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 ...

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3
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157 views

A simple question about *-homomorphism in C*-algebra

Let $A$ and $B$ be C*-algebra, $h\colon A\rightarrow B$ is *-homomorphism. If $a\in A_{\operatorname{sa}}$, then $\operatorname{sp}(h(a))\backslash \{0\}\subset \operatorname{sp}(a)\backslash\{0\}$. ...
7
votes
2answers
237 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 ...
2
votes
1answer
69 views

A confusion in C*-algebra

I am reading a book about C*-algebra. But I meet with some problems. In the book, the author says: If $I$ is an ideal in a C*-algebra $A$, then $B=I\cap I^{\ast}$ is a C*-subalgebra. However, I ...
2
votes
1answer
244 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
6
votes
1answer
255 views

Weak* operator topology and finite rank operators

We will say that ${T_i}\subset B(X,Y^*)$ converges to $T$ in W*-operator topology if $T_i(x)\rightarrow T(x)$ in W*-topology of $Y^*$( $\forall y\in Y \langle T_i(x),y\rangle \rightarrow \langle ...
5
votes
1answer
225 views

What is the relationship between spectral resolution and spectral measure?

In Kadison and Ringrose's book "FUNDAMENTALS OF THE THEORY OF OPERATOR ALGEBRAS", the author gives the following theorem. Theorem: If $A$ is a self-adjoint operator acting on a Hilbert space ...
3
votes
1answer
171 views

Confusion in Gelfand theorem in C*-algebra.

I am reading HX Lin's book, named "An introduction to the classification of amenable C*-algebras", I can not understand a corollary of Gelfand theorem(Corollary 1.3.6): If a is a normal element in a ...
2
votes
1answer
55 views

A question about compact Hausdorff space

Let $X$ be a compact Hausdorff space and $C(X)$ be the set of continuous functions on $X$. And $F$ is a closed subspace of $X$. If the $f\in C(X)$ such that $f|_{F}=0$ is only zero function( i.e. ...
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vote
0answers
48 views

A question about bounded operators on banach space [duplicate]

Let $L(X)$ denotes the Banach algebra of all bounded linear operators acting on a Banach space $X$. And $T$ is not invertible. Can we find a invertilbe bounded operator series $\{T_{n}\}$ such that ...
1
vote
1answer
182 views

A question about convex set

I need to prove the closed set $C\subseteq \mathbb{R}_{+}$ is a convex. And let $x$, $y$ be arbitrary given in $C$, I have proved that $1/2(x+y)\in C$. Then does this means $C$ is convex ?
2
votes
2answers
313 views

Norm of an inverse operator: $\|T^{-1}\|=\|T\|^{-1}$?

I am a beginner of funcional analysis. I have a simple question when I study this subject. Let $L(X)$ denote the Banach algebra of all bounded linear operators on Banach space X, $T\in X$ is ...
1
vote
1answer
46 views

Why must trivial extension of C*-algebra be split Short Exact Sequence?

Background: Suppose $0 \to B \to E -q-> A \to 0$ is a short exact sequence of C*-algebras. Since B sits as an ideal of E, there is a natural *-homomorphism from E to the multiplier algebra of B (by ...
3
votes
1answer
60 views

A bounded everywhere defined operator that is affiliated to a von Neumann algebra is in the algebra

A possibly unbounded operator $T$ on a Hilbert space $\mathcal H$ is (in my source) defined as affiliated to a von Neumann algebra $M$ if for each unitary element $u$ of $M^\prime$, $u^*Tu=T$ (or ...
3
votes
1answer
201 views

A question on multiplicative linear functional on Banach algebra.

I am reading a book about C*-algebra. But i am confused with some of its content. It says Assume $A$ is a non-unital C*-algebra and $\tilde{A}$ is its unitization (the elements of the form ...
2
votes
0answers
138 views

Does ternary operations have associative property?

Binary Operation is a function. Right? We know that all Binary operations have associative property. They must be either associative or non-associative. The condition is : $$(a*b)*c = a*(b*c)$$ ...
1
vote
0answers
188 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...
0
votes
1answer
58 views

A question about download the recent paper. [closed]

I am interested in the journal about operator theory, such as Studia Math and Operators and Matrices. However, my college do not buy some journals. How can I get the paper from these journal?
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votes
0answers
53 views

self-adjoint subalgebras of matrix algebra

Is there any classification theorem for the self-adjoint matrix subalgebras of $M_n(\mathbb{C})$ the algebra of $n \times n$ matrices over $\mathbb{C}$ ?
0
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0answers
42 views

Intuition behind a braid operator which is also a solution for Yang-Baxter equation

I am going through this paper, 'Quantum entanglement and topological entanglement' by Louis H Kauffman and Samuel J Lomonaco Jr published in New Journal of Physics 4 (2002). It started with ...
1
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1answer
78 views

Map to multiplier algebra for C*-subalgebra

We have such a claim: If $A$ is an ideal of C*-algebra $B$, then there is a unique morphism $f\colon B\to M(A)$ such that $f$ is identity on $A$, here $M(A)$ is the multiplier algebra of $A$. Now ...
1
vote
1answer
46 views

A criterion for vector states to be in the same irreducible representation

a little wish...: is there a theorem that corresponds or implies the following Let A be a C* algebra with the data of a representation in B(H). Let x,y be two vectors and call S(x,y) the set of ...
7
votes
2answers
250 views

Gelfand Naimark Theorem

The commutative Gelfand-Naimark theorem tells us that every unital commutative C* algebra is isometrically isomorphic to the space of continuous functions on its maximal ideal space. The non- ...
19
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1answer
461 views

Why study operator spaces?

I'm currently enrolled in an operator spaces course and I'm finding it difficult to understand why we study them in the first place. Functional analysis is motivated well enough for me and even though ...
6
votes
1answer
150 views

Question on irreducible representation of a Banach algebra

Let $\mathcal A$ be a Banach algebra over $\mathbb{C}$, $\mathcal X$ a irreducible left $\mathcal A$-module. If $x,y \in \mathcal X$ are linearly independent, there exists an element $a\in\mathcal A$ ...
2
votes
1answer
64 views

A question about quotient space of $R(T^{n})$

I am reading a paper about spectral theory. The author says it is easy to see the following proposition: For $T\in L(X)$, if dim$(R(T^{d})/R(T^{d+1}))<\infty$, then $R(T^{d})$ is closed if and ...
4
votes
1answer
260 views

The set of all continuous functions on a locally compact Hausdorff space.

I am reading a book about C*-algebra. There is a example that i could not understand. Let $X$ be a locally compact Hausdorff space and $C_{0}(X)$ be the set of all continuous functions vanishing at ...
1
vote
1answer
106 views

The operator norm of complex matrices

Let $M_{n}$ be the algebra of $n\times n$ complex matrices. By identifying $M_{n}$ with $B(\mathbb{C^{n}})$, the set of all bounded linear maps from the n-dimensional Hilbert space $\mathbb{C^{n}}$ to ...
7
votes
1answer
50 views

simple connectedness and abelian C* algebras

From Gelfand-Neumark Theorem, we know that topological properties of a compact Hausdorff space $X$ are encoded in the abelian $C^*$-algebra of continuous complex-valued functions on $X$ (with $||f||= ...
4
votes
1answer
64 views

Norms arising from all representations of *-algebras

It is common that in order to obtain a $C^*$-algebra from a $^*$-algebra $A$ one defines a norm on $A$ by $$\|x\|=\sup\{\|\pi(x)\|\,|\,\pi\ \text{is a }^*\text{-representation of }A\}.$$ However, I ...
1
vote
1answer
54 views

Question about operators on Hilbert space

Let $\cal{H}$ be a Hilbert space, $P_1,P_2,\cdots,P_m$ a sequence of orthonormal projections such that $P_iP_j=0$ for $i\neq j$ and $P_1+P_2+\cdots+P_m=I$. Then $\|\sum^m_{k=1}P_kTP_k\|\leq\|T\|$ for ...
2
votes
1answer
98 views

An quasi-nilpotent operator restricted to a subspace is a nilpotent?

I am reading a paper about operator theory, there is a proposition I could not understand. Let $T\in L(X)$ be a quasi-nilpotent operator and $X_{1}$ be a non-zero finite-dimensional subspace of X, ...
2
votes
1answer
70 views

When does Gelfand Naimark Theorem Hold?

I was going through the proof of the Gelfand Naimark Theorem for the Unital Commutative Banach Algebras. In proving that each character has norm equal to $1$, we used the fact that $\|e\| = 1$ where ...
1
vote
0answers
75 views

Kernel of an infinite dimensional operator

Can the Kernel of an infinite dimensional operator have $\dim=0$? I am thinking to the annihilation ($\hat E$) and creation ($\hat E^\dagger$) operators. Suppose, in fact, we have an infinite but ...
5
votes
1answer
266 views

Physical interpretation of $q$-deformation

I am currently reading the paper Quantum Group Particles and Non-Archimedean Geometry by Volovich and Aref'eva. Here they discuss the difference between $q$-deformation and $\hslash$-deformation. In a ...
1
vote
1answer
75 views

number of generators of MASA

Let $\mathcal{H}$ be an infinite-dimensional Hilbert Space. Do the maximal abelian self-adjoint subalgebras of $\mathcal{B}(\mathcal{H})$ always have infinitely many generators as an algebra ? (The ...
1
vote
1answer
93 views

Left support of an operator on a Hilbert space

The left support $l(x)$ of an operator $x$ between Hilbert spaces $\mathbb{H}$ and $\mathbb{K}$ is defined as the smallest projection $e \in \mathfrak{B}(\mathbb{H})$ such that $ex=x$. The question ...
3
votes
0answers
134 views

Projections in group $C^*$-algebras

Let $G$ be an amenable, discrete and infinite group. Cosinder its group C*-algebra $C^*(G)$ canonically represented on $B(\ell_2(G))$ by the left-regular representation $x\mapsto \delta_x$. Take the ...
7
votes
0answers
267 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
5
votes
0answers
60 views

Masas in quotients

Let $A$ be a von Neumann algebra and let $B$ be a norm-closed ideal of $A$ (but not necessarily WOT-closed). What one has to assume about $A$ and $B$ to ensure that if $M\subset A$ is a maximal ...
2
votes
3answers
291 views

universal C* algebras

Is there a standard reference which has a discussion on universal $C^*$-algebras ? (definition, properties, examples, etc) Searching on the internet has led me to tidbits of information but I would ...
1
vote
1answer
176 views

The free group $II_{\infty}$ factor isomorphism problem

Let $\Gamma$ be an infinite discrete group, and $H = l^{2}(\Gamma)$ the separable infinite dimensional Hilbert space. Let $\rho$ be the left regular representation of $\Gamma$ on $H$. Definition : ...
2
votes
1answer
179 views

How to generate algebraic span of a set of matrices (how many multiplications?)

I've got a question about matrices and matrix algebras that offhand seems difficult, I'm wondering there is any sharp solution? Or perhaps it's known to not have any solution at all? Suppose you have ...
1
vote
0answers
79 views

Lorentz group and eigenvalues

For generators of the Lorentz group ($\hat {R}_{k}$ corresponds to the generators of 3-rotations, $\hat {L}_{k}$ corresponds to the generators of the boosts) we have the following algebra: $$ [\hat ...
3
votes
1answer
188 views

Does an irreducible operator generate an exact $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible if $W^{*}(T)=B(H)$. Definition : A ...
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vote
1answer
77 views

C* identity origins

In the context of $C^*$-algebras , why is the $C^*$-identity a "natural" one to choose ? ($||a^* a||=||a||^2$). Some books try to motivate this by noting that bounded operators on a Hilbert space ...
2
votes
1answer
199 views

Does an irreducible operator generate a nuclear $C^{*}$-algebra?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : An operator $T \in B(H)$ is irreducible (Halmos 1968) if its commutant $\{ T\}'$ ...
8
votes
1answer
431 views

reference for operator algebra

I am taking a course on operator algebra this semester. My instructor has suggested a reference "Kadinson and Ringrose." Are there any other good/standard references for this subject that I can look ...
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1answer
77 views

Connection in the KK-Theory

I have some questions about the connection in the KK-Theory. 1)The definition is complicated, why? What is the motivation? 2)Does any relation bewteen the connection at here with the differential ...
2
votes
1answer
115 views

Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...
2
votes
2answers
195 views

Hahn-Banach Theorem in the C*-algebra

What is the Hahn-Banach Theorem in the C*-algebra(or W*-algebra maybe)? If B is an nondense subalgebra of C*-algebra(or W*-algebra maybe), can we get an state f of A which is always zero at the ...