# Tagged Questions

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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### A nilpotent element of an algebra which does not lie in the span of commutator elements.

What is an example of a $C^{*}$ algebra such that the span of nilpotent elements is not a sub vector space of the span of commutator elements. Obviously any such $C^{*}$ algebra would be a ...
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### Does pointwise nilpotency imply global nilpotency?

Is there a compact Haussdorf space $X$ and $C^{*}$ algebra $A$ with a continuous map $f:X\to A$, such that $f(x)\in A$ is a nilpotent element, $\forall x \in X$, but $f$ is not a ...
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### Equivalence of Definitions of Strong Operator Topology

I have a couple questions about how we define the strong operator topology on $\mathscr{B} (H)$ that I'm hoping someone can help me with. First, I thought that the strong operator topology was the ...
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### Homotopy elements in C*-algebras

Let A is a C*-algebra, A$^+$ means A$\times$$\mathbb{C} equipped with pointwise sum and with a multiplication defined by: (a, \lambda)(b, \mu) = (ab + \lambdab + \mua, \lambda$$\mu$) If x, ...
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### Atomic W*- algebra

Let $A$ be a C*-algebra. Put $z: = \sup\{ e\in A^{**} ;\text{ e is a minimal projection}\}$. Easily can see $z$ is a central projection. Set $M:= A^{**}z$. 1) Is $M$ an atomic W*-algebra in general? ...
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### $C(S^1)$ does not have a single generator

Let $S^1$ be the unit circle in the complex plain and $C(S^1)$ be the continuous function space on $S^1$.$f\in C(S^1)$ is a generator means that $\{p(f) |\text{ p is a polynomial in z}\}$ is dense in ...
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### Direct limits of simple C*-algebras are simple

Let $S$ be a non-empty set of simple C$^*$-subbalgebras of a C$^*$-algebra $A$. Let us also suppose that $S$ is upwards-directed and that the union of all element of $S$ is dense in $A$. Then $A$ is ...
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### How is the spectrum of a product of operators related to the spectrum of each term in the product?

I will use the usual notation of $\sigma(A)$ to denote the spectrum of an operator $A$. Is there a relationship between the spectrum of bounded operators (on complex Hilbert space) $A$ and $B$ and ...
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### Comparing two positive linear functionals when support$(\psi)\leq$ support$(\phi)$

The following point has been addressed in the significant paper "order ideal in C*-algebras and its dual (By E. Effros - Lemma 4.1). The reference is Dixmier's book (1957 French). Unfortunately, I ...
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### Functional calculus for unitization of an algebra?

I have been stuck on this problem for a week now: "Let $A$ be a Banach algebra without identity, let $a\in A$, and let $f$ be holomorphic on a neighborhood of $\sigma(a)$, so that $f(a)\in A^\#$ is ...
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### $C*$-algebra Identity for Matrices

What's the easiest way to see that the $n \times n$-matrices over $\mathbb{C}$ satisfy the $C^*$-algebra identity $\|aa^*\| = \|a\|^2$?
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### One-sided identities in Banach algebras

What is an example of a Banach algebra with a left identity but with no right identities? Is there an example of an operator algebra with this property?
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### Special case of Green's imprimitivity theorem and related question

Consider a locally compact group $G$ and a closed subgroup $H$ of $G$, and let $G$ act on $G/H$ by left translation. Green's imprimitivity theorem implies that the crossed product $C_0(G/H)\rtimes G$ ...
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### The reduction of nilpotency order of nilpotent elements of $C^{*}$ algebras

Assume that $A$ is a unital $C^{*}$-algebra. Let $a\in A$ be a nilpotent element with $$a^{k}=0,\;\;k>1.$$ Are there two elements $x,y\in A$ with $a=xy,\;\;(yx)^{k-1}=0$? Motivation for ...
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### Expectation value of the product of two operators

Does $\langle A^2B^2\rangle = \langle B^2A^2\rangle$ (i.e. the operators $A$ and $B$ commute) somehow imply $\langle A^2\rangle = \langle B^2\rangle$? If so, why? To add some context, my operators ...
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### A certain *-isomorphism

let $A$ be a C*-algebra and $z\in A^{**}$ the supremum of all the minimal projections in $A^{**}$. How can show $* -$ homomorphism $A\to zA\subset zA^{**}$ is injective?
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### Independence of choice of faithful representation in reduced $C^*$ crossed product

In the definition of the reduced $C^*$ crossed product associated with an action of a discrete group $G$ on a $C^*$-algebra $A$, one can begin with any faithful representation of $A$ on a Hilbert ...
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### Existence of uniform multiplicity projection in abelian Von Neumann algebras.

I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with ...
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### postliminal $C^*$-algebra
A ‎$‎‎C^*$-algebra ‎‎$‎‎A$ ‎is ‎said ‎to ‎be ‎postliminal ‎if ‎for ‎every ‎non-zero ‎irreducible ‎representation ‎‎$‎(H,‎\varphi‎)‎$ ‎we ‎have ‎‎$‎‎K(H)‎\subseteq‎ ‎‎\varphi‎(A)‎$‎ ‎ In ‎Murrphy's ‎...
Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...