$ and $A_+$ denote the positive elements. An element $a\in A_+$ is called strictly positive if $\overline{aAa}=A$. Want to prove: if $(e_n)$ is an approximate identity ...
6
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0answers
132 views

Decomposing $\mathcal{B}(H)$

Let $H$ be an infinite-dimensional Hilbert space and let $\mathcal{B}(H)$ be the (C*/W*-)algebra of bounded operators on it. Actually, you may forget about the involution in $\mathcal{B}(H)$ because I ...
5
votes
1answer
214 views

Is the centre of a C*-algebra a sub-C*-algebra?

I believe that the answer is affirmative and I would be grateful to any comments on my attempt (see below) of proving this. Let $A$ be a C*-algebra and denote by $Z(A)$ the centre of $A$. First of ...
5
votes
2answers
170 views

Induced map on $K_{1}$-group

Let $A$ be a unital $C^{\ast}$-algebra. Any automorphism $\alpha$ of $A$ induces a map on $K_{1}(A)$ by $\alpha_{\ast}[u]=[\alpha(u)]$. Let the automorphism $\alpha$ be inner, does it follow that ...
5
votes
1answer
210 views

Applications of Elliott's theorem concerning the classification of AF-algebras

An AF-algebra is a $C^* $-algebra which is the inductive limit of an inductive sequence of finite-dimensional $C^*$-algebras. Elliott's theorem concerning the classification of AF-algebras says that ...
5
votes
1answer
100 views

Definite states on C*-algebras

A state $\omega$ on a unital $C^*$ algebra $A$ is called definite at $a\in A$ self-adjoint if $\omega(a^2)=\omega(a)^2$. I proved that if we have such a definite state at $a$, then for all $b\in A$ ...
5
votes
1answer
172 views

Existence of a completely supported probability measure

Given a compact Hausdorff space $X$, does there exist a probability $\mu$ on X such that the support of $\mu$ is $X$? This is equivalent to say, for any unital commutative C*-algebra, can we show the ...
5
votes
1answer
85 views

“Refinement” of the existence of faithful representations of C*-algebras?

Conway, in a course in operator theory, brings the statement 1. below as a theorem and statement 2. below as an exercise. Still, he states that 2. refines 1., but I can't see it. Every C*-algebra ...
5
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1answer
106 views

characters of a $C^*$-algebra

I have read that a state $\rho$ on a unital $C^*$-algebra $A$ is a character (i.e. multiplicative) if and only if, for all unitary $u\in A$, $|\rho(u)| = 1$. Is there an easy proof, or can someone ...
5
votes
1answer
58 views

Analyticity of C*-algebra valued functions

Let $\mathcal{A}$ be a unital C*-algebra and consider a function $f:\mathbb{C} \rightarrow \mathcal{A}$. What is an accessible tool to prove or disprove that $f$ is analytic, i.e. can be locally ...
5
votes
1answer
120 views

Continuity of double centralizers in Banach algebras

I had some problems with a certain exercise, came up with a solution, but I'm not sure it is correct. Exercise ("MURPHY, C*-Algebras and Operator Theory", Chapter 2, exercise 1) Let $A$ be a ...
5
votes
1answer
92 views

Doubt about Proposition 2.39 in Dana Williams' crossed product book

You can see the proposition in a google books preview here. First and foremost, my question is: Question: Am I correct to interpret Proposition 2.39 as setting up a bijective correspondence ...
5
votes
1answer
116 views

Operator K-theory and the unitary group

Suppose that $A$ is a C*-algebra whose unitary group is contractible (e.g. $B(H)$ or more generally the stable multiplier algebra of any C*-algebra). It is clear from the definition that $K_1(A) = ...
5
votes
1answer
44 views

The set of all normal operators on a Hilbert space is not strongly closed

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of ...
5
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0answers
68 views

Conditional expectation onto maximal abelian subalgebras

If you take a von Neumann algebra $M$ and any its maximal abelian subalgebra (masa) $D$, then there is a norm-one projection from $M$ onto $D$ (conditional expectation). The same is true if you take ...
5
votes
1answer
257 views

application of c*algebras to PDEs

I am preparing an introductory talk about c* algebras and I'd like to motivate C*algebras or show an application of them and I'd prefer an application in the field of partial differential equations. ...