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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Two questions about ultraweak and ultrastrong topology from Dixmier

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below: Let $M$ be an ultraweakly ...
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Basis for completely bounded maps.

The set of completely bounded (CB) maps forms can be considered as a complex span of the set of completely positive (CP) maps. Can we find a basis for this complex linear space of CB maps such that ...
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149 views

reduced crossed products

Given a discrete group $G$ and a $G$-$C^*$-algebra $A$ we can form the reduced crossed product $A\rtimes_r G$. I want to define it by the closure of $C_c(G,A)$ in $\mathcal{B}(\ell^2(G,A)$ where this ...
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Dropping homomorphisms to quotients of C$^*$-algebras

Let $A$ be a C$^*$-algebra, let $\Delta:A\rightarrow A \otimes_{\min} A$ be a $*$-homomorphism, and let $\phi$ be a state on $A$. Let $(H,\pi,\xi_0)$ be the GNS construction for $\phi$; let $B=\pi(A) ...
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When does Stinespring dilation yield a faithful representation?

Let $A$ be a $C^*$-algebra, $H$ a Hilbert space, $\phi: A \to B(H)$ a completely positive map. The Stinespring construction yields a triple $(K, V, \pi)$ where $K$ is a Hilbert space, $V: H \to K$ a ...
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Eigenvalues of operators

I have a linear operator $T$ which acts on the vector space of square $N\times N$ matrices in this way: $T(A)=0.5(A-A^\mathrm{t})$ ($A^\mathrm{t}$: the transpose of a matrix $A$). I need to prove ...
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Commutant of bounded linear operators on a Hilbert space

Given a Hilbert space $H$, denote by $\mathcal{A}=\mathcal{B}(H)$ the C*-algebra of bounded linear operators on $H$. Denote further by $$\mathcal{B}(H)' := \{A\in \mathcal{B}(H) : [A,B]=0 \;\forall ...
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Gel'fand representation of a non-unital Banach space: what's wrong with this argument

My argument below is hacked together from pages 5-6 of Davidson's "$C^*$ algebras by example". Theorem: The multiplicative linear functionals on a unital abelian Banach algebra are continuous of ...
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A simple question about operator norm

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 ...
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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 ...
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Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
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Problem with spectral theorem and spectral measure.

There is a passage in a book that is not very clear to me: A is a C*Algebra and $a$ is selfadjoint. Then "Indeed identifying A with an algebra of operators on a Hilbert space $\mathcal{H}$, by the ...
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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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56 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 ...
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51 views

Subalgebras of certain C*-algebras

Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and ...
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67 views

A question on simple and unital $C^\star$-algebra

There is a quotation of a book "$C^\star$-algebras Finite-Dimensional Approximations" Definition 1.7.4. A representation $\pi: A \rightarrow B(H)$ is called essential if $\pi(A)$ contains no nonzero ...
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51 views

A theorem about conditional expectation in C*-algebra

Definition 1. Let $B\subset A$ be C*-algebra. A projection from A onto B is a linear map $E: A \rightarrow B$ such that $E(b)=b$ for every $b\in B$. A conditional expectation from A onto B is a ...
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49 views

How to explain a theorem in C*-algebra

I am reading a book "C*-algebra and finite-Dimensional Approximations". In the fundamental facts, it introduce the Noncommunicative Lusin's theorem: Let $A\in B(H)$ be a nondegenerate C*-algebra ...
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37 views

Exponential map in $C^{*}$-algebra and unitary invariance

Let $A$ be a unital $C^{*}$-algebra. Let $X$ be a closed vector subspace of $A$ which is unitarily invariant in the sense that $uXu^{*}\subseteq X$ for all unitaries $u$ of $A$. I want to show that ...
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70 views

When an invertible element in a $C^{*}$-algebra is unitary

I am trying to show that if $a$ is an invertible element of a unital $C^{*}$-algebra, and $||a||=||a^{-1}||=1$, then $a$ is unitary. I can do this if I think of $a$ as a Hilbert space operator using ...
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An exercise on C*-algebra

A representation $\pi$: $A\rightarrow B(H)$ is said to be irreducible if $\pi(A)$ has no non-trivial invariant subspace. A C*-algebra $A$ is said to be liminal if $\pi(A)=K(H_{\pi})$ for every ...
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105 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 ...
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98 views

K-theory, $K_{0}$ of algebra of compact operators

I don't understand how to define the trace of a matrix with values in operators. This occurred in the following situation: Suppose that $H$ is an Hilbert space and $K$ is the algebra of compact ...
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170 views

Unitary operator in von Neumann algebra

Let $R\subseteq B(H)$ be a von Neumann algebra, and $U\in R$ be unitary. Prove that there is a self adjoint operator $A\in R$ such that $||A||\leq \pi$, and $U=\exp(iA)$ . Any idea how to start! Thank ...
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Polar decompostion for the operator algebras

I find that most of books discussing the polar decompostion at the W*-algebras, but not C*-algebras. I guess the rough reason is that the element of W*algebras has the well supported set, but I want ...
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Dense subalgebras of von Neumann algebras and increasing nets

Let $N$ be a von Neumann algebra, and $A$ be a dense $*$-subalgebra of $N$ (in the ultraweak topology) with $A''=N$. Is it true that: For any $x\in N^+$, there exists a increasing net $(x_j)$ in ...
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70 views

Conjugates in integrals

Is it always true that $\int f(x)^*M^\dagger Mf(x) dx=\int (Mf(x))^*(Mf(x)) dx$? where $M$ is an operator. If so, is there a simple proof that this is so? THanks.
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27 views

Are contractive completely positive maps trace decreasing?

Are contractive completely positive maps trace decreasing? More precisely, suppose that $f\colon M\to N$ is a normal cpc map between von Neumann algebras with normalised normal traces. (That is ...
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20 views

Is the von neumann algebra of locally compact amenable group hyperfinite?

Let $G$ be a discrete group and $\mathcal{L}(G)$ the associated von Neumann algebra. It is well known that $G$ is amenable if and only if $\mathcal{L}(G)$ is hyperfinite. Does there exist a ...
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The comprehension of a paragraph about point-ultraweak convergence

There is a quotation below (in the book "C*-algebras and Finite-Dimensional Approximations") Remark 2.1.3. It follows from Sakai's predual uniqueness theorem that when checking point-ultra weak ...
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35 views

A question on double dual of C*-algebra

Let $A, B$ be the C*-algebra. Assume $A$ is nonunital, $B$ is unital and $\phi: A \rightarrow B$ is a contractive completely positive map. Then we consider the double adjoint map $\phi^{**}: ...
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Arveson's Extension Theorem in C*-algebra

I am reading a book C*-algebra and finite-Dimensional Approximations. There are two conclusions (in the book) below. Corollary 1.5.16. Let $E\subset A$ be an operator subsystem and $\phi: E ...
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25 views

A confusion on Stinespring dilation

There is a quotation of Stinespring dilation in a book about C*-algebra. (Stinespring dilation) Let $A$ be a unital C*-algebra and $\phi: A \rightarrow B(H)$ be a completely positive map. Then, there ...
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A funny way of expressing the identity operator

I have encountered the following trick that people in C*-algebra use; but frankly I don't understand why really this is true. Let $A$ be a unital C*-algebra acting non-degeneratly on a Hilbert space ...
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An inequality for completely positive maps.

Let $f\colon A\to B$ be a contractive completely positive, ${}^*$-preserving map between C*-algebras and take $a\in A$. How one can prove that $$0\leqslant f(a)f(a^*)\leqslant f(aa^*)?$$ Some authors ...
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Positive elements of a $C^*$ (MURPHY, ex 2-2).

I'm studying "MURPHY, $C^*$-Algebras and Operator Theory" thoroughly and got stuck in the following exercise: Exercise 2, chapter 2. Let $A$ be a unital $C^*$-algebra. (a) If $a,b$ are positive ...
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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 ...
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101 views

Prove the approximate identity from the unitization

Suppose $A$ is a $C^*$-algebra without unit, $A^+$ is a unitization of $A$, treat $A$ in the $A^+$, if $\{x_n\}$ in $A$ converge (or monotonous converge) to $1$ in $A^+$, does $\{x_n\}$ must be the ...
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1answer
61 views

Inner product on a von Neumann algebra

Let $M$ be a $\sigma$-finite von Neumann algebra (one which admits a faithful normal state) acting on a Hilbert space $H$. Denote its faithful normal state by $\omega$. We can define an inner product ...
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103 views

Determining whether an operator is Hermitian

The operator $F$ is defined by $F\psi(x)=\psi(x+a)+\psi(x-a) $, where $a$ is a nonzero constant. Determine whether or not $F$ is a Hermitian operator. If the condition for $F$ to be Hermitian is ...
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Multiplicative linear functional on algebra of limit of polynomials

Let $A$ be the space of all functions which are limit of polynomials over the unit ball $D$. Then $A$ is a commutative Banach algebra. Then how do I show that $A$ has no non zero multiplicative linear ...
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1answer
51 views

Set of states not compact

I'm looking for an example of a non-unital $C^*$-algebra $A$ whose set of states $S(A)$ is not compact (in the weak* topology, of course). I think $K(H)$, the compact operators over a Hilbert space ...
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A reference request for sums of $C^*$-algebras

Does anyone know where I can find a reference for the following well-known fact: Let $(X_i)_{i\in I}$ be a family of compact Hausdorff spaces and let $X$ be the disjoint sum of all $X_i$s. Then ...
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180 views

How to make a topological subspace Hausdorff

If $X$ is a compact topological space then $C(X)$ is a C* algebra. I'm not going to attempt to discuss the locally compact case with $C_0(X)$ because usually the definition of that space requires the ...
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138 views

The commutant of a tensor product

In this whole post beginning with the second paragraph, by "tensor product" I will mean the operation done on Von Neumann algebras where one applies the homomorphism from here Tensor products of maps ...
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73 views

An expression for the sup of projections

In $B(H)$ where $H$ is a Hilbert space, we have that if p and q are (orthogonal) projections that Inf{$p$, $q$} is in $B(H)$. This also holds if we replace $B(H)$ by the phrase "the strong closure of ...
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25 views

Unit in the image of a cp map

This is another question which looks non-trivial to me. Suppose that we have a completely positive map $f\colon M_n \to M_m$ such that $f(a) = I_m$, the identity matrix on $M_m$. Is there a positive ...
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45 views

An exercise on nuclear maps in C*-algebra

Definition 2.1.1 Let $A, B$ be the C*-algebra, a map $\theta: A\rightarrow B$ is called nuclear if there exist contractive completely positive maps $\phi_{n}: A\rightarrow M_{k(n)}(\mathbb{C})$ and ...
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29 views

Does it hold that $\{a\}'' = \{f(a) : f$ bounded, Borel measurable$\}$?

Let $H$ be an Hilbertspace. Let $a\in\mathcal{B}(H)$ be a self-adjoint operator. Does it hold that $\{a\}^{\prime\prime}=\{f(a):f\in L^{\infty}(\sigma(a))~\text{Borel measureable}\}$?
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A question about $C^\ast$-algebra

In Kadinson's book Fundamentals of The Theory of Operator Algebra, when the author proved the Theorem 7.2.1, he let $V$ be an extreme point of the unit ball of $C^\ast$-algebra $\cal{U}$, $h$ be a ...